Stability analysis for a simply supported functionally graded cylindrical panel shows the effects of mate-rial and geometric parameters as well as imperfection on buckling and postbuckli
Trang 1Nonlinear analysis of stability for functionally graded cylindrical panels under axial compression
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:
Received 6 October 2009
Received in revised form 7 December 2009
Accepted 18 December 2009
Available online 25 January 2010
Keywords:
Nonlinear analysis
Functionally Graded Materials
Postbuckling
Cylindrical panel
a b s t r a c t
This report presents an analytical approach to investigate the stability of functionally graded cylindrical panels under axial compression Equilibrium and compatibility equations for functionally graded panels are derived by using the classical shell theory taking into account both geometrical nonlinearity in von Karman–Donnell sense and initial geometrical imperfection The resulting equations are solved by Galer-kin procedure to obtain explicit expressions of buckling loads and postbuckling load–deflection curves Stability analysis for a simply supported functionally graded cylindrical panel shows the effects of mate-rial and geometric parameters as well as imperfection on buckling and postbuckling behaviors of the panel
Ó 2009 Elsevier B.V All rights reserved
1 Introduction
Flat and curved panel elements constitute a major portion of the
structure of aerospace vehicles They are found in the aircraft
com-ponents as primary load carrying structures such as wing and
fuse-lage sections as well as in spacecraft and missile structural
applications Moreover, these elements can also be found in
vari-ous industries such as shipbuilding, transportation, and building
constructions Some investigations on buckling and postbuckling
of laminated composite cylindrical panels are reported in works
[2–5] Recently, a new composite is known as Functionally Graded
Materials (FGMs) with high performance heat resistance capacity
has been developed Some works have published relating to the
stability of FGM structures such as[6–9]
In this report, the buckling and postbuckling of FGM cylindrical
panels subjected to axial compressive loads are investigated by an
analytical approach The formulation is based on the classical shell
theory with both von Karman–Donnell type of kinematic
nonlin-earity and initial geometrical imperfection are taken into
consider-ation The resulting equations are solved by Galerkin procedure to
obtain closed-form expressions of the buckling loads and
post-buckling load–deflection curves Stability analysis for a simply
sup-ported panel shows the effects of material and geometric
parameters and imperfection on the buckling and postbuckling
behaviors of the panel
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 as is shown
inFig 1 The panel is made from a mixture of ceramics and metals, and is defined in a coordinate system (x, h, z), where x and h are in the axial and circumferential directions of the panel and z is per-pendicular to the middle surface and points inwards (h/
2 6 z 6 h/2) The effective modulus of elasticity is assumed to vary only in the thickness direction according to simple power law dis-tribution as[8]
EðzÞ ¼ Emþ ðEc EmÞ 2z þ h
2h
where Ecand Emto be elastic moduli of ceramic and metal, respec-tively, andmPoisson ratio assumed to be a constant It is evident that, the inner surface (z = h/2) of the panel is ceramic-rich and the outer surface (z = h/2) is metal-rich
3 Governing equations
In the framework of the classical shell theory[1], the equilib-rium and compatibility equations are derived as follows
Dr4w f;xx=R f;yy w;xxþ w
;xx
2f;xy w;xyþ w
;xy
h
þ f;xx w;yyþ w
;yy
0927-0256/$ - see front matter Ó 2009 Elsevier B.V All rights reserved.
* Corresponding author.
E-mail address: ducnd@vnu.edu.vn (N.D Duc).
Contents lists available atScienceDirect
Computational Materials Science
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 m a t s c i
Trang 2r4f E1 w2
;xy w;xxw;yy w;xx=R þ 2w;xyw
;xy w;xxw
;yy w;yyw
;xx
where y = Rh, r4¼ @4=@x4þ 2@4=@x2@y2þ @4=@y4, and q is lateral
pressure positive inwards Also, w to be deflection of panel, w
ini-tial imperfection representing a small iniini-tial deviation of the panel
surface from a cylindrical shape, f Airy stress function, and
D ¼ E1E3 E
2
E1ð1 m2Þ; ðE1;E2;E3Þ ¼
Z h=2
h=2 EðzÞð1; z; z2Þdz ð4Þ
Eqs.(2) and (3)are the basic equations used to investigate the
sta-bility of functionally graded panels subjected to axial compressive
loading 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
stability of FGM cylindrical panels under mechanical loads The
functionally graded cylindrical panel is assumed to be simply
sup-ported on all edges and, in general case, subjected to in-plane
com-pressive loads, uniformly distributed along the edges, and lateral
pressure uniformly distributed on the outer surface of the panel
The displacement and force boundary conditions for a simply
supported panel are defined as
w ¼ Mx¼ Nxy¼ 0; Nx¼ Nx0 on x ¼ 0; a
where Nx0;Ny0are prebuckling force resultants in directions x and y,
respectively To solve two Eqs.(2) and (3)for two unknowns w and
f, we assume the following approximate solutions satisfying simply
supported boundary conditions on all edges[5]
w ¼ W sinmpx
a sin
npy b
f ¼ F sinmpx
a sin
npy
b hðxÞ kðyÞ
where m, n = 1, 2, are number of half-waves, W and F are
con-stant coefficients depending on m and n Also, h(x) and k(y) to be
preselected functions such that solutions(6)satisfy force boundary
conditions, thus
d2hðxÞ
dx2 ¼
Ny0
F ;
d2kðyÞ
dy2 ¼
Nx0
Considering the boundary conditions(5), the imperfections of the
panel are assumed as follows which are in the shape of the buckling
mode
w¼lh sinmpx
a sin
npy
where 1 6l61 to be imperfection size
Introduction of Eqs.(6)–(8)into Eqs.(2) and (3), then applying Galerkin method for the resulting equations, we obtain nonlinear algebraic equations for W and F Subsequently, eliminating variable
F from these equations gives
Dp4 m2B2
aþ n2
þE1ðmbBaÞ
4
R2
W 16E1m
3nb2B4a
þ Nx0p2b2 m2B2
aþ n2
m2B2
aþ bn2
ðW þlhÞ
þ512E1m
2n2B4 a
þ16b 4 mnp2 m2B2
aþ n2
where m, n are odd numbers, and
Eq (9) is used to investigate the buckling and postbuckling behaviors of FGM cylindrical panels subjected to various condi-tions of loading However, in the present report, simply supported FGM cylindrical panel is assumed to be under only in-plane axial compressive load p (in Pascals), uniformly distributed along curved edges x = 0, a In this case Nx0= ph, Ny0= q = 0 and Eq.(9)yields
p ¼ Dp2 m2B2
aþ n2
m2B2aB2h þ
E1B4aR2am2
p2 m2B2
aþ n2
2 6
3
W þl
16E1mnB
3
aRa 3p2Bh m2B2
aþ n2
ð3W þ 4lÞW
W þl
þ 512E1B
2
an2 9p2B2hm2B2aþ n22ðW þ 2lÞW ð11Þ
where
D ¼ D=h3; E1¼ E1=h; Bh¼ b=h; Ra¼ a=R; W ¼ W=h ð12Þ
For a perfect panel (l= 0), Eq.(11)leads to
p ¼
Dp2m2B2aþ n22
m2B2
4
aR2am2
p2 m2B2
aþ n2
16E1mnB
3
aRa
p2Bh m2B2
aþ n2
2
an2 9p2B2 m2B2
aþ n2
from which buckling compressive load may be obtained at W ¼ 0 as
pu¼
Dp2 m2B2
aþ n2
m2B2
aB2 þ E1B
4
aR2am2
p2ðm2B2
Above equation represents upper buckling compressive load of FGM panel The critical upper buckling load pucris obtained for the values
of m and n that make the preceding expression a minimum
Eq.(13) points out that the pðWÞ curve has an extremum at dp=dW ¼ 0, i.e at W0¼ 9mBaRaBh=ð64nÞ By examining the sign
d2p=dW2one obtains the conclusion that the pðWÞ curve reaches minimum at W0and
pl¼ pðW0Þ ¼Dp2m2B2aþ n22
m2B2aB2h
E1B4
aR2
am2 8p2m2B2þ n22 ð15Þ
θ
x
y z
b
a
R
h
Fig 1 Configuration and the coordinate system of the FGM cylindrical panel.
Trang 3represents lower buckling compressive load of FGM panel This
analysis shows that the panel subjected to axial compression can
exhibit a snap-through to a new equilibrium position at W0 The
intensity of the snap-through is given by the difference between
the upper and lower buckling loads, i.e 9E1m2B4
aR2
a=
8p2 m2B2
aþ n2
2
Eq.(13)determines postbuckling equilibrium paths pðWÞ of the perfect panel with values of m and n that make
buckling loads a minimum In case of imperfect panel, postbuckling
pðWÞ curves represented by Eq.(11)originate from coordinate
ori-gin This indicates imperfection sensitivity of axially loaded panel
and no bifurcation buckling point exists for imperfect panels
5 Results and discussion
To illustrate the proposed approach, we consider a
ceramic–me-tal functionally graded panel that consist of aluminum and
alu-mina with the following properties
As shown in Ref.[2], the most pronounced buckling and
post-buckling responses for deformation modes with half-wave
num-bers m = n = 1 Thus, the results presented in this section also
correspond to values of m = n = 1 Effects of some material and
geo-metric parameters on the postbuckling behavior of the perfect and
imperfect FGM cylindrical panels are shown inFigs 2–5 It is noted
that in all figures W/h denotes the dimensionless maximum
deflec-tion of the panel
Fig 2 shows the postbuckling load–deflection curves of FGM
cylindrical panels under axial compressive loads with different
val-ues of volume fraction index k (=0, 1 and 5) As can be seen, both
well-known snap-through behavior and imperfection sensitivity
of the panels are exhibited in this figure Both bifurcation-type
buckling loads and postbuckling equilibrium paths become lower
for higher values of k representing panels with the greater
percent-age of metal, as expected Furthermore, the severity of
snap-through response, which is measured by difference between upper
(bifurcation point) and lower buckling loads, is decreased when k
increases
Fig 3shows the effect of width-to-thickness ratio b/h (=20, 30
and 40) on the postbuckling behavior of the FGM panels under
ax-ial compression with k = 1 Fig 4shows the effect of
length-to-width ratio a/b (=0.75, 1.0 and 1.5) on the postbuckling behavior
Fig 3 Postbuckling paths of the panel vs b/h.
Fig 4 Postbuckling paths of the panel vs a/b.
Trang 4of the panels under similar conditions It is evident from these
fig-ures that postbuckling load carrying capacity of the panels is
con-siderably reduced when b/h and a/b ratios increase It is also seen
that panels with small a/b ratio experience a severe snap-through
response, although their buckling loads are comparatively high
Furthermore, the postbuckling equilibrium paths become more
stable, i.e exhibit a more benign snap-through behavior, for small
values of b/h or large values of a/b standing for shallower panels
The effect of panel curvature on the postbuckling response of
axially-loaded FGM cylindrical panels is illustrated inFig 5 with
three various values of length-to-radius ratio a/R (=0.2, 0.5 and
0.75) As can be observed, the buckling loads and postbuckling load
bearing capacity of the panels are increased when a/R increases
and the deflection is small and a converse trend occurs when the
deflection is sufficiently large In addition, it is shown that the
pan-els with small a/R ratio (i.e a/R = 0.2) have stable postbuckling
equilibrium paths due to its flatted configuration
6 Concluding remarks
The report presents a simple analytical approach to investigate
the buckling and postbuckling behaviors of functionally graded
cylindrical panels under axial compressive loads By using Galerkin
method, closed-form relations of buckling loads and postbuckling
load–deflection curves for a simply supported FGM cylindrical
pa-nel under axial compression, with and without imperfection, are
determined The results show the snap-through behavior, imper-fection sensitivity and complex postbuckling behavior of axially loaded panels The study also confirms that the postbuckling behaviors of FGM cylindrical panels are greatly influenced by material and geometric parameters, initial geometric imperfection
as well
Acknowledgements This report is supported by the science researching project of Vietnam National University – Hanoi, coded QGTD.09.01 The authors are grateful for this financial support
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