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Trang 1Nonlinear postbuckling of an eccentrically stiffened thin FGM plate
resting on elastic foundations in thermal environments
Vietnam National University, Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Viet Nam
a r t i c l e i n f o
Article history:
Received 29 May 2013
Received in revised form
7 September 2013
Accepted 17 October 2013
Available online 6 December 2013
Keywords:
Nonlinear postbuckling
Eccentrically stiffened P-FGM plates
Classical plate theory
Elastic foundation
Thermal environments
a b s t r a c t
This paper first time presents an analytical investigation on the nonlinear postbuckling of imperfect eccentrically stiffened thin FGM plates under temperature and resting on elastic foundation using a simple power-law distribution (P-FGM) Both of the FGM plate and stiffeners are deformed under thermal loads The formulations are based on the classical plate theory taking into account geometrical nonlinearity, initial geometrical imperfection, temperature-dependent properties and the Lekhnitsky smeared stiffeners technique with Pasternak type elastic foundation By applying Galerkin method and using stress function, effects of material and geometrical properties, temperature, elastic foundation and eccentrically stiffeners on the buckling and postbuckling loading capacity of the eccentrically stiffened FGM plate in thermal environments are analyzed and discussed Some results were compared with the one of the other authors
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1 Introduction
The FGM plates and shells, as other composite structures,
usually reinforced by stiffening member to provide the benefit of
added load-carrying static and dynamic capability with a relatively
small additional weight penalty Thus study on static and dynamic
problems of reinforced FGM plates and shells with geometrical
nonlinearity are of significant practical interest Up to date, the
investigation on static and dynamic of eccentrically stiffened FGM
structures has received comparatively little attention Recently,
Bich et al studied nonlinear postbuckling and dynamic of
eccen-trically stiffened functionally graded shallow shells[1,2], buckling
and postbuckling of an eccentrically stiffened functionally graded
cylindrical panels[3] Dung et al considered nonlinear stability of
eccentrically stiffened functionally graded imperfect plates resting
on elastic foundation [4] Duc investigated nonlinear dynamic
response of imperfect eccentrically stiffened doubly curved FGM
shallow shells on elastic foundations [5] Notice that in all the
publication mentioned above [1–5], the eccentrically stiffened
FGM plates and shells are considered without temperatures There
has been no publication on the FGM plates and shells reinforced
by eccentrically stiffeners in thermal environment The most
difficult part in this type of problem is to calculate the thermal
mechanism of FGM plates and shells as well as eccentrically
stiffeners under thermal loads
In this paper, our investigation is the first proposal for an imperfection eccentrically stiffened FGM plate in thermal environ-ments and resting on elastic foundation in which we studied the nonlinear postbuckling using a simple power-law distribution (P-FGM) We consider the eccentrically stiffened thin FGM plate
in thermal environments with temperature independent material property, i.e the Young's modulus E, thermal expansion coefficient
a, the mass densityρ, the thermal conduction K and even Poisson ratio v are independent to the temperature Those vary in the thickness direction z as well as temperature T in the two variables function of z and T The investigation under those assumptions for FGM plates is a very challenging work Moreover, the presence of the eccentrically stiffeners makes it more difficult to solve Here,
we have solved this problems taking into account all above assumptions
The formulations are based on the classical plate theory taking into account geometrical nonlinearity, initial geometrical imper-fection, temperature-dependent properties and the Lekhnitsky smeared stiffeners technique with Pasternak type elastic founda-tion Using Galerkin method and stress function, the effects of geometrical and material properties, temperature, elastic founda-tion and eccentrically stiffeners on the nonlinear response of the P-FGM plate in thermal environments are analyzed and discussed
2 Eccentrically stiffened FGM plate on elastic foundations Consider a ceramic–metal eccentrically stiffened FGM plate of length a, width b and thickness h resting on an elastic foundation
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Thin-Walled Structures
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n Corresponding author Tel.: þ84 4 37547978; fax: þ84 4 37547724.
E-mail address: ducnd@vnu.edu.vn (N Dinh Duc)
Trang 2A coordinate systemðx; y; zÞ is established in which ðx; yÞ plane on
the middle surface of the plate and z is thickness direction
ðh=2rzrh=2Þ as shown inFig 1
The volume fractions of constituents are assumed to vary
through the thickness according to the following power law
distribution (P-FGM):
VcðzÞ ¼ 2z2hþh
where N is volume fraction index (0rN o1) The material
properties of P-FGM shells have been assumed to be
temperature-dependent and graded in the thickness direction z:
½Eðz; TÞ; ρðz; TÞ; αðz; TÞ; Kðz; TÞ ¼ ½EmðTÞ; ρmðTÞ; αmðTÞ; KmðTÞ
þ½EcmðTÞ; ρcmðTÞ; αcmðTÞ; KcmðTÞ 2z2hþh
vðz; TÞ
¼ νmðTÞþνcmðTÞ 2z2hþh
N 1
ð2Þ where
EcmðTÞ ¼ EcðTÞEmðTÞ;
νcmðTÞ ¼ νcðTÞνmðTÞ;
ρcmðTÞ ¼ ρcðTÞρmðTÞ;
αcmðTÞ ¼ αcðTÞαmðTÞ;
KcmðTÞ ¼ KcðTÞKmðTÞ; N Z0; N1Z0 ð3Þ
It is evident from Eqs.(2)and(3)that the upper surface of the
plateðz ¼ h=2Þ is ceramic -rich, while the lower surface ðz ¼ h=2Þ
is metal-rich A material property Pr, such as the elastic modulus E,
Poisson ratio ν, the mass density ρ, the thermal expansion
coefficient α and coefficient of thermal conduction K can be
expressed as a nonlinear function of temperature[6–8]:
Pr¼ P0ðP 1T 1þ1þP1T1þP2T2þP3T3Þ ð4Þ
in which T¼ T0þΔTðzÞ and T0¼ 300 K (room temperature); P0,
P1, P1, P2and P3are coefficients characterizing of the constituent
materials In short, we will use T-D (temperature dependent) for
the cases in which the material properties depend on temperature
Otherwise, we use T-ID for temperature independent cases The
material properties for the later one have been determined by(4)
at room temperature, i.e T0¼ 300 K
The load–displacement relationship of the elastic foundation is
assumed as following[9–12]:
where∇2¼ ∂2=∂x2þ∂2=∂y2, w is the deflection of the FGM plate, k1
and k2are Winkler foundation stiffness and shear layer stiffness of
Pasternak foundation, respectively
3 Eccentrically stiffened thin FGM plate under temperatures
In the present study, the classical plate theory is used to obtain the equilibrium and compatibility equations as well as expressions
of buckling loads and postbuckling equilibrium paths of FGM plates The strains across the plate thickness at a distance z from the mid-plane are[13,14]
εx
εy
γxy
0 B
1 C
A ¼
ε0 x
ε0
γ0 xy
0 B
@
1 C Aþz
kx
ky
2kxy
0 B
1
whereε0
xandε0are the normal strains,γ0
xyis the shear strain at the middle surface of the plate, and kx; ky; kxyare the curvatures
In the framework of the classical plate theory, the strains at the middle surface and the curvatures are related to the displacement components u; v; w in the coordinates as[13,14]
ε0 x
ε0
γ0 xy
0 B
@
1 C
A ¼
uxþw2
x=2
vyþw2=2
uyþvxþwxwy
0 B
1 C A;
kx
ky
kxy
0 B
1 C
A ¼
wxx
wyy
wxy
0 B
1
In which u; v are the displacement components along the x; y directions, respectively
Interestingly, comparing to the other[1–5], we have assumed that the eccentrically outside stiffeners also depend on tempera-ture Hooke law for an FGM plate with temperature-dependent properties is defined as[13,14]
ðssh
x; ssh
yÞ ¼1νEðz; TÞ2ðz; TÞ½ðεx; εyÞþνðεy; εxÞð1þνÞαΔTðzÞð1; 1Þ ð8Þ
ssh
xy¼2ð1þνðz; TÞÞEðz; TÞ γxy
whereΔT is temperature rise from stress free initial state, and more generally, ΔT ¼ ΔTðzÞ; Eðz; TÞ; vðz; TÞ are the FGM plate's elastic moduli which is determined by(2)
For stiffeners in thermal environments with temperature-dependent properties, we have proposed its form adapted from
[2]as the follows:
ðsst
x; sst
yÞ ¼ E0ðεx; εyÞ E0
12ν0ðTÞα0ðTÞΔðTÞð1; 1Þ ð9Þ Here, E0¼ E0ðTÞ; ν0¼ ν0ðTÞ; α0¼ α0ðTÞ are the Young's modulus, Poisson ratio and thermal expansion coefficient of the stiffeners, respectively The FGM plate reinforced by eccentrically longitudi-nal and transversal stiffeners is shown inFig 1 E0 is elasticity modulus in the axial direction of the corresponding stiffener which is assumed identical for both types of longitudinal and transversal stiffeners In order to provide continuity between the plate and stiffeners, suppose that stiffeners are made of full metal
ðE0¼ EmÞ if putting them at the metal-rich side of the plate, and conversely full ceramic stiffenersðE0¼ EcÞ at the ceramic-rich side
of the plate (this assumptionfirst time was proposed by Bich in[1]
and has been used in[1–5]) The contribution of stiffeners can be accounted for using the Lekhnitsky smeared stiffeners technique
[1–5] In order to investigate the FGM plates with stiffeners in the thermal environment, we have not only taken into account the materials moduli with temperature-dependent properties but also
we have assumed that all elastic moduli of FGM plates and stiffener are temperature dependence and they are deformed in the presence of temperature Hence, the geometric parameters, the plate's shape and stiffeners are varied through the deforming process due to the temperature change We have assumed that the thermal stress of stiffeners is subtle which distributes uniformly through the whole plate structure, therefore, we can ignore it Lekhnitsky smeared stiffeners technique can be adapted from
Fig 1 Geometry and coordinate system of an eccentrically stiffened FGM plate on
elastic foundation.
Trang 3[1–5]for eccentrically stiffened FGM plate under temperatures as
the follows:
Nx¼ I10þE
T
0AT1
sT
1
!
ε0
xþI20ε0þðI11þCT
1ÞkxþI21kyþΦ1
Ny¼ I20ε0
xþ I10þE
T
0AT 2
sT 2
!
ε0
þI21kxþðI11þCT
2ÞkyþΦ1
Nxy¼ I30γ0
xyþ2I31kxy
Mx¼ ðI11þCT
1Þε0
xþI21ε0þ I12þE
T
0IT1
sT 1
!
kxþI22kyþΦ2
My¼ I21ε0
xþðI11þCT
2Þε0
þI22kxþ I12þE
T
0IT 2
sT 2
!
kyþΦ2
Mxy¼ I31γ0
The relation(10)is our most importantfinding, where Iijði ¼ 1; 2; 3;
j¼ 0; 1; 2Þ:
I1j¼Z h=2
h=2
EðzÞ
1νðzÞ2zjdz
I2j¼Z h=2
h=2
EðzÞνðzÞ
1νðzÞ2zjdz
I3j¼
Z h =2
h=2
EðzÞ
2½1þνðzÞz
jdz¼12ðI1jI2jÞ
ðΦ1; Φ2Þ ¼
Z h =2
h=2
EðzÞαðzÞ
1νðzÞΔTðzÞ 1; zð Þdz
IT1¼d
T
1ðhT
1Þ3
12 þAT
1ðzT
1Þ2
; IT
2¼d
T
2ðhT
2Þ3
12 þAT
2ðzT
2Þ2
CT1¼E0AT
1zT
1
sT
1
; CT
2¼E0AT
2zT 2
sT 2
zT
1¼h
T
1þhT
2 ; zT
2¼h
T
2þhT
After the thermal deformation process, the geometric shapes of
stiffeners which can be determined as follows:
dT1¼ d1ð1þαmTðzÞÞ; dT
2¼ d2ð1þαmTðzÞÞ;
hT1¼ h1ð1þαmTðzÞÞ; hT
2¼ h2ð1þαmTðzÞÞ;
zT
1¼ z1ð1þαmTðzÞÞ; zT
2¼ z2ð1þαmTðzÞÞ;
sT
1¼ s1ð1þαmTðzÞÞ; sT
2¼ s2ð1þαmTðzÞÞ ð12Þ where the coupling parameters C1; C2 are negative for outside
stiffeners and positive for inside one; s1; s2are the spacing of the
longitudinal and transversal stiffeners; I1; I2 are the second
moments of cross-section areas; z1; z2 are the eccentricities of
stiffeners with respect to the middle surface of plate; and the
width and thickness of longitudinal and transversal stiffeners are
denoted by d1; h1 and d2; h2 respectively A1; A2 are the
cross-section areas of stiffeners Although the stiffeners are deformed by
temperature, we, however, have assumed that the stiffeners keep
its rectangular shape of the cross section Therefore, it is
straight-forward to calculate AT1; AT
2 The nonlinear equilibrium equations of a perfect plate based on
the classical plate theory are given by[13,14]
Mx;xxþ2Mxy;xyþMy;yyþNxwxxþ2NxywxyþNywyyk1wþk2∇2w¼ 0
ð13cÞ calculated from Eq.(10)
ε0
x¼ A22NxA12NyþB11wxxþB12wyyðA22A12ÞΦ1
ε0¼ A11NyA12NxþB21wxxþB22wyyðA11A12ÞΦ1
γ0
xy¼ A66Nxyþ2B66wxy ð14Þ where
A11¼Δ1ðI10þE
T
0AT1
sT 1
Þ; A22¼Δ1 I10þE
T
0AT2
sT 2
!
; A12¼I20
Δ; A66¼I1
30
Δ ¼ I10þE
T
0AT1
sT 1
!
I10þE
T
0AT2
sT 2
!
I2 20
B11¼ A22ðI11þCT
1ÞA12I21; B12¼ A22I21A12ðI11þCT
2Þ
B21¼ A11I21A12ðI11þCT
1Þ; B22¼ A11ðI11þCT
2ÞA12I21
B66¼I31
Substituting once again Eq.(14)into the expression of Mij in
(10), then Mijinto the Eq.(13c)leads to
Nx ;xþNxy ;y¼ 0
B21fxxxxþB12fyyyyþðB11þB222B66Þfxxyy
D11wxxxxD22wyyyyðD12þD21þ4D66Þwxxyyþ
þNxwxxþ2NxywxyþNyw;yyk1wþk2∇2w¼ 0 where
D11¼ I12þE
T
0IT 1
sT 1
B11ðI11þCT
1ÞI21B21
D22¼ I12þE
T
0IT2
sT 2
B22ðI11þCT
2ÞI21B12
D12¼ I22B12ðI11þCT
1ÞI21B22
D21¼ I22B21ðI11þCT
2ÞI21B11
fðx; yÞ is stress function defined by
Nx¼ fyy; Ny¼ fxx; Nxy¼ fxy ð18Þ For an imperfect FGM plate, Eq.(16)are modified into form as
[12,15–17]
B21fxxxxþB12fyyyyþðB11þB222B66Þfxxyy
D11wxxxxD22wyyyyðD12þD21þ4D66Þwxxyy
þfyyðwxxþwn
xxÞ2fxyðwxyþwn
xyÞ
þfxxðwyyþwn
yyÞk1wþk2∇2
in which wnðx; yÞ is a known function representing initial small imperfection of the plate The geometrical compatibility equation for an imperfect plate is written as[12,15–17]:
ε0
x ;yyþε0
;xxγ0
xy ;xy¼ w2
xywxxwyyþ2wxywnxywxxwnyywyywnxx ð20Þ From the constitutive relation(15)in conjunction with Eq.(19)one can write
ε0
x¼ A22fyyA12fxxþB11wxxþB12wyyðA22A12ÞΦ1
ε0¼ A11fxxA12fyyþB21wxxþB22wyyðA11A12ÞΦ1
γ0
xy¼ A66fxyþ2B66wxy ð21Þ Setting Eq.(21)into Eq.(20)gives the compatibility equation of
an imperfect FGM plate as
A11fxxxxþA22fyyyyþðA662A12ÞfxxyyþB21wxxxx
þB12wyyyyþðB11þB222B66Þwxxyy
ðw2
xywxxwyyþ2wxywn
xywxxwn
yywyywn
xxÞ ¼ 0 ð22Þ Eqs.(19) and (22)are nonlinear equations in terms of variables
w and f and used to investigate the stability of FGM plate on elastic foundations subjected to mechanical, thermal and thermo-mechanical loads
Trang 4In the present study, the edges of FGM plates are assumed to be
simply supported Depending on the in-plane restraint at the
edges, three cases of boundary conditions[15–17], labeled as Case
1, 2and3, may be considered
Case 1 Four edges are simply supported and freely movable (FM)
The associated boundary conditions are
w¼ Nxy¼ Mx¼ 0; Nx¼ Nx0at x¼ 0; a
w¼ Nxy¼ My¼ 0; Ny¼ Ny0at y¼ 0; b ð23Þ
Case 2 Four edges are simply supported and immovable (IM) The
associated boundary conditions are
w¼ u ¼ Mx¼ 0; Nx¼ Nx0at x¼ 0; a
w¼ v ¼ My¼ 0; Ny¼ Ny0at y¼ 0; b ð24Þ
Case 3 The edges are simply supported Uniaxial edge loads are
applied in the direction of the x-coordinate The edges x¼ 0; a are
considered freely movable, the remaining two edges being
unloaded and immovable For this case, the boundary conditions
are defined as
w¼ Nxy¼ Mx¼ 0; Nx¼ Nx0at x¼ 0; a
w¼ v ¼ My¼ 0; Ny¼ Ny0at y¼ 0; b ð25Þ
where Nx0; Ny0 are pre-buckling force resultants in directions x
and y, respectively
To solve two Eqs.(19) and (22) for unknowns w and f, and with
the consideration of the boundary conditions (23)–(25), we
assume the following approximate solutions[15–17]
ðw; wnÞ ¼ ðW; μhÞ sin λmx sinδny ð26aÞ
f¼ A1 cos 2λmxþA2 cos 2δnyþA3sinλmx sinδnyþ12Nx0y2þ12Ny0x2
ð26bÞ whereλm¼ mπ=a, δn¼ nπ=b, W is amplitude of the deflection and
μ is parameter of imperfection; m; n are odd natural numbers The
coefficients Aiði ¼ 1=3Þ are determined by substitution of Eqs.(26a
and26b) into Eq.(22)as
A1¼ δ2
32A11λ2
m
ðW þ2μhÞW; A2¼ λ2m
32A22δ2ðW þ2μhÞW
A3¼ B21λ4
mþB12δ4þðB11þB222B66Þλ2
mδ2
A11λ4
mþA22δ4þðA662A12Þλ2
mδ2 W ð27Þ Subsequently, substitution of Eqs.(26a), (26b) into Eq (19)and
applying the Galerkin procedure for the resulting equation yield
ab
4
B21 λ 4
m þ B 12 δ 4 þ ðB 11 þ B 22 2B 66 Þλ 2
m δ 2
A 11 λ 4
m þ A 22 δ 4 þ ðA 66 2A 12 Þλ 2
m δ 2
D11λ4
mD22δ4ðD12þD21þ4D66Þλ2
mδ2k1ðλ2
mþδ2Þk2
8
>
>
9
>
>W
þ8λmδn
3 B21λ4
mþB12δ4þðB11þB222B66Þλ2
mδ2
A11λ4
mþA22δ4þðA662A12Þλ2
mδ2
WðW þμhÞ
2λmδn
3
B21
A11þB12
A22
WðW þ2μhÞab4ðNx0λ2
mþNy0δ2ÞðW þμhÞ
ab64 λ4m
A22þAδ4
11
!
Eq.(28)is used to determine nonlinear buckling and postbuckling
response of rectangular eccentrically stiffened FGM plates in
thermal environments It is not so difficult to realize that Eq
(28)is more complicated than the equation written in[4]without
the temperature andν ¼ const An interested characteristics of Eq
(28) is temperature dependent, which are displayed in the
B21; B12 ;; B11; B22; D11; D22; A11; A22; ::: coefficients as shown in Eqs
(15) and (17)
4 Nonlinear buckling and postbuckling analysis 4.1 Mechanical buckling and postbuckling analysis Consider a simply supported FGM plate with all movable edges which is rested on elastic foundations and subjected to in-plane edge compressive loads Fx; Fy uniformly distributed on edges
x¼ 0; a and y ¼ 0; b respectively In this case, buckling force resultants are given
and Eq.(28)leads to
Fx¼ b1
Wb2 W
Wþμþb
3WðW þ2μÞ
4
where the coefficients b1
1; b2
1; b3
1; b4
1 are described in detail in
Appendix B For a perfect FGM plate, Eq.(30)reduces to an equation from which buckling compressive load may be obtained as Fxb¼ b2
1 4.2 Thermal buckling and postbuckling analysis
A simply supported FGM plate with all immovable edges is considered The in-plane condition on immovability at all edges, i
e u¼ 0 at x ¼ 0; a and v ¼ 0 at y ¼ 0; b, is fulfilled in an average sense as[6–8,15–17]:
Zb 0
Z a 0
∂u
∂xdx dy¼ 0;
Z a 0
Z b 0
∂v
From Eqs.(6) and (14) one can obtain the following expressions
in which Eq.(18)and imperfection have been included
∂u
∂x¼ A22fyyA12fxxþB11wxxþB12wyyðA22A12ÞΦ112w2
xwxwnx
∂v
∂y¼ A11f;xxA12fyyþB22wyyþB21wxxðA11A12ÞΦ11
2w
2wywn
y
ð32Þ Substitution of Eqs (26a), (26b) into Eq (32) and then the result into Eq.(31)givefictitious edge compressive loads as
Nx0¼ Φ1þ 1
8ðA11A22A2
12ÞðA11λ
2
mþA12δ2ÞWðW þ2μhÞ
þ4n
mb2
1
A11A22A2
12
B12A11þB22A12
ðA11A22A2
12ÞB21λ 4
m þ B 12 δ 4 þ ðB 11 þ B 22 2B 66 Þλ 2
m δ 2
A 11 λ 4
m þ A 22 δ 4 þ ðA 66 2A 12 Þλ 2
m δ 2
2 4
3 5
Wþna4m2B11A11þB21A12
A11A22A2
12
W
Ny0¼ Φ1þ 1
8ðA11A22A2
12ÞðA12λ
2
mþA22δ2ÞWðW þ2μhÞ
þna4m2
1
A11A22A2
12
B21A22þB11A12
ðA11A22A2
12ÞB21λ 4
m þ B 12 δ 4 þ ðB 11 þ B 22 2B 66 Þλ 2
m δ 2
A 11 λ 4
m þ A 22 δ 4 þ ðA 66 2A 12 Þλ 2
m δ 2
2 4
3 5
Wþ 4n
mb2
B12A12þB22A22
A11A22A2
12
Inserting Eq.(33)into Eq.(28)gives the following expression for the thermal parameter:
Φ1¼ h b1
Wþb2 W
Wþμb
3WðW þ2μÞ
4
WðW þ2μÞ
!
ð34Þ where the coefficients b1
2; b2
2; b3
2; b4
2 are described in detail in
Appendix B
Trang 5By using Eq.(11), the thermal parameterΦ1can be expressed in
terms ofΔT:
where the coefficient L are described in detail inAppendix A
Although ΔTis included in the expression for L due to the
temperature dependence of material propertiesðT ¼ T0þΔTÞ, one
may formally expressΔTfrom Eqs.(34) and (35) as follows
ΔT ¼1L b1Wþb2 W
Wþμb
3WðW þ2μÞ
4
WðW þ2μÞ
!
ð36Þ
Eq.(36)is the analytical form to determine the non-linear relation
between the bending deflection and temperature for both of the
perfect and imperfect plates under the thermal loads (for perfect
plate, μ ¼ 0) Using Eq (36), we have derived the temperature
change, ΔTb¼ ðb2
2=LÞ, which sets them into the buckling state
under the condition W¼ 0
In case of T-D, the two hand sides of Eq.(36)are temperature
dependence which makes it very difficult to solve Fortunately, we
have applied a numerical technique using the iterative algorithm
to determine the buckling loads as well as to determine the
deflection–load relations in the postbuckling period of the FGM
plate More details, given the material parameter N, the
geome-trical parameterðb=a; b=hÞ and the value of W=h, we can use these
to determineΔT in(36)as the follows: we choose an initial step
for ΔT1 on the right hand side in Eq (36) with ΔT ¼ 0 (since
T¼ T0¼ 300 K, the initial room temperature) In the next iterative
step, we replace the known value ofΔT1 found in the previous
step to determine the right hand side of Eq.(36)ΔT2 This iterative
procedure will stop at the kth step ifΔTk satisfies the condition
ΔT ΔTkjrε Here, ΔT is a desired solution for the temperature
andε is a tolerance used in the iterative steps
4.3 Thermo-mechanical postbuckling analysis
Let us consider a simply supported eccentrically stiffened FGM
plate, with movable edges x¼ 0; a and immovable ones y ¼ 0; b
(Case3), subjected to the simultaneous action of a thermalfield
and an in-plane compressive load Fxdistributed uniformly along
the edges x¼ 0; a
From thefirst of Eq.(29)and the second of Eqs.(31) and (32),
we have
Nx0¼ FxhNy0
¼ab4
λ m
δ n
B 21 λ 4
m þ B 12 δ 4 þ ðB 11 þ B 22 2B 66 Þλ 2
m δ 2
A11λ 4
m þ A 22 δ 4 þ ðA 66 2A 12 Þλ 2
m δ 2
þB 21 λ 4
m þ B 12 δ 4 þ ðB 11 þ B 22 2B 66 Þλ 2
m δ 2
A11λ 4
m þ A 22 δ 4 þ ðA 66 2A 12 Þλ 2
m δ 2
A 12 δ n
A11λ mþB22 δ n
A11λ mþB21 λ m
A11δ n
2
6
3 7
WþA12
A11
Nx0þðA11A12Þ
A11 Φ1þ8Aδ2
11
Employing these relations in Eq.(28)yields
Fx¼ b1
Wþb2 W
ðW þμÞþb
3WðW þ2μÞ
ðW þμÞ þb
4
WðW þ2μÞ
þ ðA11A12ÞLn2ΔT
A11ðm2B2þðA12n2=A11ÞÞ ð38Þ where the coefficients b1
; b2
; b3
; b4
are described in detail in
Appendix B
Eq (38) is a crucial equation to investigate the nonlinear response of eccentrically stiffened FGM plate under both of thermal and mechanical loads
5 Numerical results and discussion Here, several numerical examples will be presented for perfect and imperfect simply supported midplane-symmetric FGM plates The typical values of the coefficients of the materials mentioned in
(4)are listed inTable 1 Unlike the other works, we here assume that all coefficients depend on both of the thickness z and temperature T Technically,
it is much more difficult to capture and solve the fundamental set
of equations In this paper, we have contribute significantly to this transformation process
The parameters for the stiffeners are
z1¼ 0:0225ðmÞ; z2¼ 0:0225ðmÞ; s1¼ 0:2ðmÞ; s2¼ 0:2ðmÞ
h1¼ 0:03ðmÞ; h2¼ 0:03ðmÞ; d1¼ 0:004ðmÞ; d2¼ 0:004ðmÞ
Table 1
Material properties of the constituent materials of the considered FGM plates [6 , 8 , 16]
ρ (kg/m 3
Fig 2 Postbuckling of FGM plates under mechanical loads (1,2: stiffened FGM plate; 3, 4: un-stiffened FGM plate).
Trang 6Fig 2compares nonlinear postbuckling response of eccentrically
stiffened FGM plate and un-stiffened FGM plate under mechanical
loads It is clear that the stiffeners can enhance the thermal
loading capacity for the imperfect and perfect FGM plates The
similar conclusion has been reported for nonlinear static and
dynamic analysis of eccentrically stiffened FGM plates and shells
[1–5]
Fig 3presents postbuckling of the eccentrically stiffened FGM
plate under thermal loads (T-D) with Poisson ratioν ¼ const and
ν ¼ ν zð Þ It is clear that there is no significant difference in case of
ν ¼ const and ν ¼ ν zð Þ In our calculation, the equations under the
conditionν ¼ ν zð Þ are much more complicated Hence, for a sake of
simplicity, the others often choose the condition ofν ¼ const
In particular case of a FGM plate without stiffeners with the conditions: A1¼ A2¼ 0and I1¼ I2¼ 0, we have compared the numerical results of un-stiffened FGM plate with Dung's results
[18](without temperatures, ν ¼ ν zð Þ) and with Duc's results[15]
(ν ¼ const,N1¼ 0) under only mechanical loads Those have been presented in Figs 4 and 5 which show the good agreement between our findings for postbuckling of FGM plates and the others
Figs 6and7present the positive influence of elastic founda-tions on the postbuckling of eccentrically stiffened FGM plates under uniaxial compressive load (all FM edges) and uniform temperature (all IM edges) The effect of Pasternak foundation K2
on the critical compressive loads and the thermal resistance of
Fig 3 Effect of Poisson's ratio on postbuckling of FGM plate.
Fig 4 Comparing the force–bending curve of FGM plate without stiffeners with
Dung [18] (ν ¼ νðzÞ).
Fig 5 Comparing the force–bending curve of FGM plate without stiffeners with Tung [15] (ν ¼ const).
Fig 6 Effects of elastic foundation on the postbuckling of eccentrically stiffened FGM plate under compression (all FM edges).
Trang 7FGM is larger than the Winkler foundation K1 This conclusion has
been also reported in others publications[9–12,17]
Figs 8and 9 present effects of thermo-mechanical loads on
nonlinear response of FGM plates in thermal environment.Fig 8
investigated effects of temperatures on postbuckling of
eccentri-cally stiffened FGM plate under compressive loads with all IM
edges and T-D properties These show us that the decrease in
temperature reduces the mechanical loading ability of the perfect
plate as well as the imperfect plate.Fig 9also studies the effects of
the compressive loads on postbuckling of eccentrically stiffened
FGM plate in thermal environment with all FM edges and T-D
properties Obviously, both of the presence of the compressive and
thermal loads reduce significantly the loading capacity of the FGM
plates
Figs 9and10show effects of boundary conditions (two edges
FM x¼ 0; a and two edges IM y ¼ 0; b) on the postbuckling of
eccentrically stiffened FGM plates under compressive loads The
curves in case of FM have been plotted using Eq (30) with the loading ratio β ¼ 0 Whereas, the curves in case of IM have been plotted using Eq.(38) with ΔT ¼ 0 This has illustrated that the boundary conditions have a significant effects on the buckling and postbuckling of FGM plates Although the perfect FGM plate have only been buckling in case of the large loads, the loading ability of the imperfect FGM plates in the postbuckling period as well as under the boundary condition of the edge y¼ 0; b is much better than those of perfect one
Fig 11presents effects of volume fraction index on the post-buckling of eccentrically stiffened FGM plates under thermal loads (all IM edges) These postbuckling curves show the loading ability
of FGM get worse with the increase of N; N1
Figs 12 and13 show effects of imperfection on postbuckling response of eccentrically stiffened FGM plate under mechanical and thermal loads In postbuckling period, those suggest us that
Fig 7 Effects of elastic foundation on the postbuckling of eccentrically stiffened
FGM plate under thermal loads (T-D; all IM edges).
Fig 8 Effects of temperatures on the postbuckling of eccentrically stiffened FGM
plate under compression (all IM edges).
Fig 9 Effects of compressive loads on the postbuckling of eccentrically stiffened FGM plate in thermal environment (all FM edges).
Fig 10 Effects of boundary conditions (FM and IM) on the postbuckling of eccentrically stiffened FGM plates under compressive loads.
Trang 8the imperfect properties have affected actively on the loading
ability in the limit of large enough W=h In other words, the
loading ability increases with μ This has been reported in the
other papers[4,5;15–17]
To validate the accuracy of the proposed approach, the
obtained numerical results for un-stiffened FGM plate and
ν ¼ const under uniaxial compression are compared with those
in [12] Computations have been carried out for the following
material and the geometrical parameters of FGM plate: Em¼
70 GPa; Ec¼ 380 GPa; ν ¼ 0:3; and a=b ¼ 1; b=h ¼ 40; N ¼ 1; m ¼
n¼ 1; β ¼ 0 It is seen that these results (inFigs 14and15) are in
good agreement to those one of Duc[12]
We have also compared ourfindings with Dung[4]for stiffened
FGM plate under uniaxial compression Computations have been
carried out with the following material and the geometrical parameters of the FGM plate: a¼ b ¼ 1:50 m; h ¼ 0:008 m and stiffeners with z1¼ z2¼ 0:019ðmÞ; s1¼ 0:15ðmÞ; s2¼ 0:15ðmÞ,h1¼
0:03ðmÞ; h2¼ 0:03ðmÞ; d1¼ 0:003ðmÞ; d2¼ 0:003ðmÞ
It is seen that obtained results inFig 16are in good agreement
to those one of Dung[4]
6 Conclusion This paper first time presents an analytical investigation on the nonlinear postbuckling for imperfect eccentrically stiffened
Fig 11 Effects of volume fraction index on the postbuckling of eccentrically
stiffened FGM plates under thermal loads (all IM edges).
Fig 12 Effect of imperfection on postbuckling of eccentrically stiffened FGM plate
under compressive edge loads (all FM edges).
Fig 13 Effect of imperfection on postbuckling of eccentrically stiffened FGM plate under thermal loads (all IM edges).
Fig 14 Comparison of postbuckling curves for un-stiffened FGM plates under uniaxial compression with the results of Duc [12]
Trang 9thin FGM plates using a simple power-law distribution (P-FGM)
under temperatures Both of FGM plate and stiffeners are
deformed under thermal loads The formulations are based on
the classical plate theory taking into account geometrical
non-linearity, initial geometrical imperfection, temperature-dependent
properties and the Lekhnitsky smeared stiffeners technique with
Pasternak type elastic foundation Using Galerkin method and
stress function, effects of material and geometrical properties,
elastic foundation and eccentrically outside stiffeners on the
buckling and postbuckling loading capacity of the imperfect
eccentrically stiffened P-FGM plate in thermal environments are
analyzed and discussed Some results were compared with the
ones of the other authors
Acknowledgment This work was supported by the National Foundation for Science and Technology Development of Viet Nam– NAFOSTED under Grant number 107.02-2013.06
Appendix A
L¼ Z 1
0
Emαm
1vmvcmuN 1duþZ 1
0
αmEcmþαcmEm
1vmvcmuN 1 du
"
þZ 1
0
αcmEcmu2N
1vmvcmuN 1du
#
Appendix B
b1¼32mn3 B
2
B2 B
4m4B21þn4B12þ B 11þB222B66
B2m2n2
B4m4A11þn4A22þ A662A12
B2m2n2
2 4
3 5 1
m2B2þn2β
b2¼
B4m4B21 þ n 4 B12þ B 11 þ B 22 2B 66
B 2 m 2 n 2
B 4 m 4 A11þ A 22 n 4 þ A 66 2A 12
B 2 m 2 n 2
D11B4
am4D22n4 D12þD21þ4D66
B2
am2n2
K 1 B 4 D 11
π 4 D 11 K 2 B 2
π 2 B2am2þn2
2 6 6 6
3 7 7
2
B2 B2
am2þβn2
b3¼8mnB
2
3B2
B21
A11
þB12
A22
! 1
m2B2þβn2
b4¼ π2 16B2
m4B4
A22
þn4
A11
! 1
m2B2þβn2
b2¼
B4a m 4 B 21 þ n 4 B 12 þ B 11 þ B 22 2B 66
B 2
a m 2 n 2
π 4
A11B 4
a m 4 þ A 22 n 4 þ A 66 2A 12
B 2
a m 2 n 2
D11B4m4π4D22n4π4 D 12þD21þ4D66
B2m2n2π4
B2
K1D11B2
K2π2D11B2m2þn2
2 6 6 6
3 7 7 7
B2m2þn2
B2π2
b1¼ B2
B2m2þn2
2 6
4 32mnB 2 a
3B 4 h
B 4
a m 4 B21þ n 4 B12þ B 11 þ B 22 2B 66
m 2 n 2 B 2 a
A 11 B 4
a m 4 þ A 22 n 4 þ A 66 2A 12
m 2 n 2 B 2 a
4nmB
2
B4
B12A11þ A 12 B22
A22A11 A 2 12
B 4
a m 4 B21þ n 4 B12þ B 11 þ B 22 2B 66
m 2 n 2 B 2 a
A11B 4
a m 4 þ A 22 n 4 þ A 66 2A 12
m 2 n 2 B 2 a
0 B B
@
1 C C A
B11A11þA12B21
A22A11A2
12
4
nB4 4mnB
2
B4
A22B21þ A 12 B11
A 22 A 11 A 2 12
B 4
a m 4 B 21 þ n 4 B 12 þ B 11 þ B 22 2B 66
m 2 n 2 B 2 a
A11B 4
a m 4 þ A 22 n 4 þ A 66 2A 12
m 2 n 2 B 2 a
2 6 6 4
3 7 7 5
Fig 15 Comparison of postbuckling curves for un-stiffened FGM plates under
uniform temperature rise with the results of Duc [12]
Fig 16 Comparison of present postbuckling curves for stiffened FGM plates under
uniaxial compression with the results of Dung [4]
Trang 104 A12B12þA22B22
A22A11A2
12
mB4
3 7
b3¼ 8mnB
2
3 B 2m2þn2
B2
B21
A11
þB12
A22
!
b4¼ π2
B2m2þn2
1
16B 2
h
B 4
a m 4
A22 þn 4
A11
þ m 2 B 2
a
8B 2
h A 22 A 11 A 2
12
m2B2A11þn2A12
8B 2
h A 22 A 11 A 2
12
A 12m2B2þA22n2
2
6
6
3 7 7
b1¼ 32mnB 2
a
3B 2
h
1
m 2 B 2
a þ A12 n2
A11
B 21 m 4 B 4
a þ B 12 n 4 þ ðB 11 þ B 22 2B 66 Þm 2 n 2 B 2
a
A11m 4 B 4
a þ A 22 n 4 þ ðA 66 2A 12 Þm 2 n 2 B 2
a
þ4n2Ba
B2
1
m2B2þA 12 n 2
A11
A 12 n
A11B a mB a m
n
21 m 4 B 4
a þ B 12 n 4 þ ðB 11 þ B 22 2B 66 Þm 2 n 2 B 2
a
A11m 4 B 4
a þ A 22 n 4 þ ðA 66 2A 12 Þm 2 n 2 B 2
a
þ B 22 n
A11BamþB a B 21 m
A11n
2
6
3 7
b23¼ π2
B2 m2B2þA 12 n 2
A11
B21 m 4 B 4
a þ B 12 n 4 þ ðB 11 þ B 22 2B 66 Þm 2 n 2 B 2
a
2
A11m 4 B 4
a þ A 22 n 4 þ ðA 66 2A 12 Þm 2 n 2 B 2
a
D11m4B4D22n4ðD12þD21þ4D66Þm2n2B2
K1D11B 4
a
π 4 m 2B2þn2K
2 D11B 2 a
π 2
8
>
>
>
>
9
>
>
>
>
b3¼8mnB
2
3B2
1
m2B2þA12n 2
A 11
A11
þB12
A22
!
16 m2B2þn 2 A12
A 11
B2
m4B4
A22
þn4
A11
!
þ n4π2 8A11B2
1
m2B2þn 2 A12
A 11
A11¼ hA11; A12¼ hA12; A22¼ hA22; A66¼ hA66
B11¼B11
h; B12¼B12
h ; B21¼B21
h; B22¼B22
h; B66¼B66
h
D11¼D11
h3; D22¼D22
h3; D12¼D12
h3; D21¼D21
h3; D66¼D66
h3
K1¼k1a4
D11; K2¼k2a2
D11; W ¼W
h; Ba¼b
a; Bh¼b
h; β ¼Fy
Fx
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