Nonlinear buckling of imperfect eccentrically stiffenedwith temperature-dependent properties in thermal environments Vietnam National University, Ha Noi, 144 Xuan Thuy, Cau Giay, Ha Noi,
Trang 1Nonlinear buckling of imperfect eccentrically stiffened
with temperature-dependent properties in thermal environments
Vietnam National University, Ha Noi, 144 Xuan Thuy, Cau Giay, Ha Noi, Viet Nam
a r t i c l e i n f o
Article history:
Received 11 November 2013
Received in revised form
10 January 2014
Accepted 20 January 2014
Available online 6 February 2014
Keywords:
Nonlinear buckling
S-FGM with metal–ceramic–metal layers
Eccentrically stiffened cylindrical shells
Imperfection
Elastic foundation
Thermal environment
a b s t r a c t
In this paper, an analytical approach is presented to investigate the nonlinear static buckling for imperfect eccentrically stiffened functionally graded thin circular cylindrical shells with temperature-dependent properties surrounded on elastic foundation in thermal environment Both shells and stiffeners are deformed simultaneously due to temperature Material properties are graded in the thickness direction according to a Sigmoid power law distribution in terms of the volume fractions of constituents (S-FGM) with metal–ceramic–metal layers The Lekhnitsky smeared stiffeners technique with Pasternak type elastic foundation, stress function and the Bubnov–Galerkin method are applied Numerical results are given for evaluating effects of temperature, material and geometrical properties, elastic foundations and eccentrically outside stiffeners on the buckling and post-buckling of the S-FGM shells The obtained results are validated by comparing with those in the literature
& 2014 Elsevier Ltd All rights reserved
1 Introduction
The material has variable mechanical property with
interna-tional name Funcinterna-tionally Graded Material and often abbreviated
FGM was developed and named by a group of material scientists at
Sendai Institute of Japan in 1984[1,2] This material is a type of new
generation composite, intelligent composite, appears as a result of
actual demands for a material that can overcome the disadvantages
of traditional metals and laminated normal composites This
func-tionally graded material is formed from two component materials of
ceramic and metal in which the volume ratio of each composition
varies smoothly and continuously from this side to the other side
according to the structure wall thickness in order to be suitable for
the characteristic strength of the component materials
The cylindrical shell is a structure that is used popularly in the
industry, national defense and in the modern engineering
indus-tries Since FGM was researched and developed, the shell
calcula-tions need to be expanded and go into more details However, due
to the non-slope of circular cylindrical shells and complexity in
calculation, the nonlinear stability researches of them are still very
limited in comparison with the structures of plate or other kinds of
shells A few case studies on the stability of FGM cylindrical shells are introduced below: Lanhe et al.[3]have used the uncoupled
equations system to study the problem of the linear stability perfect FGM cylindrical shells under thermal loads Li and Lin[5]
studied buckling and postbuckling of anisotropic laminated cylind-rical shell subjected to external pressure loads Huang and Han[6]
discussed nonlinear postbuckling and buckling behaviors of FGM cylindrical shells subjected to combined axial and radial pressure
In this analysis, the nonlinear strain–displacement relations of large deformation and the Ritz energy method were used Iqbal
et al.[7]studied free vibration of thin FGM cylindrical shells by using wave propagation approach based on the classical shell theory Li and Batra[8]investigated buckling of axially compressed thin cylindrical shell with FGM middle layer Najafizadeh et al.[9]
used analytical approach and displacement functions to investi-gate buckling behavior of functionally graded stiffened cylindrical shells reinforced by rings and stringer subjected to axial compres-sion The buckling analysis of short cylindrical shells surrounded
by an elastic medium was carried out by Naili and Oddou [10]
imperfect FGM cylindrical shells under axial compression in thermal environment They used the Galerkin method, leading
to the closed form solutions for critical buckling load Van der Neut [12] pointed out the importance role of the eccentricity of
Contents lists available atScienceDirect
International Journal of Mechanical Sciences
http://dx.doi.org/10.1016/j.ijmecsci.2014.01.016
0020-7403 & 2014 Elsevier Ltd All rights reserved.
n Corresponding author Tel.: þ84 4 37547978; fax: þ84 4 37547424.
E-mail address: ducnd@vnu.edu.vn (N.D Duc).
International Journal of Mechanical Sciences 81 (2014) 17–25
Trang 2stiffeners in the buckling of isotropic cylindrical shells under axial
and linear buckling of FGM cylindrical shells based on a
two-dimensional higher order shear deformation theory Huang and
Han[14,15]studied the buckling and post-buckling of unstiffened
FGM cylindrical shells under axial compression, radial pressure
and combined axial compression and radial pressure based on
the Donnell shell theory and the nonlinear strain–displacement
relations of large deformation
theories for buckling analysis of the perfect and imperfect
cylind-rical shells[16–18] Shen[19]employed the theory of Reddy and
Liu to study postbuckling of shear deformable cross-ply laminated
cylindrical shells under combined external pressure and axial
geometric imperfections on the buckling and postbuckling of
composite laminated cylindrical shells subjected to combined axial
compression and uniform temperature rise using Reddy's higher
order shear deformation shell theory and employing a von Karman
type of kinematic nonlinearity Sheng and Wang[21]investigated
the buckling and dynamic stability of FGM cylindrical shells
embedded in an elastic medium and subjected to mechanical
and thermal loads based on thefirst-order shear deformation shell
theory The post-buckling analysis of pressure-loaded functionally
graded cylindrical shells without stiffeners based on the classical
shell theory with von Karman–Donnell-type of kinetic
deformation theory, this author[23]continued to investigate the
post-buckling of FGM hybrid cylindrical shells in thermal
environ-ments under axial loading Shen [24] studied the postbuckling
response of a shear deformable functionally graded cylindrical
shell offinite length embedded in a large outer elastic medium
and subjected to axial compressive loads in thermal environments,
this author also researched on the thermal postbuckling response
of a shear deformable functionally graded cylindrical shell offinite
length embedded in a large outer elastic medium[25]
For dynamic analysis of FGM cylindrical shells, Ng et al.[26]and Darabi et al.[27]presented respectively linear and nonlinear para-metric resonance analyses for un-stiffened FGM cylindrical shells Jiang and Olson [28] extended a super element to the nonlinear static and dynamic analysis of orthogonally stiffened cylindrical shells Sofiyev et al.[29,30]obtained critical parameters for unstif-fened cylindrical thin shells under linearly increasing dynamic torsional loading and under a periodic axial impulsive loading by using the Galerkin technique together with the Ritz type variation method Recently, Bich et al.[31]investigated nonlinear static and dynamic buckling analysis of imperfect eccentrically stiffened func-tionally graded circular cylindrical thin shells (P-FGM) under axial compression, but without elastic foundations and temperature Duc and Quan[32] have studied the P-FGM metal–ceramic–layer doubled curved shells with stiffeners in a temperature-changing environment When stiffened shells are affected with temperature, both the shells and the stiffeners are deformed, therefore, calcula-tions become complex Duc and Thang[33] studied an analytical approach to investigate the nonlinear static buckling and postbuck-ling for imperfect eccentrically stiffened functionally graded thin circular cylindrical shells surrounded on elastic foundation with
compression
Unlike circular cylindrical shell P-FGM in Bich's research[31], in this paper, we research the nonlinear stability of imperfect eccentrically stiffened S-FGM thin circular cylindrical shells with metal–ceramic–metal layers and temperature-dependent proper-ties in thermal environments, which are symmetric through the middle surface by Sigmoid-law distribution and surrounded on elastic foundations The formulations are based on the Donnell shells theory taking into account geometrical nonlinearity, initial geometrical imperfection, temperature-dependent properties and the Lekhnitsky smeared stiffeners technique with Pasternak type elastic foundation Using the Galerkin method and stress function, the effects of geometrical and material properties, temperature, elastic foundation and eccentrically stiffeners on the nonlinear
Nomenclature
respectively
CT; CT
sT; sT
y spacing of the stringer and ring stiffeners, respectively
AT; AT
y cross-section areas of stiffeners
IT; IT
y moment of inertia of stiffeners cross section relative to the shell middle surface
zT; zT
y eccentrically of stiffeners with respect to the middle surface the shell
dT; dT
y width of the stringer and ring stiffened, respectively
hTx; hT
y height of the stringer and ring stiffeners, respectively
Fig 1 Configuration of an eccentrically stiffened S-FGM circular cylindrical shell.
Trang 3response of the eccentrically stiffened S-FGM shell in thermal
environments are analyzed and discussed
2 Eccentrically stiffened S-FGM cylindrical shells on elastic
foundations
Consider a functionally graded thin circular cylindrical shell with
R; L; h – are the radius, the length and the thickness of the shell,
respectively (Fig 1)[31,34,35]
assumed by the Sigmoid power-law distribution (S-FGM)[34]
VcðzÞ ¼
2z þ h
h
2z þ h
h
8
<
with N is the volume-fraction index The subscripts c and m are
ceramic and metal constituents respectively
According to the mentioned law, the material coefficients of the
S-FGM shell can be expressed in the form
½Eðz; TÞ;νðz; TÞ;ρðz; TÞ;αðz; TÞ; Kðz; TÞ
¼ ½EmðTÞ;νmðTÞ;ρmðTÞ;αmðTÞ; KmðTÞ
þ½EcmðTÞ;νcmðTÞ;ρcmðTÞ;αcmðTÞ; KcmðTÞ
2z þ h
h
2z þ h
h
8
<
where
EcmðTÞ ¼ EcðTÞEmðTÞ;ρcmðTÞ ¼ρcðTÞρmðTÞ;
νcmðTÞ ¼νcðTÞνmðTÞ;αcmðTÞ ¼αcðTÞαmðTÞ;
From Eq (2) we can see that for S-FGM (Fig 1): E¼ Em at
A material coefficient Pr such as the elastic modulus E, Poisson
ratioν, the mass density ρ, the thermal expansion coefficientα
and coefficient of thermal conduction K can be expressed as a
nonlinear function of temperature[36–38]
In which T¼ T0þΔTðzÞ and T0¼ 300 K (room temperature);
P 1; P0; P1; P2; P3are coefficients characterizing of the
constitu-ent materials The material properties for the later one have been
determined by(4)at room temperature, i.e T0¼ 300 K
The shell–foundation interaction is represented by the
Paster-nak model as
where∇2¼ ∂2=∂x2þ∂2=∂y2, w is the deflection of the shell, k1is
Winkler foundation modulus and k2is the shear layer foundation
stiffness of the Pasternak model
3 Theoretical formulation
The strains at the middle surface relating to the displacement
non-linearity assumption are of the form[39,40]
ε0x¼ u;xþ12ðw;xÞ2; ε0y¼ v;ywRþ12ðw;yÞ2; γ0xy¼ u;yþv;xþw;xw;y
ð6Þ According to the Donnell shell theory, the nonlinear strain–
displacement relations from the middle surface for a thin circular
cylindrical shell have the form[39,40]
εx¼ε0xþzkx; εy¼ε0yþzky; γ ¼γ þ2zkxy
In which ε0x;ε0y are the normal strains andε0xy is the shear strain at the middle surface of the shell and kx; ky, kxy are the curvatures and twist
Hooke law for an FGM shell with temperature-dependent properties is defined as
ðssh
x;ssh
yÞ ¼1Eðz; TÞν2ðz; TÞ½ðεx;εyÞþνðεy;εxÞð1þνÞαΔTðzÞð1; 1Þ;
ssh
where ΔT is temperature rise from stress free initial state, and more generally, ΔT¼ΔTðzÞ; Eðz; TÞ;νðz; TÞ are the FGM shell's elastic moduli which are determined by(2)
For stiffeners in thermal environments with temperature-dependent properties, we have proposed its form adapted from Ref.[32]as follows:
ðsst
x;sst
yÞ ¼ E0ðεx;εyÞ E0
here, E0¼ E0ðTÞ; ν0¼ν0ðTÞ; α0¼α0ðTÞ are Young's modulus, Poisson ratio and thermal expansion coefficient of the stiffeners, respectively Where E0 is Young's modulus of stringers and rings stiffeners with E0¼ Em
We have assumed that the thermal stress of stiffeners is subtle which distributes uniformly through the whole shell structure Therefore, we can ignore it and Lekhnitsky smeared stiffeners technique can be adapted from Ref.[41–44] as follows:
Nx¼ I10þET0AT
sT
!
ε0xþI20ε0yþðI11þCTÞkxþI21kyþΦ1;
Ny¼ I20ε0xþ I10þE
T
0ATy
sT y
!
ε0yþI21kxþðI11þCT
yÞkyþΦ1;
Nxy¼ I30γ0xyþ2I31kxy;
Mx¼ ðI11þCT
Þε0xþI21ε0yþ I12þE
T
0IT
sT
!
kxþI22kyþΦ2;
My¼ I21ε0xþðI11þCT
yÞε0yþI22kxþ I12þE
T
0ITy
sT y
!
kyþΦ2;
Iijði ¼ 1; 2; 3; j ¼ 0; 1; 2Þ:
I1j¼Z
h =2
h=2
EðzÞ
1νðzÞ2zjdz; j ¼ 0; 2
I2j¼Z h=2
h=2
EðzÞνðzÞ
1νðzÞ2zjdz; j ¼ 0; 2
I3j¼Z h=2
h=2
EðzÞ
2½ð1þνðzÞzjdz¼
1
2ðI1jI2jÞ; j ¼ 0; 2
IT¼d
T
ðhT
Þ3
ðzTÞ2; IT
y¼d
T
yðhT
yÞ3
yðzT
yÞ2;
CT¼E0ATzT
sT ; CT
y¼E0ATyzT
y
sT y
;
zT¼h
T
þhT
y¼h
T
yþhT
AT¼ dT
sT; AT
y¼ dT
ysT
y:
ðΦ1;Φ2Þ ¼ Z h=2
h=2
EðzÞαðzÞ
N.D Duc, P.T Thang / International Journal of Mechanical Sciences 81 (2014) 17–25 19
Trang 4where the coupling parameters Cx; Cy are negative for outside
stiffeners and positive for inside one; Ix; Iy are the second
moments of cross-section areas; sx; sy are the spacing of the
longitudinal and transversal stiffeners; zx; zyare the eccentricities
of stiffeners with respect to the middle surface of shell; and the
width and thickness of longitudinal and transversal stiffeners are
denoted by dx; hx and dy; hy respectively Ax; Ay are the
cross-section areas of stiffeners Although the stiffeners are deformed
by temperature, we, however, have assumed that the stiffener
keep its rectangular shape of the cross section Therefore, it is
straightforward to calculate AT; AT
y After the thermal deformation process, the geometric shapes of
stiffeners which can be determined as follows[32]:
dTx¼ dx½1þαmTðzÞ; dT
y¼ dy½1þαmTðzÞ;
hT¼ hx½1þαmTðzÞ; hT
y¼ hy½1þαmTðzÞ;
zT¼ zx½1þαmTðzÞ; zT
y¼ zy½1þαmTðzÞ;
sT
¼ sx½1þαmTðzÞ; sT
Interestingly, in this paper, from Eqs.(9) and (12), we can see
that the material properties of eccentrically outside stiffeners also
depend on temperature
The nonlinear equilibrium equations of the perfect S-FGM
cylindrical shells based on the classical shell theory are[39,40]
Mx;xxþ2Mxy;xyþMy;yyþNy
RþNxw;xxþ2Nxyw;xyþNyw;yy
Calculated from Eq.(10)
ε0x¼ J22NxJ12NyþG11w;xxþG12w;yyðJ22J12ÞΦ1;
ε0y¼ J12NxþJ11NyþG21w;xxþG22w;yyðJ11J12ÞΦ1;
γ0xy¼ J66Nxyþ2G66w;xy; ð14Þ
where
J11¼Δ1 I10þE
T
0AT
sT
!
; J12¼I20
Δ;
J22¼Δ1 I10þE
T
0ATy
sT
y
!
; J66¼I1
30;
G11¼ J22ðI11þCT
ÞJ12I21; G22¼ J11ðI11þCT
yÞJ12I21;
G12¼ J22I21J12ðI11þCT
yÞ; G21¼ J11I21J12ðI11þCT
Þ;
G66¼I31
and
Δ¼ I10þE
T
0ATx
sT
!
I10þE
T
0ATy
sT y
!
I2
20: Substituting once again Eq.(14)into the expression of Mijin
(10), then Mijinto Eq.(13c)leads to
Nx ;xþNxy ;y¼ 0;
Nxy ;xþNy ;y¼ 0;
G21ϕ;xxxxþðG11þG222G66Þϕ;xxyyþG12ϕ;yyyy
D11w;xxxxD22w;yyyyðD12þD21þ4D66Þw;xxyyþ⋯
Ny
RþNxw;xxþ2Nxyw;xyþNyw;yyþqk1wþk2∇2w¼ 0; ð16Þ
where
D11¼ I12þE
T
0AT
sT G21I21ðI11þCT
ÞG11;
D22¼ I12þE
T
0ATy
sT
y G12I12ðI22þCT
yÞG22;
D12¼ I22G22I21ðI11þCT
ÞG12;
D21¼ I22G11I21ðI11þCT
yÞG21;
ϕðx; yÞ is stress function defined by
For an imperfect S-FGM circular cylindrical shell Eq (16) is modified into form as
G21ϕ;xxxxþðG11þG222G66Þϕ;xxyyþG12ϕ;yyyyD11w;xxxxD22w;yyyy
ðD12þD21þ4D66Þw;xxyy
þϕ;xx
R þϕ;yyðw;xxþwn
;xxÞ2ϕ;xyðw;xyþwn
;xyÞþϕ;xxðw;yyþwn
;yyÞ
In which wnðx; yÞ is a known function representing initial small imperfection of the shell The geometrical compatibility equation for imperfect cylindrical shells written as
ε0x ;yyþε0y ;xxγ0xy ;xy¼ 1
Rw;xxþw2
;xyw;xxw;yyþ2w;xywn
;xy
From the constitutive relations Eq.(14)in conjunction with Eq
(18)one can write
ε0x¼ J22ϕ;yyJ12ϕ;xxþG11w;xxþG12w;yyðJ22J12ÞΦ1
ε0y¼ J12ϕ;yyþJ11ϕ;xxþG21w;xxþG22w;yyðJ11J12ÞΦ1
γ0xy¼ J66ϕ;xyþ2G66w;xy ð21Þ Setting Eq.(21)into Eq.(20)gives the compatibility equation of an imperfect S-FGM shell as
J11ϕ;xxxxþðJ662J12Þϕ;xxyyþJ22ϕ;yyyyþG21w;xxxxþG12w;yyyy þðG11þG222G66Þw;xxyy
;xyw;xxw;yyþ2w;xywn;xyw;xxwn;yyw;yywn;xxwxx
R
¼ 0 ð22Þ Eqs.(19) and (22)are nonlinear equations in terms of variables w andϕand used to investigate the nonlinear buckling of imperfect eccentrically stiffened functionally graded thin circular cylindrical
metal layers (S-FGM) and subjected mechanical and thermal loads
following approximate solutions[41,42]
f¼ A1 cos 2λmxþA2 cos 2δnyþA3 sinλmx sinδnyþð1=2ÞNx0y2
ð24Þ
λm¼ mπ=L;δn¼ n=R, W are amplitude of the deflection andμis imperfection parameter The coefficients Aiði ¼ 1=3Þ are deter-mined by substitution of Eqs.(23) and (24)into Eq.(22)as
A1¼ δ2 n
32J11λ2 m
WðW þ2μhÞ; A2¼ λ2
m
32J22δ2 n
WðW þ2μhÞ;
m
R½J11λ4
mþJ22δ4
þðJ662J12Þλ2
mδ2
½G21λ4
mþG12δ4
nþðG11þG222G66Þλ2
mδ2
n
½J11λ4
mþJ22δ4
nþðJ662J12Þλ2
mδ2
Trang 5Substitution of Eqs.(23) and (24)into(19)and applying the
Galerkin procedure for the resulting equation yield
1
λmδn
2λ2
m
R
½G21λ4
mþG12δ4
þðG11þG222G66Þλ2
mδ2
½J11λ4
mþJ22δ4
nþðJ662J12Þλ2
mδ2
n
½G21λ4
mþG12δ4
nþðG11þG222G66Þλ2
mδ2
n2
½J11λ4
mþJ22δ4
nþðJ662J12Þλ2
mδ2
n
λ4
m
R2
1
½J11λ4
mþJ22δ4þðJ662J12Þλ2
mδ2
D11λ4
mD22δ4
nðD12þD21þ4D66Þλ2
mδ2
nk1ðλ2
mþδ2
nÞk2
2
6
6
6
6
6
6
6
4
3 7 7 7 7 7 7 7 5 W
16λmδn
λ4
m
J22þδ4
J11
!
ðW þμhÞWðW þ2μhÞλm
δn
Nx0ðW þμhÞ ¼ 0;
ð26Þ where m; n are odd numbers This equation will be used to analyze
the buckling behaviors of eccentrically stiffened S-FGM shells
under mechanical and thermal loads
4 Nonlinear buckling analysis
4.1 Thermal buckling analysis
A simply supported S-FGM circular cylindrical shell on two
immovable edges and under steadily increasing temperature is
considered (Table 1) The condition expressing the immovability
on the boundary edges of the shell, i.e u¼ 0 at x ¼ 0; L is justified
in an average sense as
Z 2πR
0
Z L
0
∂u
From Eqs.(6) and (14)one can obtain the following expression
in which Eq.(18)and imperfect have been included
∂u
∂x¼ J22ϕ;yyJ12ϕ;xxþG11w;xxþG12w;yy12w2
;xw;xwnxðJ22J12ÞΦ1
ð28Þ Substitution of Eqs.(23) and (24)into(28)and then the result into
Eq.(27)givefictitious edge compressive loads as
Nx0¼ J22J12
J22
Φ1þ 1
8J22λ2
By using Eq.(11), the thermal parameterΦ1can be expressed
in terms ofΔT:
in which
0
Ecαc
1ðvcþvmctNÞdtþ
Z 1 0
ðEcαmcþEmcαcÞtN
1ðvcþvmctNÞ dt
"
þZ 1
0
Emcαmct2N
1ðvcþvmctNÞdt
#
ð31Þ
Although ΔT is included in the expression for L due to the
temperature dependence of material propertiesðT ¼ T0þΔTÞ, one
may formally expressΔT from Eqs.(26) and (30)as follows:
ΔT¼1P A22
b1
W
where
b1¼
m2π2L2R2
½B21m4π4þB12n4L4þðB11þB222B66Þm2n2π2L22
½A11m4π4þA22n4L4þðA662An
12Þm2n2π2L2 þ
þR2
h
½B21m4π4þB12n4L4þðB11þB222B66Þm2n2π2L2
½A11m4π4þA22n4L4þðA662An
12Þm2n2π2L2
½A11m4π4þA22n4L4þðA662An
12Þm2n2π2L2þ
Dn11m2π2
L2R2 Dn22n4L2
m2π2R2n2ðDn
12þDn
21þ4Dn
66Þ
R2 k1L2R2
m2π2 m2π2þn2L2
m2π2 k2
2 6 6 6 6 6 6 6 6
3 7 7 7 7 7 7 7 7
b2¼ 1 PðA22A12Þ
A22
16m2π2L2R2
m4π4
An22 þn4L4
An11
!
m2π2
8L2R2
ð33Þ
Eq (32) is the analytical form to determine the non-linear relation between the bending deflection and temperature for both
of the perfect and imperfect shells under the thermal loads (for perfect shellμ¼ 0) Using Eq.(32), we have derived the tempera-ture change,ΔTb¼1 A 22
ðA 22 A 12 Þ
b1, which sets them into the buck-ling state under the condition W¼ 0
Eq.(32)is 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 buckling period of the S-FGM shells To be more specific, given the material parameter
N, the geometrical parameterðLR; RhÞ and the value of W=h, we can use these to determineΔT in(32)as follows: we choose an initial step forΔT1on the right hand side in Eq.(32)withΔT¼ 0 (since
T¼ T0¼ 300 K, the initial room temperature) In the next iterative step, we replace the known value ofΔT found in the previous step
to determine the right hand side of Eq (32), ΔT2 This iterative procedure will stop at the kth-steps if ΔTk satisfies the con-ditionjΔTΔTkjrε Here,ΔT is a desired solution for the tem-perature andεis a tolerance used in the iterative steps
4.2 Mechanical buckling analysis
To clarify the effects of buckling load of the S-FGM shell with
metal–ceramic layers, in this section, we consider the effects of shell under axial compression without temperature, and afterward compare the results with those of Bich et al.[31]
Suppose that an imperfect S-FGM circular cylindrical shell is simply supported and subjected to axial compressive load
Nx0¼ Pxh, where Px is the average axial stress on the shell's end sections, positive when the shells subjected to axial compres-sion The boundary conditions considered in this paper are
And Eq.(26)leads to
Px¼ a1
W
where
a1¼
1
m2π2L2R2
½G21m4π4þG12n4L4þðG11þG222G66Þm2n2π2L22
½J11m4π4þJ22n4L4þðJ662J12Þm2n2π2L2
2
Rh
½G21m4π4þG12n4L4þðG11þG222G66Þm2n2π2L2
½J11m4π4þJ22n4L4þðJ662J12Þm2n2π2L2
½J11m4π4þJ22n4L4þðJ662J12Þm2n2π2L2
þDn11m2π2
L2R2 þDn22n4L2
m2π2R2þn2ðDn
12þDn
21þ4Dn
66Þ
R2 þk1L2R2
m2π2 þm2π2þn2L2
m2π2 k2
2 6 6 6 6 6 6 6 6
3 7 7 7 7 7 7 7 7
N.D Duc, P.T Thang / International Journal of Mechanical Sciences 81 (2014) 17–25 21
Trang 6a2¼ 1
16m2π2L2R2
m4π4
J22 þn4L4
J11
!
LR¼RL; Rh¼Rh; W ¼Wh ; k2¼k2
h; k1¼ k1h; G21¼G21
h ;
G12¼G12
h ; G11¼G11
h ; G22¼G22
h ; G66¼G66
h ; J11¼ J11h;
J22¼ J22h; J12¼ J12h; J66¼ J66h; D11¼D11
h3; D22¼D22
h3;
D12¼D12
h3; D21¼D21
h3; D66¼D66
For a perfect cylindrical shellsμ¼ 0 Eq.(35)leads to
5 Numerical result and discussion
To illustrate, we consider a symmetric S-FGM circular
cylind-rical shell with the parameters as follows:
L¼ 0:75 m; R ¼ 0:5 m; h ¼ R=80;
sT¼2πR
ns ; sT
y¼nL
r; ns¼ 20; nr¼ 70;
Fig 2 Nonlinear response of the un-stiffened imperfect S-FGM and P-FGM circular
cylindrical shells (without elastic foundations).
Fig 3 Nonlinear response of the stiffened S-FGM and P-FGM circular cylindrical
Fig 4 Effects of N index on the nonlinear response of the S-FGM circular cylindrical shells under mechanical load.
Fig 5 Effects of N index on the nonlinear response of the S-FGM circular cylindrical shells under thermal load.
Fig 6 Effect of imperfection on buckling of eccentrically stiffened S-FGM circular
Trang 7Fig 7 Effects of the stiffeners on the nonlinear response of the S-FGM circular
cylindrical shells under mechanical load.
Fig 8 Effects of R/h index on the nonlinear response of S-FGM circular cylindrical
shells under mechanical load.
Fig 9 Effects of R/h index on the nonlinear response of S-FGM circular cylindrical
shells under thermal load.
Fig 10 Effects of imperfection and elastic foundation on the nonlinear response of S-FGM circular cylindrical shells under mechanical load.
Fig 11 Effects of ratio L/R on the nonlinear response of S-FGM circular cylindrical shells under mechanical load.
Fig 12 Effects of ratio L/R on the nonlinear response of S-FGM circular cylindrical N.D Duc, P.T Thang / International Journal of Mechanical Sciences 81 (2014) 17–25 23
Trang 8hT¼ hT
y¼ 0:01 m; dT
¼ dT
y¼ 0:0025 m;
k1¼ 100; k2¼ 30;
where ns and ns are the number of strings, rings of the shells,
respectively
Figs 2 and 3show a comparison between the present results
for the S-FGM shell and Bich's results[31]for the P-FGM shell with
imperfect shell without stiffeners and elastic foundation with
N¼ 2, we can see that the solid line is much higher than the dash
line, revealing the loading capacity of the S-FGM circular
cylind-rical shell with metal–ceramic–metal layers is higher than P-FGM
shell with metal–ceramic layers InFig 3, we consider the shells
with stiffeners, imperfection but without the elastic foundation
and realized that the solid line (μ¼ 0 – perfect shell) is always
higher than dash line (μ¼ 0:1 – imperfect shell), and the loading
capacity of the shell with stiffeners (Fig 3) is better than the shell
without stiffeners (Fig 2)
Fig 4 and Fig 5 show the influence of the volume ratio and
imperfection on buckling behavior of S-FGM cylindrical shell
under mechanical and thermal loads, respectively From two
figures, we can see that when N is increased, the curve becomes
lower; this means the weaker loading capacity of the shells This is
right because when N is increased, the metal ratio is increased;
however, elastic module of metal is lower than ceramic (EmoEc)
We also see that at the same point of deflection, the loading
capacity of the perfect shell is a little better than imperfect one
Fig 6shows the influence of imperfection of initial shape on
buckling behavior of S-FGM shell under mechanical load It
indicates that the loading capacity of the shell is decreased when
compressive cylinder shells with stiffeners and without stiffeners
From thefigure, we can see that in both cases, the perfect ðμ¼ 0Þ
and imperfectðμ¼ 0:1Þ cylindrical shells with stiffeners can
with-stand higher compression than the ones without the stiffeners
This clearly shows the better effectiveness of stiffeners
Figs 8 and 9show the influence of radius ratio on the thickness
R=h ¼ ð100; 150; 200Þ on buckling behavior of S-FGM cylindrical
shell under mechanical and thermal loads From these twofigures,
we can see that when R=h is increased, the curve becomes lower
This is right because when R=h is increased, the circular cylindrical
shell becomes thinner and the load capacity is decreased
Fig 10 presents the effects of the elastic foundations on
buckling behavior of perfect (μ¼ 0) and imperfect (μ¼ 0:1)
S-FGM circular cylindrical shells under mechanical load Obviously,
buckling load is enhanced due to the presence of elastic
founda-tions and the effect of Pasternak foundation k2 on the loading
capacity is higher than the Winkler foundation k
Figs 11 and 12show the influence of the ratio of the length on radius L=R on buckling behavior of S-FGM cylindrical shell under mechanical and thermal loads As shown inFig 11, the mechan-ical loading capacity of the shell is increased when the ratio of
L=R is increased.Fig 12shows that the thermal loading capacity
of the shell is decreased when the ratio of L=R is increased
6 Concluding remarks This paper presents an analytical investigation on the nonlinear buckling response for imperfect eccentrically stiffened S-FGM thin circular cylindrical shells with metal–ceramic–metal layers sur-rounded on elastic foundation in thermal environment The shell subjected to axial compression and thermal loads Both S-FGM shell and stiffeners are deformed by temperature The formula-tions are based on the Donnell shell theory taking into account
temperature-dependent properties and the Lekhnitsky smeared stiffeners technique with Pasternak type elastic foundation Using the Galerkin method and stress function, effects of material and geometrical properties, temperature, elastic founda-tion and eccentrically outside stiffeners on the buckling loading capacity of the imperfect eccentrically stiffened S-FGM shell in thermal environment are analyzed and discussed The results showed that the addition of stiffeners increases the mechanical and thermal loading capacity of the FGM shell, and the loading capacity of the S-FGM shell with metal–ceramic–metal layers is higher than P-FGM shell with metal–ceramic layers with the same geometrical parameters Some results were compared with the ones of the other authors
Acknowledgments This paper was supported by Project “Nonlinear analysis on stability and dynamics of functionally graded shells with special shapes”of Vietnam National University, Hanoi The authors are grateful for this support
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