DSpace at VNU: Nonlinear dynamic response and vibration of shear deformable imperfect eccentrically stiffened S-FGM circular cylindrical shells surrounded on elastic foundations

Accepted Manuscript
Nonlinear dynamic response and vibration of shear deformable
imperfect eccentrically stiffened S-FGM circular cylindrical shells
surrounded on elastic foundations
Nguyen Dinh Duc, Pham Toan Thang

PII:
DOI:
Reference:

S1270-9638(14)00230-2
10.1016/j.ast.2014.11.005
AESCTE 3163

To appear in:

Aerospace Science and Technology

Received date: 4 September 2014
Revised date:
1 November 2014
Accepted date: 9 November 2014

Please cite this article in press as: D.D. Nguyen, T.T. Pham, Nonlinear dynamic response and vibration of
shear deformable imperfect eccentrically stiffened S-FGM circular cylindrical shells surrounded on elastic
foundations, Aerosp. Sci. Technol. (2014), />
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Nonlinear dynamic response and vibration of shear deformable imperfect
eccentrically stiffened S-FGM circular cylindrical shells
surrounded on elastic foundations
Nguyen Dinh Duc*, Pham Toan Thang
Vietnam National University, Ha Noi, 144 XuanThuy – Cau Giay – Ha Noi – Viet Nam
Email: ; , Tel: +84-4-37547978; Fax: +84-4-37547424
Abstract: This paper presents an analytical approach to investigate the nonlinear dynamic response
and vibration of imperfect eccentrically stiffened functionally graded thick circular cylindrical shells
surrounded on elastic foundations using both of the first order shear deformation theory and stress
function with full motion equations (not using Volmir's assumptions). Material properties are graded in
the thickness direction according to a Sigmoid power law distribution (S-FGM) in terms of the volume
fractions of constituents with metal - ceramic - metal layers. The S-FGM shells are subjected to
mechanical and damping loads. Numerical results for dynamic response of the shells are obtained by
Runge-Kutta method. The results show the influences of geometrical parameters, the volume fractions
of metal – ceramic – metal layers, imperfections, theelastic foundations, eccentrically stiffeners, pre–
loaded axial compression and damping loads on the nonlinear dynamic response and nonlinear
vibration of functionally graded cylindrical shells. The proposed results are validated by comparing
with other results reported in literature.
Keywords: Nonlinear dynamic response, vibration, Sigmoid FGM thick circular cylindrical shells, the
first order shear deformation theory, elastic foundations.

1. Introduction
The idea of FGMs was first introduced in 1984 by a group of Japanese material scientists
[1]. Functionally graded materials (FGMs) are composite materials obtained by combining and
mixing two or more different constituent materials, which are distributed along the thickness in
______________________
Corresponding author: Duc.N.D E-mail address:

accordance with a volume fraction law. The FGM have received considerable attention in
recent years due to their high performance heat resistance capacity and excellent characteristics
in comparison with conventional composites.
Regarding to the dynamic and vibration of FGM plates and shells, Loy et al. [2]
analyzed the vibrations of the FGM cylindrical shells. They found that the natural frequencies
are affected by the constituent volume fractions and configurations of the constituent materials.
Pradhan et al. [3] studied the vibration characteristics of FGM cylindrical shells made of
stainless steel and zirconia under different boundary conditions. Free vibration analysis of
functionally graded cylindrical shells with holes was researched in [4]. Ebrahimi and
Najafizadeh [5] investigated the free vibration of a two-dimensional functionally graded
circular cylindrical shell. The equations of motion are based on the Love’s first approximation
classical shell theory. Shen [6] researched the large amplitude vibration behavior of a shear
deformable FGM cylindrical shell of finite length embedded in a large outer elastic medium
and in thermal environments. Najafizadeh and Isvandzibaei [7,8] studied free vibration of FGM
cylindrical shells with ring support by using Ritz method based on the first order and higher
order shear deformation shell theories. Haddadpour et al. [9]considered free vibration of simply
supported FGM cylindrical shells with four sets of in-plane boundary conditions by using
Galerkin method based on the classical shell theory. Alibeigloo et al. [10] presented the
numerical free vibration analysis for FGM cylindrical shell embedded thin piezoelectric
layers. Sofiyev and Kuruoglu [11] focused the torsional vibration and buckling of un-stiffened
cylindrical shell with functionally graded coatings surrounded by an elastic medium. Bich and
Nguyen [12] used the displacement functions to investigate the nonlinear vibration of FGM unstiffened cylindrical shells subjected to axial and transverse mechanical loads. Their results
shown that the Volmir’s assumption can be used for nonlinear dynamic analysis with an
acceptable accuracy. Shariyat [13] studied the dynamic buckling of imperfect FGM cylindrical
shells with integrated surface-bonded sensor and actuator layers subjected to some complex
combinations of thermo-electro-mechanical loads. Shen [14-16] presented a postbuckling
analysis of FGM cylindrical thin shells and FGM panels subjected to axial compression or
external pressure in thermal environments. Shen and Noda [17] obtained the postbuckling
analysis for FGM cylindrical shells with piezoelectric actuators subjected to lateral pressure in

thermal environments. Loy et al. [18] investigated the vibration of FGM cylindrical shells
composed of stainless steel and nickel, considering the influence of the constituent volume
fractions and the effects of the constituent materials on the frequencies. Meiche et al. [19]
proposed a new hyperbolic shear deformation theory taking into account transverse shear
deformation effects for the buckling and free vibration analysis of thick functionally graded
sandwich plates. Benachour et al [20] used the four variable refined plate theory for free
vibration analysis of plates made of functionally graded materials with an arbitrary gradient.
Hebali et al. [21] developed a new quasi-three-dimensional (3D) hyperbolic shear deformation
theory for the bending and free vibration analysis of functionally graded plates. Bessaim et al.
[22] studied a new higher-order shear and normal deformation theory for the bending and free
vibration analysis of sandwich plates with functionally graded isotropic face sheets. Larbi et al.
[23] investigated an efficient shear deformation beam theory based on neutral surface position
is developed for bending and frees vibration analysis of functionally graded beams.
Bouremanaet al. [24] studied a new first-order shear deformation beam theory based on neutral
surface position is developed for bending and free vibration analysis of functionally graded
beams. Meziane et al. [25] proposed an efficient and simple refined shear deformation theory is
presented for the vibration and buckling of exponentially graded material sandwich plate
resting on elastic foundations under various boundaries. Draiche et al [26] investigated the use
of trigonometric four variable plate theory for free vibration analysis of laminated rectangular
plate supporting a localized patch mass.
Today, functionally graded shells involving circular cylindrical shells are widely used in
many important details of space vehicles, aircrafts, nuclear power plants and many other
engineering applications. For example, the strategic missiles using solid materials, they
capable fly far beyond the continent with great velocity, so their hull could stand very high
strength and high temperatures. To satisfy it, the shell of the strategic missiles usually is made
of composite carbon-carbon or functionally graded materials (FGM). FGM circular cylindrical
shell also could be used as the shell of a nuclear reactor or special engineering
pipes,...Regarding to the static and dynamic analysis of the FGM circular cylindrical shells,

Duc and Thang [27] studied an analytical approach to investigate the nonlinear static buckling
and postbuckling for imperfect eccentrically stiffened functionally graded thin circular


cylindrical shells surrounded on elastic foundation with ceramic–metal–ceramic layers and
subjected to axial compression. Duc and Thang [28] also investigated the nonlinear static
buckling for imperfect functionally graded thin circular cylindrical shells reinforced by
stiffeners in thermal environment.
Some researchers have used the first-order and high-order shear deformation theories for
buckling analysis of the perfect and imperfect thick composite cylindrical shells [29-31]. Sheng
and Wang [32] studied dynamic behavior for the functionally graded cylindrical shell with
surface-bonded PZT piezoelectric layer under moving loads. Shahsiah and Eslami [33]
investigated the thermal buckling of FGM cylindrical shells under two types of thermal loads
based on the first order shear deformation shell theory. Shen [34] researched the large
amplitude vibration behavior of a shear deformable FGM cylindrical shell of finite length
embedded in a large outer elastic medium and in thermal environments. Shahsiah and Eslami
[35] presented the buckling temperature of simply supported FGM cylindrical shells under two
cases of thermal loading using the first order shear deformation shell theory. Bouderba et al.
[36] studied the thermomechanical bending response of functionally graded plates resting on
Winkler-Pasternak elastic foundations.Tounsi et al. [37] proposed a refined trigonometric shear
deformation theory (RTSDT) taking into account transverse shear deformation effects is
presented for the thermoelastic bending analysis of functionally graded sandwich plates.
Bourada et al. [38] performed the use of a new four-variable refined plate theory for thermal
buckling analysis of functionally graded material (FGM) sandwich plates. Belabed et al. [39]
presented an efficient and simple higher order shear and normal deformation theory for
functionally graded material (FGM) plates. Bouiadjra [40] studied the nonlinear behavior of
functionally graded material (FGM) plates under thermal loads using an efficient sinusoidal
shear deformation theory. Fekrar et al. [41] developed a new sinusoidal higher-order plate
theory for bending of exponential graded plates. Bousahla et al [42] proposed a new
trigonometric higher-order theory including the stretching effect for the static analysis of

advanced composite plates such as functionally graded plates.Saidi et al. [43] included an
analytical solution to the thermo-mechanical bending analysis of functionally graded sandwich
plates by using a new hyperbolic shear deformation theory. Houari et al. [44]developeda new
higher order shear and normal deformation theory to simulate the thermoelastic bending of


F
FGM sandw
wich plates. Sadoune et
e al. [45] prresented a nnew simplee first-orderr shear defoormation
ttheory for laminated
l
composite
c
plates.
p
Notee that all thee publicatioons mentionned above [[29-45],
tthe authorss used the first order shear deformation theeory with ddisplacement functionns while
studying no
onlinear dyn
namic and vibration
v
off thick FGM
M shells.
In thiss paper, wee research the nonlinear dynamiic and nonnlinear vibrration of im
mperfect
eccentricallly stiffened
d functionallly graded thick
t
circulaar cylindriccal shells w

with metal-cceramicm
metal layerrs, which arre symmetric through the middle surface byy Sigmoid-laaw distribuution (SF
FGM) and surrounded
d on elastic foundation
ns using the first order shear deforrmation theeory and
V
assu
umption is not approppriate due too the fact thhat the rightt side of
stress functtion. The Volmir's
equations of
o motion doesn’t eq
qual to Zerro [46]. Fuurthermore, in this paaper, we toook into
account off the presen
nce of stifffeners and elastic
e
founndations. T
Therefore, tthe calculatting has
bbecome more
m
compliicated. Thee Galerkin method aand Runge--Kutta metthod are uused for
dynamic analysis
a
of the cylind
drical shells to give expressionn of naturaal frequenccies and
nnonlinear response. Nu
umerical reesult shows the effects of characteeristics of ffunctionallyy graded
m
materials, geometrical
g
l and materrial propertties, elastic foundationns and ecceentrically sttiffeners

on the dynaamical behaavior of the shells.
2. Theoretiical formulations
2.1. Eccenttrically stifffened S-FGM thick circular cyylindrical sshells surrounded on
n elastic
foundation
ns.

Fig.1. Configuratio
C
on of an ecccentrically stiffened S--FGM thickk circular cyylindrical shhell.


For an S-FGM cylindrical shell made of two different constituent materials with metalceramic-metal layers, the volume fractions Vc z
and Vm z
can be written in the Sigmoid
power law distribution as [27-28,47]

£¦ 2 z
h ¬N
¦¦žžž
­­ ,
¦
­
Ÿ
®
h
¦
Vc z
 ¤
¦¦ 2 z

h ¬N
­ ,
¦¦žž
ž h ­­®
¦Ÿ
¦¥
Vm z

Vc z
 1,

h
N p 0,  b z b 0,
2
h
0b z b ,
2

(1)

where the Young’s modulus E , the Poisson’s ratio O are expressed as:

£¦ 2 z
h ¬N
h
Ecm  Ec - Em , N p 0,  b z b 0,
¦¦¦žžž
­­­ ,
Ÿ h ®
2

  E z
, S z
¯  < Em , Sm >
< Ecm , Scm >¤¦
¢
±
¦¦ 2 z
h ¬N
(2)
­­ , 0 b z b h , S  S - S .
¦¦žž
cm
c
m
ž h ­®
¦Ÿ
2
¦¥
O  const.
with volume fraction index N dictates the material variation profile through the S-FGM shell
thickness, the subscripts m and c are metal and ceramic constituents respectively.
The shell-foundation interaction is represented by Pasternak model as
q  k1w  k2‹2 w, (3)

where ‹2  s 2 / sx 2
s 2 / sy 2 , w is the deflection of the shell, k1 is Winkler foundation
modulus and k2 is the shear layer foundation stiffness of Pasternak model.


Fig. 2.Geeometry and coordinatte system off a circular cylindricall shell surroounded on eelastic

foundations
2.2. Governing equattions
A
According to the Redd
dy’s first-orrder shear deformation
d
n (FSDT) inn [48] whicch assumes that the
ttransverse normal stress is neglligible and normal doo not remaiin perpenddicular to thhe midwithin the shell is
surface after deformaation. The displacemeent field foor an arbitrrary point w
o be
assumed to

u x, y, z
 u0 ( x, y )
zGx x, y
,
v x, y, z
 v0 ( x, y )
zGy x, y
,

(4)

w x, y, z
 w0 ( x, y ),

w
where u0 , v0 , w0 are thee displacem
ment compo
onent alongg the x, y, z

coordinaate directioons, and
tthe quantitiies Gx and Gy denote the
t mid-plaane rotationns of two trransverse noormal abouut the y
and x axes.
o the FS
SDT and von-Karma
v
an nonlineaar strains-ddisplacemennt relationn which
Based on
accounts fo
or the modeerately largeer deflection
n and smalll strain, we obtain


sG
sGx
sv
, Fx  F0 y 0
z y ,
sx
sy
sy
 sG
sG ¬
H xy  H 0 xy
z žž x
y ­­­ ,
žŸ sy
sx ®­

Fx  F0 x
z

H yz  Gy


(5)

sw
sw
, H zx  Gx

,
sy
sx

with
2

2
su 1 ž sw ¬­
sv w 1 ž sw ¬­

ž ­­ , F0 y