Nonlinear mechanical, thermal and thermo-mechanical postbucklingof imperfect eccentrically stiffened thin FGM cylindrical panels on elastic foundations a Vietnam National University, Han
Trang 1Nonlinear mechanical, thermal and thermo-mechanical postbuckling
of imperfect eccentrically stiffened thin FGM cylindrical panels on
elastic foundations
a
Vietnam National University, Hanoi, 144 Xuan Thuy-Cau Giay, Hanoi, Vietnam
b
The University of Melbourne, Parkville, VIC 3010, Australia
a r t i c l e i n f o
Article history:
Received 24 April 2015
Received in revised form
19 July 2015
Accepted 3 August 2015
Keywords:
Nonlinear mechanical and thermal
post-buckling
Eccentrically stiffened FGM cylindrical
pa-nels
Imperfection
Elastic foundations
a b s t r a c t
This paper presents an analytical approach to investigate the nonlinear stability analysis of eccentrically stiffened thin FGM cylindrical panels on elastic foundations subjected to mechanical loads, thermal loads and the combination of these loads The material properties are assumed to be temperature-dependent and graded in the thickness direction according to a simple power law distribution Governing equations are derived basing on the classical shell theory incorporating von Karman–Donnell type nonlinearity, initial geometrical imperfection, the Lekhnitsky smeared stiffeners technique and Pasternak type elastic foundations Explicit relations of load–deflection curves for FGM cylindrical panels are determined by applying stress function and Galerkin method The effects of material and geometrical properties, im-perfection, elastic foundations and stiffeners on the buckling and postbuckling of the FGM panels are discussed in detail The obtained results are validated by comparing with those in the literature
& 2015 Elsevier Ltd All rights reserved
1 Introduction
Composite panels are commonly used in aerospace, mechanics,
naval and other high-performance engineering applications due to
their light weight, high specific strength and stiffness, excellent
thermal characteristics At high temperatures, composite panels
are found to buckle without the application of mechanical loads
Therefore, the buckling and postbuckling response of composite
panels have to be well understood Recently, a new class of
com-posite materials known as functionally graded materials (FGMs)
attracts special attention of a lot of authors in the world FGM is a
new generation of composite material in which its mechanical
properties vary smoothly and continuously from one surface to the
other Functionally graded structures such as cylindrical panels in
recent years, play the important part in the modern industries As
a result, static response of FGM cylindrical panels has been the
subject of many studies for a long period of time Shen and Wang
[1]presented thermal postbuckling analysis for FGM cylindrical
panels resting on elastic foundations They [2]also studied the
nonlinear bending analysis of simply supported FGM cylindrical
panel resting on an elastic foundation in thermal environments
Lee et al.[3]investigated the thermomechanical behaviors of FGM panels in hypersonic airflows Alibeigloo and Chen[4]developed the three-dimensional elasticity solution for static analysis of a FGM cylindrical panel with simply supported edges Tung and Duc [5] studied the nonlinear response of thick FGM doubly curved shallow panels resting on elastic foundations and subjected to some conditions of mechanical, thermal, and thermomechanical loads They[6]also investigated the nonlinear response of pres-sure-loaded FGM cylindrical panels with temperature effects Aghdam et al.[7]considered bending of moderately thick clamped FGM conical panels subjected to uniform and non-uniform dis-tributed loadings Du et al [8]studied the nonlinear forced vi-bration of infinitely long functionally graded cylindrical shells is using the Lagrangian theory and the multiple scale method A semi-analytical solution for static response of fully clamped shear-deformable FGM doubly curved panels is presented by Shahman-souri et al.[9] Kiani et al.[10]focused on the static, dynamic and free vibration analysis of a FGM doubly curved panel Bich et al [11]researched the linear buckling of FGM truncated conical pa-nels subjected to axial compression, external pressure and the combination of these loads Static and dynamic stabilities of FGM panels which are subjected to combined thermal and aerodynamic loads are investigated in work of Sohn and Kim[12]based on the first order shear deformation theory Yang et al.[13]published the
Contents lists available atScienceDirect
journal homepage:www.elsevier.com/locate/tws Thin-Walled Structures
http://dx.doi.org/10.1016/j.tws.2015.08.005
0263-8231/& 2015 Elsevier Ltd All rights reserved.
n Corresponding author.
E-mail address: ducnd@vnu.edu.vn (N.D Duc).
Trang 2results on thermo-mechanical postbuckling analysis of FGM
cy-lindrical panels with temperature-dependent properties Recently,
in 2014, Duc[14]published a valuable book“ Nonlinear static and
dynamic stability of functionally graded plates and shells”, in which
the results about nonlinear static stability of shear deformable
FGM panels are presented Tung [15] introduced an analytical
approach to investigate the effects of tangential edge constraints
on the buckling and postbuckling behavior of FGM flat and
cy-lindrical panels subjected to thermal, mechanical and
thermo-mechanical loads and resting on elastic foundations
However, since this area is relatively new, there are very little
researches on nonlinear static problems of FGM cylindrical panels
and cylindrical shells reinforced by stiffeners Duc and Quan[16]
investigated the nonlinear response of eccentrically stiffened FGM
cylindrical panels on elastic foundations subjected to mechanical
loads Najafizadeh et al [17] considered the elastic buckling of
FGM stiffened cylindrical shells by rings and stringers subjected to
axial compression loading Dung et al.[18]analyzed the nonlinear
buckling and postbuckling of FGM stiffened thin circular
cylind-rical shells surrounded by elastic foundations in thermal
en-vironments and under torsional load Bich et al.[19]presented an
analytical approach to investigate the nonlinear static and
dy-namic buckling of imperfect eccentrically stiffened FGM thin
cir-cular cylindrical shells subjected to axial compression
To the knowledge of the authors, there is limited publication on
the stability of FGM structures reinforced by eccentrically
stiffen-ers in thermal environments The most difficult part in this type of
problem is to calculate the thermal mechanism of FGM structures
as well as stiffeners under thermal loads Duc et al [20,21]
in-vestigated the nonlinear postbuckling of an eccentrically stiffened
thin FGM plate and circular cylindrical shell resting on elastic
foundation in thermal environments Development of the results
in these researches, this paper deals with the nonlinear
post-buckling of imperfect eccentrically stiffened thin FGM cylindrical
panels on elastic foundations under mechanical loads, thermal
loads and the combination of these loads The material properties
are assumed to be temperature-dependent and graded in the
thickness direction according to a simple power law distribution
Both of the panels and the stiffeners are assumed to be deformed
due to the presence of temperature Using Galerkin method and
stress function, the effects of geometrical and material properties,
imperfection, elastic foundations and stiffeners on the nonlinear
response of the imperfect eccentrically stiffened FGM cylindrical
panels are analyzed
2 Problem statement Consider an eccentrically stiffened functionally graded cylind-rical panel with the radii of curvature, thickness, axial length and arc length of the panel areR,h,aandb, respectively and is defined
in coordinate system(x y z, , ),as shown inFig 1 The panel is re-inforced by eccentrically longitudinal and transversal stiffeners The width and thickness of longitudinal and transversal stiffeners are denoted byd h x, xandd h y, yrespectively;s s x, yare the spacings
of the longitudinal and transversal stiffeners The quantitiesA x,A y
are the cross-section areas of stiffeners and I I z z x, y, x, y are the second moments of cross-section areas and the eccentricities of stiffeners with respect to the middle surface of panel, respectively
E0is Young's modulus of ring and stringer stiffeners In order to provide continuity between the panel and stiffeners, suppose that stiffeners are made of full metal(E0=E m)
The panel is made from a mixture of ceramic and metal, and the material constitution is varied gradually by a simple power law distribution, in which the volume fractions of the ceramic and metal are expressed as
⎛
⎝
⎞
⎠
2
m
N
( ) = + ( ) = − ( )
( )
where N is volume fraction index (0 ≤N< ∞), subscripts m
and c stand for the metal and ceramic constituents, respectively Effective propertiesPr effof FGM panel, such as the elastic modulus
Eand the thermal expansion coefficientαare determined by linear rule of mixture as
Preff( ) =z Pr V z c c( ) +Pr V z m m( ), ( )2
in which Pr denotes a temperature-dependent material prop-erty The effective properties of the FGM panel are obtained by substituting Eq.(1)into Eq.(2)as
⎡⎣ ⎤⎦
⎡⎣ ⎤⎦⎛⎝ ⎞⎠
h
c c
mc mc
N
α
[ ( ) ( )] = ( ) ( )
+ ( ) ( ) +
( )
where
E mc(z T, ) =E m(z T, ) −E z T c(, ), α mc(z T, ) =α m(z T, ) −α c(z T, ), ( )4
and the Poisson's ratio is assumed to be constant
z v const
ν( ) = =
A material propertyPrcan be expressed as a nonlinear function
of temperature[1,2,13]
Fig 1 Configuration and the coordinate system of an eccentrically stiffened cylindrical panel on elastic foundations.
Trang 3Pr P0 P T1 1 1 P T P T P T , 5
in which T=T0+ ΔT, Δ T is the temperature increment in the
environment containing the panel and T0=300 K (room
tem-perature),P P0, − 1,P P1, 2andP3are coefficients characterizing of the
constituent materials
The panel–foundation interaction of Pasternak model is given
by
where∇ = ∂ ∂2 2/ x2+ ∂ ∂2/ y2,wis the deflection of the panel,k1is
Winkler foundation modulus andk2is the shear layer foundation
stiffness of Pasternak model
3 Theoretical formulation
Taking into account the von Karman–Donnell geometrical
nonlinearity terms, the strains at the middle surface and
curva-tures relating to the displacement components u v w, , based on
the classical thin shell theory are[22,23]
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎛
⎝
⎜
⎜
⎜⎜
⎞
⎠
⎟
⎟
⎟⎟
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎛
⎝
⎜
⎜⎜
⎞
⎠
⎟
⎟⎟
k k k
w w w
/2
7
x
y
xy
x x
y x x y
x y xy
xx yy xy
0
0
0
, ,2
, , ,
ε
ε
γ
=
+
− +
+ +
=
−
−
−
( )
where x0
ε and y0
ε are normal strains, xy0
γ is the shear strain at the middle surface of the panel andk ij, ij=x y xy, , are the curvatures
The strain components across the panel thickness at the
dis-tancez from the mid-plane are given by
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
z
k k k
2
8
x
y
xy
x
y
xy
x y xy
0
0
0
ε
ε
γ
ε
ε
γ
( )
Hooke's law for cylindrical panel taking into account the
tem-perature-dependent properties is defined as
E z T
E z T
, 1 , 1, 1 , ,
x sh y sh x y
y x
xy sh xy
2
(σ σ )
σ
ν γ
= ( )
− [( )
+ ( ) − ( + ) ( ) ( )]
= ( )
and for stiffeners
x st y st 0 x y 0 0
ν α
( ) = ( )( ) − ( )
where E T0( ), α0( )T are the Young's modulus and thermal
ex-pansion coefficient of the stiffeners, respectively Unlike other
publications, in this paper, material properties of the eccentrically
outside stiffeners are assumed to depend on temperature
All elastic moduli of FGM panels and stiffeners are assumed to
be temperature dependence and they are deformed in the
pre-sence of temperature Therefore, the geometric parameters, the
panel's shape and stiffeners vary through the deforming process
due to the temperature change However, because the thermal
stress of stiffeners is subtle which distributes uniformly through
the whole panel structure, we can ignore it The contribution of
stiffeners can be accounted for using the Lekhnitsky smeared
stiffeners technique Then integrating the stress–strain equations
and their moments through the thickness of the panel, the
ex-pressions for force and moment resultants of an eccentrically
stiffened FGM cylindrical panel are obtained as
,
,
2 ,
,
,
x
y y
x
y y
12 0 22
66 0 66
0
2
66 0 66
γ
γ
= ( + ) + + ( + ) + +
= + ( + ) + + ( + ) +
= ( + ) + + ( + ) + +
= + ( + ) + + ( + ) +
where
h
1 2
/2
/2
∫
ν
ν
ν
ν
ν
ν
= =
− = − = ( + )
= =
− = − = ( + )
= =
− = − = ( + ) ( ) = −
− − ( ) ( )Δ ( ) ( )
and
⎛
⎝
⎞
⎠
⎡
⎣⎢
⎤
⎦⎥
E A z s
1,
1 2
1
2 1 ,
12
1 3
1 2
1
x x x x x y
y y
y y
x x x x y
y y y
x x
T y y T
c mc
3
3
3 3
= ( ) + ( ) = ( ) + ( )
= +
+ = + − ( + )
+ − + + ( + ) ( )
with the geometric shapes of stiffeners after the thermal de-formation process in Eq.(13)can be determined as the follows:
α
α
= ( + ( )) = ( + ( ))
= ( + ( ))
= ( + ( )) = ( + ( ))
= ( + ( )) = ( + ( ))
The nonlinear equilibrium equations of FGM cylindrical panels based on classical shell theory are given as[22,23]
x xx xy xy y yy
y
x xx xy xy y yy
where q is an external pressure uniformly distributed on the surface of the panel
The geometrical compatibility equation for an imperfect FGM cylindrical panel is written as[22,23]
Trang 4w w w w w w w
R
2
16
x yy y xx xy xy xy xx yy xy xy xx yy
yy xx
xx
,
0
,
0
,
0
( )
⁎
The first two equations of the nonlinear motion Eqs (15a),
(15b) are automatically satisfied by choosing the stress function
f x y( , )as
Substituting relation(17)into Eq.(11), we obtain
, ,
0
0
0
66 , 66 ,
γ
where
⎛
⎝
⎞
⎠
⎛
⎝
⎠
⎟
⎛
⎝
⎞
⎠
⎛
⎝
⎠
⎟
A
s A
E A
E A
A
1
1
,
19
x x
y y
x x
y y
66
0
12 22 12 12 22 21 11 12 12 11
66
66
66
Δ
=
( )
⁎
⁎
Substituting once again Eq (18) into the expression of
M M M x, y, xyin Eq.(11), then M M M x, y, xyinto the Eq.(15c)leads to
2
0,
20
xxxx yyyy
xxyy x xx xy xy
y yy
y
11 , 22 ,
+ + ( + − )
− ( + + ) + +
+ + + − + ∇ =
( )
where
,
,
, ,
x
x
x
y
y
y
x
y
22 22
0
66 66 66 66
For an imperfect cylindrical panel, Eq (20) is modified into
form as
2 4 2
yy xx xy xy xx yy xx
,
1
2 2
+ + ( + − )
where w x y⁎( , )is a known function representing initial small
im-perfection of the panel
Setting Eq.(18)into Eq.(16)gives the compatibility equation of
an imperfect eccentrically stiffened FGM cylindrical panel as
⎜
⎟
⎛
⎝
⎞
⎠
w R
2 2 2
0
23
xxxx yyyy xxyy xxxx
xy xx yy xy xy xx yy yy xx
xx
,
( )
Eqs.(22) and (23) are nonlinear equations in terms of variables
wand f and they are used to investigate the nonlinear stability of FGM eccentrically stiffened cylindrical panels on elastic foundations
4 Solution procedures
In the present study, the edges of eccentrically stiffened FGM cylindrical panel are assumed to be simply supported Depending
on the in-plane restraint at the edges, three cases of boundary conditions, labeled asCases 1, 2 and 3will be considered[5] Case 1 Four edges of the FGM cylindrical panel are simply sup-ported and freely movable (FM) The associated boundary condi-tions are
0, at 0,
0
0
Case 2 Four edges of the FGM cylindrical panel are simply sup-ported and immovable (IM) In this case, boundary conditions are
0, at 0,
0 0
Case 3 All edges of the FGM cylindrical panel are simply sup-ported Two edges x=0,a are freely movable, whereas the re-maining two edges y=0, b are immovable For this case, the boundary conditions are defined as
0, at 0,
0
0
where N x0, N y0 are in-plane compressive loads at movable edges (i.e.Case 1 and the first of Case 3) or are fictitious com-pressive edge loads at immovable edges (i.e.Case 2and the second
ofCase 3)
The mentioned conditions(24)–(26)can be satisfied identically
if the panel deflectionwis chosen by[5,15,16]
whereλ m=m a π/ ,δ n=n b π/ ,m n, =1, 2, are natural numbers representing the number of half waves in thexand ydirections, respectively;Wis the amplitude of deflection
Concerning with the initial imperfection w⁎, we introduce an assumption it has the same form like the panel deflection w, i.e
where the coefficient μvarying between 0and1represents im-perfection size
Introduction of Eqs (27) and (28) into the compatibility Eq (23), we define the stress function as
Trang 5f A x A y A x y N y
N x
cos 2 cos 2 sin sin 1
2 1
y
0 2
+
( )
with
A
A
2 2
n
m
m n m
1
2
11 2
2 2
22 2 3
2
11 4 22 4 66 12 2 2
21 4 12 4 11 22 66 2 2
11 4 22 4 66 12 2 2
δ
λ
λ
=
+ + ( − )
− + + ( + − )
Setting Eqs.(25)–(27)into Eq.(22)and applying the Galerkin
procedure for the resulting equation we obtain equation for
de-termining nonlinear static analysis of eccentrically stiffened FGM
cylindrical panels on elastic foundations
⎧
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎫
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎭
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎡
⎣
⎦
⎥
⎛
⎝
⎠
⎟
mn
R
W
A R
B A
B
mn
mn
N
R
q
4
2 2 2 1 2 4
8
3
1 2 2 2
6
2
4
0,
31
m n
m
m n
m n
m
n
m
m n
m n
m n
m n
x m y n
m n
y
m n
2
2
21
4
12 4
11 22 66
2 2
11
4
22 4
66 12
2 2
21 4 12 4 11 22 66 2 2
2
11
4
22 4
66 12
2 2
4
2
11 4 22 4 66 12 2 2
11 4 22 4 12 21 66 2 2
11
4
22 4 66 12
2 2 2
21
4
12
4
11 22 66
2 2
11
4
22 4
66 12
2 2
11
21
11 12
22
22
4
11 2
0 2 0 2
0
π
λ δ
λ
λ δ
λ
μ
δ
π
λ δ
π
+ + ( + − )
+ + ( − )
− + + ( + − )
+ + ( − )
−
+ + ( − )
− − − ( + + )
+
+ + ( − )
− + + ( + − )
+ + ( − )
( + )
( )
⁎
⁎
⁎
⁎
⁎
wherem n, are odd numbers Hereafter, we will consider in detail
three problems corresponding to three mentioned loading types
4.1 Mechanical stability analysis Consider a simply supported eccentrically stiffened FGM cylindrical panel with all movable edges and resting on elastic foundations Two cases of mechanical loads will be analyzed
4.1.1 Eccentrically stiffened FGM cylindrical panel under uniform external pressure
Consider an eccentrically stiffened FGM cylindrical panel with movable edges and only subjected to uniform external pressure on the upper surface of the panel In this case,N x0=N y0=0, and Eq (31)leads to
2
4
= ¯ + ¯ ¯ + + ¯ ¯ +
where
⎛
⎝
⎜⎜ ⎞⎠⎟⎟
⎛
⎝
⎠
⎟⎟
B
mn B D K
B
m n B R B
m n B R
mn B
A B
B
B A
B A
B
m B A
n A
4 16
8
2 2
16
2
2
a h
a h
a
h
a b h
a b
h
a b
b h
a h
h a
1
2 4
11 1 4
4 2
11 2 4
2 2 2
5 6 4
11 5 6 22 3 3 6 2 12 21 66
4
3 4 2 3 21
4 4
2 2 2
12 4
11 4 4 66 12 2 2 2 22 4
5 2 4 2 2
11
4 4
2 2 2
22 4 6
4 21
4 4
2 2 2
12
42
11 4 4 66 12 2 2 2 22 4
2
4 2 2 4 3
11 4 4 66 12 2 2 2 22 4
2 2 4 2
21
4 4
2 2 2
12 4 4
11 4 4 66 12 2 2 2 22 4
3
2 2
11 3
2 2 4 2 4 21 11 12 22
4 6 4
4 4 22 4 11
π
π
π
π
π
π
−
+ ( + − ) + + ( − ) +
+
+ ( − ) +
+
+ ( + − ) + + ( − ) +
= −
+ ( − ) +
+
+ ( + − ) + + ( − ) +
( )
⁎
⁎
⁎
⁎
⁎
and
k a
B
B
B h
D
D
D h
D
, , / , / , / ,
b
1 1 4 11
2
66 66 11 113 22 223 12 123
21 213 66 663
⁎
⁎
⁎
⁎
⁎
⁎
⁎
⁎
⁎
⁎
⁎
⁎
⁎
⁎
⁎
⁎
Eq (32) may be used to trace postbuckling load–deflection curves of FGM cylindrical panels resting on elastic foundations subjected to uniform external pressure
For a perfect panel( = )μ 0 ,Eq.(32)leads to
q=b W1 ¯ + (b2+b W3) ¯2+b W4 ¯3 (35)
Trang 64.1.2 Eccentrically stiffened FGM cylindrical panel under axial
compressive loads
A movable edges eccentrically stiffened cylindrical panel
sup-ported by elastic foundations and subjected to axial compressive
loadsF xuniformly distributed at two curved edgesx=0,ain the
absence of external pressure and thermal loads is considered
In this case, the prebuckling force resultants are
The introduction of Eq.(36)into Eq.(31)gives
W W
2
2 ,
37
μ
μ
= ¯
¯ + + ¯ +
¯ ¯ +
¯ + + ¯ ¯ + ( )
where
⎛
⎝
⎠
⎟
⎛
⎝
⎠
⎟⎟
B
m B B
B
m B R
m B B
R
B
m B
B D K
n
mB
n mB
B A
B A b
m B B
m B
A
n A
4
2
2 2
2
, 32
32
3
2
2
3
8
a
a b
a h
b
h
a
h
a h
a b
h
b
a h
a
1
2 2 2
11
2
4 2 22
2 2 2
2 2
2
2 2 2 2
11
4 4
2 2 2
22 4 2
2 2 2
21
4 4
2 2 2
12
42
11 4 4 66 12 2 2 2 22 4
21 4 4 11 22 66 2 2 2 12 4
11 4 4 66 12 2 2 2 22 4
11
2 2 2
2
2 2
2
11 1
2 2 2
2
2 2
11 4 4 66 12 2 2 2 22 4
2
21 4 4 11 22 66 2 2 2 12 4
11 4 4 66 12 2 2 2 22 4
11
2 21 11 12 22
4
2
2 2 2
4 4 22 4 11
π
π
π
π
π
π
+
+ ( − ) +
+
+ ( + − ) + + ( − ) +
+ ( − ) +
= −
+ ( − ) +
+
+ ( + − ) + + ( − ) +
( )
⁎
⁎
⁎
⁎
⁎
Eq (37) is employed to trace postbuckling load–deflection
curves of the imperfect eccentrically stiffened FGM panel subjected
to axial compressive loads
For a perfect cylindrical panel ( = )μ 0 only subjected to axial
compressive load F x, Eq.(37)leads to
2
From which upper buckling compressive load may be obtained
withW→0asF x=b1
4.2 Thermal stability analysis
A simply supported eccentrically stiffened FGM cylindrical
pa-nel on elastic foundations with all immovable edges is considered
The panel is subjected to uniform external pressure q and
si-multaneously exposed to temperature environments The in-plane
condition on immovability at all edges, i.e u=0 at x=0,a and
v=0at y=0,b, is fulfilled in an average sense as[5]
u
x dxdy
v
y dydx
40
From Eqs.(7) and (18) one can obtain the following expressions
in which initial imperfection has been included
u
v
R
1
1
x x x
y y y
y
,2 , ,
,2 , ,
Φ
Φ
∂
∂
( )
⁎
⁎
Substitution of Eqs.(27)–(29)into Eq.(41)and then the result into Eq.(40)givefictitious edge compressive loads
⎧
⎨
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎡
⎣
⎤
⎦
⎫
⎬
⎪
⎪
⎪
⎭
⎪
⎪
⎪
42
Nx mn
m
R n
R W
0 1 42
1
114 224 66 212 2 2
2 2
214 124 11 22 266 2 2
114 224 66 212 2 2
2
1
11 22 122
11 11 21 12 2 12 11 22 12 2 12 1
8 11 22 122 11
2
122 2 ,
( )
Φ π
λ δ
δ
= +
⁎ + ⁎ + (⁎ − ⁎)
−
⁎ + ⁎ + (⁎ + ⁎ − ⁎)
⁎ + ⁎ + ( ⁎ − ⁎)
(⁎ ⁎ − ⁎ )
⁎ ⁎ + ⁎ ⁎) + (⁎ ⁎ + ⁎ ⁎) − ⁎
+ (⁎ ⁎ − ⁎ )
⁎ + ⁎ ) ( + )
⎧
⎨
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎡
⎣
⎤
⎦
⎫
⎬
⎪
⎪
⎪
⎭
⎪
⎪
⎪
43
Ny mn
m R
m
R W
0 1 42
1
114 224 66 212 2 2
4
214 124 11 22 266 2 2
114 224 66 212 2 2
2
1
11 22 122
11 12 21 22 2 12 12 22 22 2 22 1
8 11 22 122 12
2
222 2 .
( )
Φ π
λ
λ
= +
⁎ + ⁎ + (⁎ − ⁎)
−
⁎ + ⁎ + (⁎ + ⁎ − ⁎)
⁎ + ⁎ + ( ⁎ − ⁎)
(⁎ ⁎ − ⁎ )
(⁎ ⁎ + ⁎ ⁎) + (⁎ ⁎ + ⁎ ⁎) −
⁎
+ (⁎ ⁎ − ⁎ )
⁎ + ⁎ ) ( + )
Introducing N x0,N y0at Eqs.(42), (43)into Eq.(31)gives
⎧
⎨
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎫
⎬
⎪
⎪
⎪
⎪
⎭
⎪
⎪
⎪
⎪
⎡
⎣
⎢
⎢
⎢
⎢
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎧
⎨
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎡
⎣
⎤
⎦
⎫
⎬
⎪
⎪
⎪
⎭
⎪
⎪
⎪
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎡
⎣
⎢
⎢
⎛
⎝
⎜
⎜⎜
⎞
⎠
⎟
⎟⎟
⎤
⎦
⎥
m
m R
m
W
n mn
m R
mn
m
R n
R W
mn n
A R m B A B
m
A m A n
A A
A m A n
1 1
2 2 21 4
124 11 22 266 2 2
114 224 66 212 2 2
214 12 4 11 22 266 2 2
2
114 22 4 66 212 2 2 4
2
1
114 224 66 212 2 2
114 224 12 21 4 66 2 2 2 2 2 1
32 2
3 2
1
114 224 66 212 2 2
2
214 124 11 22 266 2 2
114 224 66 212 2 2
4 2
1
114 224 66 212 2 2
2 2
214 124 11 22 266 2 2
114 224 66 212 2 2
2
1
11 22 122
11 11 21 12 2 12 11 22 12 2 12
611 2
2 3 21 11 12 22
1
16 2
114 224
11 22
112 122
8 11 22 122
2
( )
Φ λ
λ
μ
δ π
λ
π
λ δ
δ
π δ λ
μ
λ
μ
=
⁎ + ⁎ + (⁎ + ⁎ − ⁎)
⁎ + ⁎ + ( ⁎ − ⁎)
−
⁎ + ⁎ + ( ⁎+ ⁎ − ⁎)
⁎ + ⁎ + (⁎ − ⁎)
−
⁎ + ⁎ + (⁎ − ⁎)
− ⁎ − ⁎ − ( ⁎ + ⁎ + ⁎) − + −
( + )
+
⁎ + ⁎ + (⁎ − ⁎)
−
⁎ + ⁎ + (⁎ + ⁎ − ⁎)
⁎ + ⁎ + ( ⁎ − ⁎)
−
⁎ + ⁎ + (⁎ − ⁎)
−
⁎ + ⁎ + (⁎ + ⁎ − ⁎)
⁎ + ⁎ + ( ⁎ − ⁎)
+ (⁎ ⁎ − ⁎)
⁎ ⁎ + ⁎ ⁎) + (⁎ ⁎ + ⁎ ⁎) − ⁎
⁎
⁎ +
⁎
⁎ ) (( ++ ))
−
⁎ + ⁎
⁎ ⁎ +
(⁎ + ⁎ ) (⁎ ⁎ − ⁎ )
( + )
In this paper, the eccentrically stiffened FGM cylindrical pa-nel is exposed to temperature environments uniformly raised from stress free initial stateT itofinal valueT fand temperature increment Δ =T T f−T i is considered to be independent from thickness variable The thermal parameter is obtained from Eq (12)as
1
where
Trang 7⎣⎢
⎤
⎦⎥
P
N
E N
1
mc c c mc mc mc
=
+
Setting Eq.(45)into Eq.(44)gives
W W
2
2 ,
47
μ
μ
Δ = ¯
¯ + + ¯ +
¯ ¯ +
¯ + + ¯ ¯ + ( )
in which specific expressions of coefficients b i3 ( =i 1, 4) are
given inAppendix A
Eq.(47)shows the relationship of thermal load–deflection of
the eccentrically stiffened FGM panel in postbuckling state and
used to trace postbuckling curves of the FGM panel under
thermal load The two sides of Eq (47) are temperature
de-pendence which makes it very complex The iterative algorithm
is used to determine the deflection–load relations in the
buck-ling period of the FGM panel To be more specific, given the
volume fraction index N, the geometrical parameters
b a/ , b h/ , b R/
( )and the value ofW h/ , we can use these values to
determine ΔT in Eq (47)as the follows: we choose an initial
step for ΔT1 on the right side in Eq (47) with Δ =T 0
(T=T0=300 K) In the next iterative step, we replace the known
value of ΔT found in the previous step to determine the right
side of Eq.(47), ΔT2 This iterative procedure will stop at the
kth-steps ifΔT ksatisfies the condition|Δ − Δ | ≤T T k ε Here,ΔTis a
desired solution for the temperature andεis a tolerance used in
the iterative steps
If the imperfection μ=0 and W→0, from above expression
(47)givesΔ =T b1
4.3 Thermo-mechanical stability analysis
The simply supported FGM cylindrical panel with tangentially
restrained edges is assumed to be subjected to external pressureq
uniformly distributed on the outer surface of the panel and
ex-posed to uniformly raised temperaturefield
Subsequently, setting Eq.(45)into Eqs.(42) and (43)then the
result into Eq.(31)give
2
= ¯ + ¯ ¯ + + ¯ ¯ +
in which specific expressions of coefficientsb i4 i 5
( = )are given in Appendix B
Eq (48) expresses explicit relation of pressure-deflection
curves for eccentrically stiffened FGM cylindrical panels
rested on elastic foundations and under combined action of
uniformly raised temperature field and uniform external
pressure
5 Numerical results and discussion
5.1 Validation of the present approach
To validate the present study, firstly, Fig 2 compares the
results of this paper for an unstiffened FGM cylindrical panel
under axial compressive loads with the results given in work of
Tung[15]with different values of elastic foundation stiffnessK1
andK2in the case of temperature independent properties.F xis
found from Eq (37) and the data base in this case is taken:
b a/ =1, b h/ =50, b R/ =0.1, N=1, μ=0.1, K1=K2=0
Secondly,Fig 3compares the present results with those of Duc
et al.[16]for stiffened and unstiffened FGM cylindrical panel
un-der uniform external pressure based on classical shell theory in
the case of temperature independent properties The input
Fig 2 Comparisons of nonlinear load–deflection curves with results of Tung [15] for the unstiffened FGM cylindrical panel under axial compressive loads.
Fig 3 Comparisons of nonlinear load–deflection curves with results of Duc et al [16] for the unstiffened and stiffened FGM cylindrical panel under uniform external
Trang 8parameters are: b/a¼1, b/h¼50, b/R¼0.5, K1¼10, K2¼30 where q
is found from Eq.(32)
Finally, the thermal postbuckling behavior of Si3N4/SUS304
unstiffened FGM cylindrical panels under uniform temperature
rise with two values of Winkler elastic foundation stiffness are
compared in Fig 4 with the theoretical results of Shen and
Wang [1] based on the higher order shear deformation shell
theory The temperature dependent properties are taken
N 0.5, m n 1, h 5mm b h, / 40, a b/ 1.2, a R/ 0.5,
0.05
μ
=
The elastic foundation stiffness in this comparison are identified as
K k b , K
E h
k b
E h
1
2 3
As can be seen that good agreements are obtained in these
three comparisons
Next, we will investigate the effects of the volume fraction
in-dex, the geometrical dimensions, elastic foundations,
imperfec-tions and stiffeners on the nonlinear response of the eccentrically
stiffened FGM cylindrical panel
The effective material properties with dependent temperature
in Eq.(5)are listed inTable 1 [1,2,13] The Poisson's ratio isv=0.3
The parameters for the stiffeners are[16]
0.4 m, 0.0225 m, 0.003 m,
0.004 m
1 2
= =
5.2 Effects of temperature Figs 5–7show effects of temperature increment ΔT on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under uniform external pressure (movable edges), axial compressive loads (movable edges) and uniform external pres-sure (immovable edges), respectively Obviously, the load-car-rying capacity of the panel with temperature independent properties is higher than the one of the panel with temperature dependent properties Moreover, the increase of temperature increment leads to the reduction of load-carrying capacity of the panel
5.3 Effects of elastic foundations and initial imperfection Figs 8–11 indicate effects of elastic foundations on the nonlinear response of the eccentrically stiffened FGM cylind-rical panels under uniform external pressure (movable edges), axial compressive loads (movable edges), uniform tempera-ture rise (immovable edges) and uniform external pressure (immovable edges), respectively Obviously, the load-carrying capacity of the panel becomes considerably higher due to the support of elastic foundations In addition, the beneficial effect
Fig 4 Comparisons of thermal postbuckling behavior with results of Shen and
Wang [1] for the unstiffened FGM cylindrical panels under uniform temperature
rise.
Table 1
Material properties of the constituent materials of the considered FGM panel.
K1
ρ(kg/m 3
K 1
Fig 5 Effects of temperature increment on the nonlinear response of the eccen-trically stiffened FGM cylindrical panels under uniform external pressure (movable edges).
Trang 9of the Pasternak foundation on the postbuckling response of
the eccentrically stiffened FGM cylindrical panels is better
than the Winkler one The effects of initial imperfection with
the coefficientμon the nonlinear response of the eccentrically
stiffened FGM cylindrical panels under different type of loads
are also shown inFigs 8–11 It can be seen that the perfect
cylindrical panel has a better mechanical and thermal loading
capacity than those of the imperfect panel
5.4 Effects of volume fraction index
Figs 12–15 show effect of volume fraction index Non the
nonlinear response of the imperfect and perfect eccentrically stiffened FGM cylindrical panels under uniform external pressure (movable edges), axial compressive loads (movable edges), uni-form temperature rise (immovable edges) and uniuni-form external pressure (immovable edges), respectively As expected, the load-carrying capacity of the FGM panel gets better if the volume N
increases This is reasonable because when N is increased, the ceramic volume fraction is increased; however, elastic module of ceramic is higher than metal (E >E ) The results from these
Fig 6 Effects of temperature increment on the nonlinear response of the
eccen-trically stiffened FGM cylindrical panels under axial compressive loads (movable
edges).
Fig 7 Effects of temperature increment on the nonlinear response of the
eccen-trically stiffened FGM cylindrical panels under uniform external pressure
(im-movable edges).
Fig 8 Effects of elastic foundations on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under uniform external pressure (movable edges).
Fig 9 Effects of elastic foundations on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under axial compressive loads (movable edges).
Trang 10figures also show that the buckling and postbuckling load is very sensitive to the change of initial imperfection
5.5 Effects of stiffeners The influences of stiffeners as well as initial imperfection on the nonlinear postbuckling response of FGM cylindrical panels under uniform external pressure (movable edges), axial com-pressive loads (movable edges), uniform temperature rise (im-movable edges) and uniform external pressure (im(im-movable edges)
Fig 10 Effects of elastic foundations on the nonlinear response of the eccentrically
stiffened FGM cylindrical panels under uniform temperature rise (immovable
edges).
Fig 11 Effects of elastic foundations on the nonlinear response of the eccentrically
stiffened FGM cylindrical panels under uniform external pressure (immovable
edges).
Fig 12 Effect of volume fraction index Non the nonlinear response of the ec-centrically stiffened FGM cylindrical panels under uniform external pressure (movable edges).
Fig 13 Effect of volume fraction index Non the nonlinear response of the ec-centrically stiffened FGM cylindrical panels under axial compressive loads (mo-vable edges).