Mahayni [2] established the large-deflection equations for a thin shallow shell and treated buckling and postbuckling behaviors of a simply sup-ported isotropic cylindrical panel subjecte
Trang 1Nonlinear response of pressure-loaded functionally graded cylindrical panels
with temperature effects
a
University of Engineering and Technology, Vietnam National University, Ha Noi, Viet Nam
b
Faculty of Civil Engineering, Hanoi Architectural University, Ha Noi, Viet Nam
a r t i c l e i n f o
Article history:
Available online 5 December 2009
Keywords:
Nonlinear analysis
Functionally graded materials
Imperfection
Temperature effects
Cylindrical panel
a b s t r a c t
This paper presents an analytical approach to investigate nonlinear response of functionally graded cylin-drical panels under uniform lateral pressure with temperature effects are incorporated Material proper-ties are assumed to be temperature-independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents Equilibrium and compat-ibility equations for cylindrical panels are derived by using the classical shell theory with both geomet-rical nonlinearity in von Karman–Donnell sense and initial geometgeomet-rical imperfection are taken into consideration The resulting equations are solved by Galerkin method to determine explicit expressions
of nonlinear load-deflection curves Stability analysis for a simply supported panel shows the effects of material and geometric parameters, in-plane restraint and temperature conditions, and imperfection
on the nonlinear response of the panel
Ó 2009 Elsevier Ltd All rights reserved
1 Introduction
Curved panel elements such as cylindrical panels constitute a
major portion of the structure of aerospace vehicles They are
found in the aircraft components as primary load carrying
struc-tures such as fuselage sections as well as in spacecraft and missile
structural applications Moreover, these elements can also be
found in various industries such as shipbuilding, transportation,
and building constructions Therefore, problems relating to
stabil-ity of such structural elements have practical importance and
at-tracted many attentions of science community Mahayni [2]
established the large-deflection equations for a thin shallow shell
and treated buckling and postbuckling behaviors of a simply
sup-ported isotropic cylindrical panel subjected to thermal loads
Pal-azotto and his co-workers reported a series of investigations on
stability of cylindrical panels made of laminated composite
mate-rials[3–8] They considered many different effects such as
mois-ture and temperamois-ture [3], higher order shear deformation, and
nonlinearity[5–7]on the stability of cylindrical panels which are
assumed to be subjected to different loads such as transverse
con-centrated and distributed loads, shear and dynamic loads as well
Yamada and Croll[9]studied buckling of isotropic cylindrical
pan-els under uniform external pressure by using a fully nonlinear Ritz
solution procedure Birman and Bert [10]investigated effects of
temperature on buckling and postbuckling of reinforced and
unstiffened plates and shells subjected to the simultaneous action
of a thermal field and an axial loading Geier and Singh[11] pre-sented a simple analytical solution for computing bifurcation buck-ling loads of thin and moderately thick orthotropic cylindrical shells and panels subjected to compression and normal pressure Jaunky and Knight [12] have assessed the accuracy of different shell theories, i.e Sanders-Koiter, Love and Donnell shell theories for buckling of anisotropic and isotropic curved panels Bukling loads using these shell theories were obtained by using a Ray-leigh–Ritz method and compared with finite element results The dynamic stability of simply supported, isotropic cylindrical panels under combined static and periodic axial forces has been investi-gated by Ng et al.[13]using the generalized Donnell’s shell theory Librescu and Chang[14,15]and Librescu and Lin[16]used analyt-ical approach to investigate postbuckling behavior of laminated composite flat and curved panels under various loading conditions such as axial, transverse, and combined loads with effect of shear deformation, imperfection, and elastic foundation are included This approach is extended by Hause et al.[17]for anisotropic flat and curved sandwich panels Librescu et al [18] presented an excellent analytical investigation on nonlinear response of flat and curved panels subjected to thermomechanical loads Shen
[20]reported a postbuckling analysis of a shear deformable lami-nated cylindrical panel with piezoelectric actuators subjected to the combined action of mechanical, electric and thermal loads Functionally Graded Materials (FGMs) are microscopically com-posites usually made from a mixture of metals and ceramics By gradually varying the volume fraction of constituent materials, their material properties exhibit a smooth and continuous change from one surface to another, thus eliminating interface problems
0263-8223/$ - see front matter Ó 2009 Elsevier Ltd All rights reserved.
* Corresponding author.
E-mail address: htung0105@gmail.com (H Van Tung).
Contents lists available atScienceDirect
Composite Structures
j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / c o m p s t r u c t
Trang 2and mitigating thermal stress concentrations Due to high
perfor-mance heat resistance capacity, FGMs are now developed as
struc-tural components in ultrahigh temperature environments and
extremely large thermal gradients such as aircraft, space vehicles,
nuclear plants, and other engineering applications Despite the
evi-dent importance in practical applications, investigations in the
buckling and postbuckling of FGM cylindrical panels are still
lim-ited in number A postbuckling analysis of axially loaded FGM
cylindrical panels in thermal environments was reported by Shen
[21] He used higher order shear deformation shell theory in
con-junction with a boundary layer theory of shell buckling and a
asymptotic perturbation technique to determine buckling loads
and postbuckling paths of FGM cylindrical panels Both initial
imperfection and temperature-dependent properties are
ac-counted for in his work Following this direction, Shen and Leung
[22] analyzed postbuckling behavior of FGM cylindrical panels
subjected to uniform lateral pressure on point of view of the
exis-tence of bifurcation-type buckling Yang et al.[23]studied
post-buckling of FGM cylindrical panels subjected to combination of
axial load and a uniform temperature change They employed the
classical shell theory and a differential quadrature iteration
algo-rithm to predict critical bifurcation-type buckling load (which only
occur for completely clamped panels) and to trace postbuckling
paths of perfect and imperfect panels with temperature-dependent
properties Recently, the nonlinear response of functionally graded
cylindrical shell panels under mechanical and thermal loads has
been investigated by Zhao and Liew[24]using the element-free
kp-Ritz method and the formulation is based on a modified version
of Sanders nonlinear shell theory
In this paper, the nonlinear response of FGM cylindrical panels
under uniform lateral pressure with and without temperature
ef-fects is investigated by an analytical approach Formulation is
based on the classical shell theory with both von Karman–Donnell
type of kinematic nonlinearity and initial geometrical imperfection
are taken into consideration The resulting equations are solved by
Galerkin procedure to obtain closed-form expressions of nonlinear
load-deflection curves Stability analysis for a simply supported
pa-nel shows the effects of material and geometric parameters,
in-plane restraint and temperature conditions, and imperfection on
the nonlinear response of the panels
2 Functionally graded cylindrical panels
Consider a functionally graded cylindrical panel with radius of
curvature R, thickness h, axial length a and arc length b The
panel is made from a mixture of ceramics and metals, and is
de-fined in coordinate system ðx; h; zÞ, where x and h are in the axial
and circumferential directions of the panel and z is perpendicular
to the middle surface and points inwards ðh=2 6 z 6 h=2Þ
Sup-pose that the material composition of the panel varies smoothly
along the thickness in such a way that the inner surface is
cera-mic-rich and the outer surface is metal-rich by following a simple
power law in terms of the volume fractions of the constituents as
[23]
VcðzÞ ¼ 2z þ h
2h
k
; VmðzÞ ¼ 1 VcðzÞ ð1Þ
where Vcand Vm are the volume fractions of ceramic and metal
constituents, respectively, and volume fraction index k is a
nonneg-ative number that defines the material distribution
We assume that the effective properties Peff of functionally
graded panel, such as the modulus of elasticity E, the coefficient
of thermal expansiona, and the coefficient of thermal conduction
K, change in the thickness direction z and can be determined by the
linear rule of mixture as
PeffðzÞ ¼ PcVcðzÞ þ PmVmðzÞ; ð2Þ
where P denotes a temperature-independent material property, and subscripts m and c stand for the metal and ceramic constituents, respectively
From Eqs.(1) and (2)the effective properties of FGM cylindrical panel can be written as follows in which Poisson’s ratiomis as-sumed to be constant
½EðzÞ;aðzÞ; KðzÞ ¼ ½Em;am;Km þ ½Ecm;acm;Kcm 2z þ h
2h
k
; mðzÞ ¼m;
ð3Þ
where
Ecm¼ Ec Em; acm¼acam; Kcm¼ Kc Km: ð4Þ
It is evident that E ¼ Ec; a¼ac; K ¼ Kc at z ¼ h=2 and
E ¼ Em; a¼am; K ¼ Kmat z ¼ h=2
3 Governing equations
In the present study, the classical shell theory is used to obtain the equilibrium and compatibility equations as well as expressions
of buckling loads and nonlinear load-deflection curves of FGM cylindrical panels
The strains across the panel thickness at a distance z from the mid-plane are
ex¼exmþ zkx; ey¼eymþ zky; cxy¼cxymþ 2zkxy; ð5Þ
whereexmandeymare the normal strains,cxymis the shear strain at the middle surface of the panel, and kijare the curvatures According to the classical shell theory, the strains at the middle surface and the curvatures are related to the displacement compo-nents u; v; w in the x; y; z coordinate directions as[1]
exm¼ u;xþ w2
;x=2; eym¼v;y w=R þ w2
;y=2;
cxym¼ u;yþv;xþ w;xw;y; kx¼ w;xx;
ky¼ w;yy; kxy¼ w;xy; ð6Þ
where y ¼ Rh, and geometrical nonlinearity in von Karman–Donnell sense is accounted for, also, subscript (,) indicates the partial deriv-ative Hooke law for a panel is defined as
ðrx;ryÞ ¼ ½E=ð1 m2
Þ½ðex;eyÞ þmðey;exÞ ð1 þmÞa DTð1; 1Þ;
sxy¼ ½E=2ð1 þmÞcxy: ð7Þ
The force and moment resultants of a panel are expressed in terms of the stress components through the thickness as
ðNij;MijÞ ¼
Z h=2
h=2 rijð1; zÞdz; ij ¼ x; y; xy: ð8Þ
Substituting Eqs.(3), (5) and (7)into Eq.(8)gives the constitu-tive relations
Nx¼ E1
1 m2ðexmþmeymÞ þ E2
1 m2ðkxþmkyÞ Um
1 m;
Ny¼ E1
1 m2ðeymþmexmÞ þ E2
1 m2ðkyþmkxÞ Um
1 m; ð9Þ
Nxy¼ E1 2ð1 þmÞcxymþ E2
1 þmkxy;
Mx¼ E2
1 m2ðexmþmeymÞ þ E3
1 m2ðkxþmkyÞ Ub
1 m;
My¼ E2
1 m2ðeymþmexmÞ þ E3
1 m2ðkyþmkxÞ Ub
1 m; ð10Þ
Mxy¼ E2 2ð1 þmÞcxymþ E3
1 þmkxy;
Trang 3E1¼ Emh þ Ecmh=ðk þ 1Þ; E2¼ Ecmh2½1=ðk þ 2Þ 1=ð2k þ 2Þ;
E3¼ Emh3=12 þ Ecmh3½1=ðk þ 3Þ 1=ðk þ 2Þ þ 1=ð4k þ 4Þ;
ðUm;UbÞ ¼
Z h=2
h=2
Emþ Ecm
2z þ h 2h
k
amþacm
2z þ h 2h
k
DTð1; zÞdz:
ð11Þ
The nonlinear equilibrium equations of a perfect cylindrical
pa-nel based on the classical shell theory are given by[1]
Nx;xþ Nxy;y¼ 0;
Nxy;xþ Ny;y¼ 0;
Mx;xxþ 2Mxy;xyþ My;yyþ Ny=R þ Nxw;xx
þ 2Nxyw;xyþ Nyw;yyþ pðx; yÞ ¼ 0; ð12Þ
where pðx; yÞ is lateral pressure positive inwards When Eqs (9),
(10)are substituted into Eqs.(12), the equilibrium equations can
be written in terms of deflection variable w and force resultants as
Nx;xþ Nxy;y¼ 0;
Nxy;xþ Ny;y¼ 0;
Dr4w Ny=R ðNxw;xxþ 2Nxyw;xyþ Nyw;yyÞ pðx; yÞ ¼ 0;
ð13Þ
wherer2
¼ @2=@x2þ @2=@y2, and
D ¼ E1E3 E
2
For an imperfect panel, let wðx; yÞ denotes a known
small imperfection This parameter represents a small initial
deviation of the panel surface from a cylindrical shape When
imperfection is considered, the equilibrium Eqs.(13)is modified
into form as
Nx;xþ Nxy;y¼ 0;
Nxy;xþ Ny;y¼ 0;
Dr4w Ny=R Nxðw;xxþ w
;xxÞ 2Nxyðw;xyþ w
;xyÞ;
Nyðw;yyþ w
;yyÞ pðx; yÞ ¼ 0:
ð15Þ
Considering the first two of Eqs.(15), a stress function f may be
defined as
Nx¼ f;yy; Ny¼ f;xx; Nxy¼ f;xy: ð16Þ
Substituting Eq.(16)into the third of Eqs.(15)leads to
Dr4w f;xx=R f;yyðw;xxþ w
;xxÞ þ 2f;xyðw;xyþ w
;xyÞ
f;xxðw;yyþ w
;yyÞ pðx; yÞ ¼ 0: ð17Þ
The Eq.(17)includes two dependent unknowns, w and f To
ob-tain a second equation relating these two unknowns, the
compat-ibility equation may be used
The geometrical compatibility equation of a cylindrical panel is
written as[1]
exm;yyþeym;xxcxym;xy¼ w2;xy w;xxw;yy w;xx=R: ð18Þ
For a imperfect panel, the above equation may be modified into
form as
exm;yyþeym;xxcxym;xy¼ w2
;xy w;xxw;yy w;xx=R þ 2w;xyw
;xy
w;xxw
;yy w;yyw
From the constitutive relations(9), one can write
ðexm;eymÞ ¼ 1
E1½ðNx;NyÞ mðNy;NxÞ E2ðkx;kyÞ þUmð1; 1Þ;
cxym¼ 2
E ½ð1 þmÞNxy E2kxy:
ð20Þ
Substituting the above equations into Eq.(19), with the aid of Eqs.(6) and (16), leads to the compatibility equation of an imper-fect FGM cylindrical panel as
r4f E1ðw2;xy w;xxw;yy w;xx=R þ 2w;xyw
;xy w;xxw
;yy w;yyw
;xxÞ ¼ 0: ð21Þ
Eqs.(17) and (21)are the basic equations used to investigate the stability of functionally graded cylindrical panels They are nonlinear equations in terms of two dependent unknowns w and f
4 Stability analysis
In this section, an analytical approach is used to investigate the nonlinear stability of FGM cylindrical panels under mechanical transverse and combined thermomechanical loads In general case, the FGM cylindrical panel is assumed to be simply supported on all edges and subjected to in-plane compressive loads, uniformly distributed along the edges, and lateral pressure uniformly distrib-uted on the outer surface of the panel Depending on the in-plane behavior at the edges, three cases of boundary conditions, labelled Cases (1), (2) and (3) will be considered[14,17,25]
Case (1) The edges are simply supported and freely movable
(FM) The associated boundary conditions are
w ¼ Mxx¼ Nxy¼ 0; Nx¼ Nx0 on x ¼ 0; a;
w ¼ Myy¼ Nxy¼ 0; Ny¼ Ny0 on y ¼ 0; b: ð22Þ
Case (2) The edges are simply supported and immovable (IM)
The associated boundary conditions are
w ¼ u ¼ Mxx¼ 0; Nx¼ Nx0 on x ¼ 0; a;
w ¼v¼ Myy¼ 0; Ny¼ Ny0 on y ¼ 0; b: ð23Þ
Case (3) The edges are simply supported Axial edge loads are
applied in the two curved edges The curved edges
x ¼ 0; a are considered freely movable, the remaining two straight edges being unloaded and immovable For this case, the boundary conditions are
w ¼ Mxx¼ Nxy¼ 0; Nx¼ Nx0 on x ¼ 0; a;
w ¼v¼ Myy¼ 0; Ny¼ Ny0 on y ¼ 0; b; ð24Þ
where Nx0; Ny0are prebuckling force resultants in directions x and
y, respectively, for Case (1) and the first of Case (3), and are fictitious compressive edge loads rendering the edges immovable for Case (2) and the second of Case (3) To solve two Eqs.(17) and (21)for un-knowns w and f, and with the consideration of the boundary condi-tions(22)–(24), we assume the following approximate solutions
[14,15]
w ¼ W sin kmx sinlny;
f ¼ A1cos 2kmx þ A2cos 2lny þ A3sin kmx sinlny
þ A4cos 2kmx cos 2lny þ1
2Nx0y
2
þ1
2Ny0x
where km¼ mp=a;ln¼ np=b; m; n ¼ 1; 2; are number of half-waves in x and y directions, respectively, and W is amplitude of deflection Also, Aiði ¼ 1 4Þ are coefficients to be determined When only the third term and the last two ones in expression of f
is retained, this solution form is consistent with that in[19] Consid-ering the boundary conditions(22)–(24), the imperfections of the panel are assumed as[14,15,25]
Trang 4w¼lh sin kmx sinlny; m; n ¼ 1; 2; ð26Þ
where the coefficientlvarying between values 1 and 1 represents
imperfection size
By substituting Eqs.(25), (26)into Eq.(21), the coefficients Ai
are determined as
A1¼E1l 2
32k2mWðW þ 2lhÞ; A2¼E1k
2 m
32l2WðW þ 2lhÞ;
A3¼ E1k
2
m
Rðk2mþl2Þ2W; A4¼ 0:
ð27Þ
Introduction of Eqs (25) and (26)into Eq.(17) and applying
Galerkin method for the resulting equation yield
Dðk2mþl2Þ2W þ ½2k2ml2ðA1þ A2Þ þ k2mNx0þl2Ny0ðW þlhÞ
32k
2
ml2
3mnp2A3ðW þlhÞ 1
R
64k2mA1
3mnp2 k2mA3þ16Ny0
mnp2
!
16p
mnp2¼ 0:
ð28Þ
Eq.(28), derived for odd values of m; n, is used to determine
buckling loads and nonlinear equilibrium paths of FGM cylindrical
panels under uniform lateral pressure with and without effect of
temperature
4.1 Mechanical stability analysis
The simply supported FGM cylindrical panel with freely movable
edges (that is, Case (1)) is assumed to be subjected to lateral pressure
q (in Pascals), uniformly distributed on the outer surface of the panel
in the absence of edge loads and temperature In this case, we have
p ¼ q; Nx0¼ Ny0¼ 0: ð29Þ
Introduction of Eqs.(27) and (29)into Eq.(28)gives
q ¼mnp2
16B4 Dp4ðm2B2
aþ n2Þ2þE1m
4B6aB2hR2a
ðm2B2
aþ n2Þ2
W
2E1p 2m4n2B5
aRa
3B3hðm2B2aþ n2Þ2WðW þlÞ E1p 2n2BaRa
24B3h WðW þ 2lÞ
þE1p 6mnðm4B4aþ n4Þ
256B4 WðW þlÞðW þ 2lÞ; ð30Þ
where
Ba¼ b=a; Bh¼ b=h; Ra¼ a=R; D ¼ D=h3;
E1¼ E1=h; W ¼ W=h: ð31Þ
For a perfect panel,l¼ 0, Eq.(30)leads to
q ¼mnp2
16B4 Dp4ðm2B2
aþ n2Þ2þE1m
4B6aB2hR2a
ðm2B2
aþ n2Þ2
W
2E1p 2m4n2B5
aRa
3B3hm2B2aþ n22þE1p 2n2BaRa
24B3
2
6
3 7 5W2
þE1p 6mnðm4B4
aþ n4Þ
Eqs (30) and (32)show that there exists no bifurcation-type
buckling for both perfect and imperfect FGM cylindrical panels
un-der uniform lateral pressure, and the cylindrical panels only
exhi-bit extremum-type buckling when the material and geometric
parameters satisfy specific conditions In other words, loss of
sta-bility occurs at a limit point rather than at a bifurcation point for
panels subjected to lateral pressure This behavior of FGM panels
is consistent with Chang and Librescu’s discussions[15]that the
lateral pressure plays on the postbuckling response of cylindrical panels the role of an initial geometric imperfection in the sense that the structure undergoes bending at the onset of loading For perfect panels, the limit points of qðWÞ curves in Eq.(32)are deter-mined by condition
dq
dW¼ A 2BW þ CW
which yields
W1;2¼B
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
B2 AC p
where
A ¼mnp2
16B4h Dp4ðm2B2aþ n2Þ2þE1m
4B6
aB2R2 a
ðm2B2aþ n2Þ2
;
B ¼ 2E1p 2m4n2B5aRa
3B3 m2B2
aþ n2
2þE1p 2n2BaRa
24B3 ; C ¼3E1p 6mnðm4B4aþ n4Þ
256B4 :
ð35Þ
By examining the sign d2q=dW2one obtains the conclusion that the curve represented by Eq.(32) reaches maximum at W1 with corresponding value
qu¼ qðW1Þ ¼ 1
3C2½Bð3AC 2B
2
Þ þ 2ðB2 ACÞ3=2; ð36Þ
and reaches minimum value at W2with
ql¼ qðW2Þ ¼ 1
3C2½Bð3AC 2B
2
Þ 2ðB2 ACÞ3=2; ð37Þ
provided
If the material and geometric parameters of the panel are such that condition(38)is satisfied, snap-through behavior of the panel may be predicted Specifically, Eqs.(36) and (37)represent upper and lower limit buckling loads of perfect FGM panel under lateral pressure, respectively, and the intensity of the snap-through re-sponse is given by the difference between two these values, i.e 4ðB2
ACÞ3=2=3C2 In case of B2
¼ AC, the qðWÞ curve has only one stationary point at W0¼ B=C which be an inflexion point as can be shown For the panels such that B2<AC, there exists only one equilibrium shape and load-deflection curve is stable For imperfect panels, a similar discussion can be given
4.2 Thermomechanical stability analysis
A simply supported FGM cylindrical panel with immovable edges (that is, Case (2)) under simultaneous action of uniform lat-eral pressure q (in Pascals) and thermal loads is considered The condition expressing the immovability on the edges, u ¼ 0 (on
x ¼ 0; a) andv¼ 0 (on y ¼ 0; b), is fulfilled on the average sense
as[14,15]
Z b
0
Z a
0
@u
@xdxdy ¼ 0;
Z a
0
Z b
0
@v
@ydydx ¼ 0: ð39Þ
From Eqs.(6) and (9)one can obtain the following relations in which Eq.(16)and imperfection have been included
@u
@x¼
1
E1
ðf;yymf;xxÞ þE2
E1
w;xx1
2w
2
;x w;xw
;xþUm
E1
;
@v
@y¼
1
E ðf;xxmf;yyÞ þE2
E w;yy
1
2w
2
;y w;yw
;yþw
Rþ
Um
E : ð40Þ
Trang 5Substituting Eqs.(25) and (26)into Eqs.(40)and then into Eqs.
(39)yield
Nx0¼ Um
1 m
p2mnRð1 m2Þ
E1ð1 m2Þk2
ml2
ðk2mþl2Þ2 mE1þ E2Rðk2
mþml2Þ
W
þ E1
8ð1 m2Þðk
2
mþml2
ÞWðW þ 2lhÞ;
Ny0¼ Um
1 m
p2mnRð1 m2Þ
E1ð1 m2Þk4
m
ðk2mþl2Þ2 E1þ E2Rðl2þmk2mÞ
W
þ E1
8ð1 m2Þ l2þmk2mÞWðW þ 2lhÞ;
ð41Þ
which represent the compressive stresses making the edges
immov-able and depending on thermal parameter and prebuckling
deflec-tion Obviously, when prebuckling deflection is ignored, i.e a
membrane state analysis, only the first terms in Eqs.(41)are retained
Subsequently, a simply supported panel with movable edges
x ¼ 0; a and immovable edges y ¼ 0; b (that is, Case (3)) and
sub-jected to combined action of uniform lateral pressure q (in Pascals)
and thermal loads in the absence of edge loads is considered
Employing Eqs.(25) and (26)in the second of Eqs.(40)and then
introduction of the result into the second of Eqs.(39)yield
Nx0¼ 0; Ny0¼ Umþ 4
p2mnR
E1k2mðk2mml2Þ
ðk2mþl2Þ2 E1þ E2Rl2
W þE1l 2
In what follows, specific expressions of thermal parameter and
the nonlinear response of FGM cylindrical panels under uniform
external pressure and two types of thermal loads will be analyzed
4.2.1 Uniform temperature rise
Under mentioned boundary conditions, environment
tempera-ture can be uniformly raised from initial value Ti to final one Tf
and temperature differenceDT ¼ Tf Tiis a constant
The thermal parameterUmcan be expressed in terms of theDT from
Eqs.(11) Subsequently, employing this expression ofUmin Eqs.(41)
and (42)and then introduction of the results into Eq.(28)yield
q ¼p2mn
16B4h Dp4ðm2B2aþ n2Þ2þE1m
4B6
aB2R2 a
ðm2B2aþ n2Þ2
"
64E1m
2B6
aB2R2 a
p4n2ðm2B2aþ n2Þ2
þ 64E1B
2
aB2hR2a
p4m2n2ð1 m2Þ
64E2BaBhRaðmm2B2aþ n2Þ
p2m2n2ð1 m2Þ
# W
p2
4B4hð1 m2Þ
2E1ð1 m2Þm4n2B5
aBhRa
3ðm2B2aþ n2Þ2
"
þ E1BaBhRaðmm2B2aþ n2Þ
E2p 2ðm4B4aþ n4þ 2mm2n2B2a
# WðW þlÞ
E1p 2BaRa½3mm2B2
aþ ð4 m2Þn2 24B3
ð1 m2Þ WðW þ 2lÞ
þ E1p 6mn
256B4hð1 m2Þ½ð3 m2
Þðm4B4
aþ n4Þ
þ 4mm2n2B2
aWðW þlÞðW þ 2lÞ
p4mnP
16B2
ð1 mÞ ðm
2B2aþ n2ÞðW þlÞ 16BaBhRa
p4mn
for FGM cylindrical panels with all immovable edges, and
q ¼p2mn 16B4h Dp4
ðm2B2
aþ n2Þ2þE1m
4B6aB2hR2a
ðm2B2
aþ n2Þ2
"
þ64E1B
2
aB2R2
a½n4þ ð2 þmÞm2n2B2
a
p4m2n2ðm2B2aþ n2Þ2
64E2BaBhRa
p2m2
# W
p2
4B4
E1m2n2B3
aBhRað5m2B2
aþ 3mn2Þ 3ðm2B2aþ n2Þ2 þ E1n
2BaBhRa E2p 2n4
WðW þlÞ E1p 2n2BaRa
6B3h WðW þ 2lÞ þE1p 6mn
256B4h ðm
2B2aþ 3n2Þ
WðW þlÞðW þ 2lÞ p4mnP
16B2 n2ðW þlÞ 16BaBhRa
p4mn
DT; ð44Þ
for panels with two edges y ¼ 0; b are immovable, where
P ¼ EmamþEmacmþ Ecmam
k þ 1 þ
Ecmacm
2k þ 1; E2¼ E2=h
2
: ð45Þ
4.2.2 Through the thickness temperature gradient
In this case, the temperature through thickness is governed by the one-dimensional Fourier equation of steady-state heat conduction
d
dz KðzÞ
dT dz
¼ 0; Tðz ¼ h=2Þ ¼ Tc; Tðz ¼ h=2Þ ¼ Tm; ð46Þ
where Tcand Tmare temperatures at ceramic-rich and metal-rich surfaces, respectively The solution of Eq.(46)can be obtained by means of polynomial series Taking the first seven terms of the ser-ies, the solution for temperature distribution across the panel thick-ness becomes[25]
TðzÞ ¼ TmþDT
rP5 n¼0
ðr k K cm =K m Þn nkþ1
P5 n¼0
ðK cm =K m Þ n nkþ1
where r ¼ ð2z þ hÞ=2h andDT ¼ Tc Tmis defined as the tempera-ture difference between ceramic-rich and metal-rich surfaces of the panel
Assuming the metal surface temperature as reference tempera-ture and substituting Eq.(47)into Eq.(11)give
Um¼ HDTh; H ¼
P5 n¼0
ðK cm =K m Þ n nkþ1
E m a m nkþ2þE m a cm þE cm a m ðnþ1Þkþ2 þ E cm a cm
ðnþ2Þkþ2
P5 n¼0
ðK cm =K m Þn nkþ1
: ð48Þ
By following the same procedure as the preceding loading case
we obtain explicit expressions of qðWÞ curves for two cases of in-plane restraints as Eqs.(43) and (44), provided P is replaced by
H Such detail expressions are omitted here for sake of brevity
5 Results and discussion
In this section, the nonlinear response of the cylindrical panels made of functionally graded materials is analyzed The panels are assumed to be simply supported on all edges and, unless otherwise stated, edges are freely movable In characterizing the behavior of the panels, deformations in which the central region of a panel moves towards the plane that contains the four corners of the pa-nel are referred to as inward deflections Deformations in the oppo-site direction are referred to as outward deflection[18] As shown
in references[15,17,18], the most pronounced buckling and post-buckling responses for deformation modes with half-wave
Trang 6num-bers m ¼ n ¼ 1 Thus, the results presented in this section also
cor-respond to values of m ¼ n ¼ 1
To validate the present formulation in the nonlinear analysis of
pressure-loaded FGM cylindrical panels, the nonlinear response of
a simply supported FGM cylindrical panel under uniform external
pressure is analyzed, which was considered by Zhao and Liew[24]
using the element-free kp-Ritz method and modified version of
Sander’s nonlinear shell theory The nonlinear load-deflection
curves of perfect panels made of zirconia ðZrO2Þ and Aluminum
ðAlÞ with different values of volume fraction index k are compared
inFig 1with Zhao and Liew’s results As can be observed, a very
good agreement is obtained in this comparison study
To illustrate the proposed approach, we consider a
ceramic-me-tal functionally graded panel that consist of zirconia and aluminum
with the following properties[24]
Em¼ 70 GPa; am¼ 23:106 C1
; Km¼ 204 W=mK;
Ec¼ 151 GPa; ac¼ 10:106 C1; Kc¼ 2:09 W=mK; ð49Þ
and Poisson’s ratio is chosen to be 0.3 Effects of material and
geo-metric parameters, temperature and in-plane boundary conditions
as well as imperfection on the nonlinear response of the perfect
and imperfect FGM cylindrical panels are graphically shown inFigs
2–12in whichFigs 2–6, 11are plotted for case of mechanical
sta-bility analysis, i.e in the absence of temperature It is noted that in
all figures W=h denotes the dimensionless maximum deflection of
the panel
Fig 2gives the nonlinear load-deflection curves for FGM
cylin-drical panels under uniform lateral pressure and with three
differ-ent values of volume fraction index k ð¼ 0; 1 and 5Þ A benign
snap-through behavior of the panels is shown in this figure It is
also seen that nonlinear curves become higher for smaller values
of k representing panels with the greater volume percentage of
zir-conia, as expected
Fig 3 shows the effect of width-to-thickness ratio b/h (=
20, 30 and 40) on the nonlinear behavior of the FGM panels with
k ¼ 1:0 Fig 4 shows the effect of length-to-width ratio
a/b (= 0.75, 1.0 and 1.5) on the nonlinear behavior of the panels
un-der similar conditions It is evident from two these figures that
loading carrying capacity of the panels is considerably reduced
when b=h and a=b ratios increase Furthermore, the nonlinear
equi-librium paths become more stable for larger values of a=b standing
for shallower panels
The effect of panel curvature on the nonlinear response of
pres-sure-loaded FGM panels is illustrated inFig 5with three various
values of length-to-radius ratio a/R (= 0.5, 0.75 and 1.0) As can
be seen, the load bearing capability of the panels is increased when
a=R ratio increases and the deflection is small and a converse trend
occurs when the deflection is sufficiently large In addition, the
re-sults indicate that the panels with large a=R exhibit a benign
snap-through response and the panels with small one have stable
equi-librium paths due to its flatted configuration
The effect of in-plane boundary conditions on the nonlinear
re-sponse of FGM panels under uniform lateral pressure is depicted in
Fig 6 In this figure, the nonlinear equilibrium paths of the panels
with all freely movable edges (Case (1)) are plotted in comparison
with their counterparts when all edges are immovable (Case (2))
and two edges y ¼ 0; b are immovable (Case (3)) As can be
ob-served, the load bearing capability of the panel in the Case (2) is
higher than that of panels in two the remaining cases
Further-more, the panel in Case (3) experiences a snap-through response
in contrast to comparatively stable behavior of the panels in Cases
(1) and (2).Fig 7shows the effect of temperature field on the
non-linear response of FGM cylindrical panels under uniform lateral
pressure with all immovable edges The counterpart of the case
in this figure for panels with two immovable edges is illustrated
inFig 8 As can be seen, in the presence of temperature field the pressure-loaded panels exhibit a bifurcation behavior with no deflection occuring until a bifurcation point is reached This is con-trast to their isothermal counterparts in which curves originate from coordinate origin This behavior may be explained that the temperature field produce the deformations that cause the central region of a panel to deflect outward (negative deflection) prior to applying the mechanical loads With the application of lateral pres-sure, the outward deflection is reduced and when lateral pressure exceeds bifurcation point loads, which are represented by the last terms with W ¼ 0 in Eqs (43) and (44) respectively, a inward deflection occurs It is also seen that the increase in buckling loads and postbuckling load carrying capacity in small range of deflec-tion due to the presence of temperature as compared with their isothermal counterparts is paid by an unstable postbuckling behavior, that is, a snap-through response The intensity of the snap-through response is more severe when environment temper-ature is increased Furthermore, the panels with two immovable edges have weaker capacity of load carrying accompanied by more intense snap-through response in comparison with the panels with all immovable edges
The effect of through the thickness temperature gradient on the nonlinear response of pressure-loaded panels is plotted inFig 9
0 0.01 0.02 0.03 0.04 0.05
W/h
Ref [24], k = 0.2 Ref [24], k = 1.0 Ref [24], Al Present, k = 0.2 Present, k = 1.0 Present, Al
q (GPa)
a/b = 1.0, b/h = 20, a/R = 0.2
Fig 1 Comparisons of nonlinear load-deflection curves for FGM cylindrical panels.
0 0.5 1 1.5 2 2.5
3
x 10−3
W/h
imperfect (μ = 0.1) perfect
q (GPa)
a/b = 1.1, b/h = 50, a/R = 0.5
k = 0
k = 1
k = 5
Fig 2 Effect of volume fraction index on the nonlinear response of FGM cylindrical
Trang 70 0.5 1 1.5 2 2.5 3
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
W/h
imperfect (μ = 0.1) perfect
q (GPa)
a/b = 1.0, a/R = 0.5, k = 1.0
1: b/h = 20
2: b/h = 30
3: b/h = 40
1
2 3
Fig 3 Effect of width-to-thickness ratio on the nonlinear response of FGM
cylindrical panels.
0
0.01
0.02
0.03
0.04
0.05
W/h
imperfect (μ = 0.1) perfect
q (GPa)
b/h = 30, a/R = 0.5, k = 1.0
1: a/b = 0.75
2: a/b = 1.0
3: a/b = 1.5
1
2 3
Fig 4 Effect of length-to-width ratio on the nonlinear response of FGM cylindrical
panels.
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
W/h
imperfect (μ = 0.1) perfect
q (GPa)
1 2 3
a/b = 1.0, b/h = 30, k = 1.0
1: a/R = 0.5
2: a/R = 0.75
3: a/R = 1.0
Fig 5 Effect of length-to-radius ratio on the nonlinear response of FGM cylindrical
0 0.005 0.01 0.015 0.02 0.025 0.03
W/h
imperfect (μ = 0.1) perfect
q (GPa)
a/b = 1.0, b/h = 50,
1: IM (all edges) 2: IM (y = 0, b)
2 3
a/R = 0.5, k = 1.0
Fig 6 Effect of in-plane boundary conditions on the nonlinear response of FGM cylindrical panels.
0 0.005 0.01 0.015 0.02 0.025 0.03
W/h
imperfect (μ = 0.1) perfect
q (GPa)
a/b = 1.0, b/h = 50, a/R = 0.5, k = 1.0 1:ΔT = 0
2:ΔT = 200o
C 3:ΔT = 400o
C
1
1
2 2
3
3
Fig 7 Effect of temperature field on the nonlinear response of FGM cylindrical panels (all immovable edges).
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
W/h
imperfect (μ = 0.1) perfect
q (GPa)
1 2 3
a/b = 1.0, b/h = 50, a/R = 0.5, k = 1.0
1:ΔT = 0 2:ΔT = 200o
C 3:ΔT = 400o
C
Fig 8 Effect of temperature field on the nonlinear response of FGM cylindrical
Trang 8with different values of temperature Tc at ceramic-rich surface when metal-rich surface temperature is retained at Tm¼ 27C (room temperature) It seems that through the thickness tempera-ture gradient produce smaller outward deflection and the panels have more stable postbuckling behavior, that is, more benign snap-through response under this temperature condition
Fig 10shows the effect of type of through the thickness tem-perature gradient on the nonlinear response of pressure-loaded FGM perfect cylindrical panels Specifically, together the present FGM cylindrical panel having inner surface (i.e concave side) is ceramic-rich, another panel with interchanged properties, which possesses ceramic-rich outer surface (i.e convex side), is consid-ered The temperature is conducted from ceramic pure surface to metal pure one through the thickness As can be observed, although lower bifurcation point and postbuckling curves in small range of deflection, temperature gradient from the outer surface of panel makes the postbuckling behavior of the panel to be more stable
Finally, the effect of initial imperfection on the nonlinear re-sponse of panels is considered.Fig 11depicts the nonlinear load-deflection curves of FGM panels under uniform lateral pressure
A similar consideration is shown in Fig 12 for panels with all immovable edges and temperature field is included In two these figures, the nonlinear equilibrium paths are plotted for various val-ues of imperfection sizelin which, as mentioned in[18], negative
or positive imperfections produce perturbations in the panel geometry that move the central region of a panel outward or in-ward, respectively As can be seen, the nonlinear load-deflection curves become lower whenlincreases from 0.5 to 0.5 and the deflection is small However, a inverse trend occurs when the deflection exceeds a specific value As above, the nonlinear equilib-rium paths start from bifurcation point rather than coordinate ori-gin due to the presence of temperature field
6 Concluding remarks
An analytical study of the nonlinear response of functionally graded cylindrical panels subjected to uniform lateral pressure with and without temperature effects has been presented The formula-tion is based on the classical shell theory with both von Karman– Donnell nonlinear terms and initial geometrical imperfection are included By using Galerkin method, explicit expressions of nonlin-ear load-deflection curves for a simply supported panel under men-tioned loads are determined From these explicit expressions, the
0
0.005
0.01
0.015
0.02
0.025
W/h
ceramic−rich inner suface ceramic−rich outer suface
q (GPa)
a/b = 1.0, b/h = 50, a/R = 0.5,
1: T
c = 400oC 2: T
c = 800oC
k = 1.0, μ = 0
1
2
Fig 10 Effect of type of temperature gradient on the nonlinear response of FGM
cylindrical panels.
0
0.005
0.01
0.015
0.02
0.025
W/h
1:μ = −0.5 2:μ = −0.2 3:μ = 0.0 4:μ = 0.2 5:μ = 0.5
q (GPa)
a/b = 1.0, b/h = 40, a/R = 0.5, k = 1.0
1 2 3 4 5
0 0.02 0.04 0.06 0.08 0.1
W/h
1:μ = −0.5 2:μ = −0.2 3:μ = 0.0 4:μ = 0.2 5:μ = 0.5
q (GPa)
a/b = 1.0, b/h = 40, a/R = 0.5
k = 1.0, ΔT = 300o
C
1 2 3 4 5
Fig 12 Effect of imperfection on the nonlinear response of FGM cylindrical panels with all immovable edges.
0
0.005
0.01
0.015
0.02
0.025
0.03
W/h
imperfect (μ = 0.1) perfect
a/b = 1.0, b/h = 50, a/R = 0.5, k = 1.0
1: T
c = 27oC 2: T
c = 400oC 3: T
c = 800oC
1
1 2
2 3
3
q (GPa)
Fig 9 Effect of temperature gradient on the nonlinear response of FGM cylindrical
panels.
Trang 9nonlinear response of the panels is analyzed and the results are
given in graphic form The results show that the nonlinear response
of the FGM cylindrical panels is greatly influenced by in-plane
restraint and temperature conditions Furthermore, the study also
confirm that the nonlinear response of pressure-loaded panels are
complex and significantly influenced by the material and geometric
parameters, initial imperfection as well
Acknowledgements
This work is supported by the research project of Vietnam
Na-tional University-Ha Noi, coded QGTD.09.01 The authors are
grate-ful for this financial support
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