Nonlinear buckling of imperfect eccentrically stiffenedmetal–ceramic–metal S-FGM thin circular cylindrical shellswith temperature-dependent properties in thermal environments

Nguyen Dinh Duc n, Pham Toan ThangVietnam 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 2013Received in revised form10 January 2014Accepted 20 January 2014Available online 6 February 2014

Keywords:Nonlinear bucklingS-FGM with metal–ceramic–metal layersEccentrically stiffened cylindrical shellsImperfectionElastic foundationThermal environment

a b s t r a c t

In this paper, an analytical approach is presented to investigate the nonlinear static buckling forimperfect eccentrically stiffened functionally graded thin circular cylindrical shells with temperature-dependent properties surrounded on elastic foundation in thermal environment. Both shells andstiffeners are deformed simultaneously due to temperature. Material properties are graded in thethickness direction according to a Sigmoid power law distribution in terms of the volume fractions ofconstituents (S-FGM) with metal–ceramic–metal layers. The Lekhnitsky smeared stiffeners techniquewith 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-FGMshells. 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 Functionally Graded Material and often abbreviatedFGM was developed and named by a group of material scientists atSendai Institute of Japan in 1984 [1,2]. This material is a type of newgeneration composite, intelligent composite, appears as a result ofactual demands for a material that can overcome the disadvantagesof traditional metals and laminated normal composites. This func-tionally graded material is formed from two component materials ofceramic and metal in which the volume ratio of each compositionvaries smoothly and continuously from this side to the other sideaccording to the structure wall thickness in order to be suitable forthe characteristic strength of the component materials.

The cylindrical shell is a structure that is used popularly in theindustry, 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, dueto the non-slope of circular cylindrical shells and complexity incalculation, the nonlinear stability researches of them are still verylimited in comparison with the structures of plate or other kinds of

shells. A few case studies on the stability of FGM cylindrical shellsare introduced below: Lanhe et al. [3] have used the uncoupledequation and Shahsiah and Eslami [4] have used couple ofequations system to study the problem of the linear stabilityperfect 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 FGMcylindrical shells subjected to combined axial and radial pressure.In this analysis, the nonlinear strain–displacement relations oflarge deformation and the Ritz energy method were used. Iqbalet al. [7] studied free vibration of thin FGM cylindrical shells byusing wave propagation approach based on the classical shelltheory. Li and Batra [8] investigated buckling of axially compressedthin 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 cylindricalshells reinforced by rings and stringer subjected to axial compres-sion. The buckling analysis of short cylindrical shells surroundedby an elastic medium was carried out by Naili and Oddou [10].Mirzavand and Eslami [11] presented the buckling analysis ofimperfect FGM cylindrical shells under axial compression inthermal environment. They used the Galerkin method, leadingto the closed form solutions for critical buckling load. Van derNeut [12] pointed out the importance role of the eccentricity of

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International Journal of Mechanical Sciences

http://dx.doi.org/10.1016/j.ijmecsci.2014.01.0160020-7403 & 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

stiffeners in the buckling of isotropic cylindrical shells under axialcompressive load. Matsunaga [13] examined the free vibrationand linear buckling of FGM cylindrical shells based on a two-dimensional higher order shear deformation theory. Huang andHan [14,15] studied the buckling and post-buckling of unstiffenedFGM cylindrical shells under axial compression, radial pressureand combined axial compression and radial pressure based onthe Donnell shell theory and the nonlinear strain–displacementrelations of large deformation.

Some researchers have used the first-order and high-ordertheories for buckling analysis of the perfect and imperfect cylind-rical shells [16–18]. Shen [19] employed the theory of Reddy andLiu to study postbuckling of shear deformable cross-ply laminatedcylindrical shells under combined external pressure and axialcompression. Shen and Li [20] studied influence of the localgeometric imperfections on the buckling and postbuckling ofcomposite laminated cylindrical shells subjected to combined axialcompression and uniform temperature rise using Reddy's higherorder shear deformation shell theory and employing a von Karmantype of kinematic nonlinearity. Sheng and Wang [21] investigatedthe buckling and dynamic stability of FGM cylindrical shellsembedded in an elastic medium and subjected to mechanicaland thermal loads based on the first-order shear deformation shelltheory. The post-buckling analysis of pressure-loaded functionallygraded cylindrical shells without stiffeners based on the classicalshell theory with von Karman–Donnell-type of kinetic nonlinear-ity is presented by Shen [22]. By using higher order sheardeformation theory, this author [23] continued to investigate thepost-buckling of FGM hybrid cylindrical shells in thermal environ-ments under axial loading. Shen [24] studied the postbucklingresponse of a shear deformable functionally graded cylindricalshell of finite length embedded in a large outer elastic mediumand subjected to axial compressive loads in thermal environments,this author also researched on the thermal postbuckling responseof a shear deformable functionally graded cylindrical shell of finitelength embedded in a large outer elastic medium [25].

For dynamic analysis of FGM cylindrical shells, Ng et al. [26] andDarabi 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 nonlinearstatic and dynamic analysis of orthogonally stiffened cylindricalshells. Sofiyev et al. [29,30] obtained critical parameters for unstif-fened cylindrical thin shells under linearly increasing dynamictorsional loading and under a periodic axial impulsive loading byusing the Galerkin technique together with the Ritz type variationmethod. Recently, Bich et al. [31] investigated nonlinear static anddynamic buckling analysis of imperfect eccentrically stiffened func-tionally graded circular cylindrical thin shells (P-FGM) under axialcompression, but without elastic foundations and temperature.Duc and Quan [32] have studied the P-FGM metal–ceramic–layerdoubled curved shells with stiffeners in a temperature-changingenvironment. 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 analyticalapproach to investigate the nonlinear static buckling and postbuck-ling for imperfect eccentrically stiffened functionally gradedthin circular cylindrical shells surrounded on elastic foundation withceramic–metal–ceramic layers (S-FGM) and subjected to axialcompression.

Unlike circular cylindrical shell P-FGM in Bich's research [31], inthis paper, we research the nonlinear stability of imperfecteccentrically stiffened S-FGM thin circular cylindrical shells withmetal–ceramic–metal layers and temperature-dependent proper-ties in thermal environments, which are symmetric through themiddle surface by Sigmoid-law distribution and surrounded onelastic foundations. The formulations are based on the Donnellshells theory taking into account geometrical nonlinearity, initialgeometrical imperfection, temperature-dependent properties andthe Lekhnitsky smeared stiffeners technique with Pasternak typeelastic 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

h thickness of shellm number of half waves axial directionn number of wave in circumferential directionN volume-fraction indexL length of the shellR radius of the shellEðzÞ; Em; Ec Young's modulus of shell, metal, ceramic

respectively

CTx ; C

Ty coupling parameters

sTx ; sTy spacing of the stringer and ring stiffeners, respectively

ATx ; A

Ty cross-section areas of stiffeners

ITx ; ITy moment of inertia of stiffeners cross section relative to

the shell middle surfacezTx ; z

Ty eccentrically of stiffeners with respect to the middle

surface the shelldTx ; d

Ty width of the stringer and ring stiffened, respectively

hTx ; hTy height of the stringer and ring stiffeners, respectively

μh known imperfect amplitude.

Fig. 1. Configuration of an eccentrically stiffened S-FGM circular cylindrical shell.

N.D. Duc, P.T. Thang / International Journal of Mechanical Sciences 81 (2014) 17–2518

response of the eccentrically stiffened S-FGM shell in thermalenvironments are analyzed and discussed.

2. Eccentrically stiffened S-FGM cylindrical shells on elasticfoundations

Consider a functionally graded thin circular cylindrical shell withR; L;h – are the radius, the length and the thickness of the shell,respectively (Fig. 1) [31,34,35].

The volume fractions of metal and ceramic, Vm and Vc isassumed by the Sigmoid power-law distribution (S-FGM) [34]

VcðzÞ ¼2zþh

h

� �N; NZ0; �h

2rzr0

�2zþhh

� �N; 0rzrh

2

8<: ; VmðzÞþVcðzÞ ¼ 1 ð1Þ

with N is the volume-fraction index. The subscripts c and m areceramic and metal constituents respectively.

According to the mentioned law, the material coefficients of theS-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�

�2zþh

h

� �N; NZ0; �h

2rzr0

�2zþhh

� �N; 0rzrh

2

8<: ð2Þ

where

EcmðTÞ ¼ EcðTÞ�EmðTÞ;ρcmðTÞ ¼ ρcðTÞ�ρmðTÞ;νcmðTÞ ¼ νcðTÞ�νmðTÞ;αcmðTÞ ¼ αcðTÞ�αmðTÞ;KcmðTÞ ¼ KcðTÞ�KmðTÞ; ð3Þ

From Eq. (2) we can see that for S-FGM (Fig. 1): E¼ Em atz¼ �h=2 and z¼ h=2 (metal) and E¼ Ec at z¼ 0 (ceramic).A material coefficient Pr such as the elastic modulus E, Poissonratio ν, the mass density ρ, the thermal expansion coefficient αand coefficient of thermal conduction K can be expressed as anonlinear function of temperature [36–38]

Pr¼ P0ðP�1T�1þ1þP1T

�1þP2T2þP3T

3Þ; ð4ÞIn which T ¼ T0þΔTðzÞ and T0 ¼ 300 K (room temperature);

P�1; P0; P1; P2; P3 are coefficients characterizing of the constitu-ent materials. The material properties for the later one have beendetermined by (4) at room temperature, i.e. T0 ¼ 300 K.

The shell–foundation interaction is represented by the Paster-nak model as

q¼ k1w�k2∇2w; ð5Þwhere ∇2 ¼ ∂2=∂x2þ∂2=∂y2, w is the deflection of the shell, k1 isWinkler foundation modulus and k2 is the shear layer foundationstiffness of the Pasternak model.

3. Theoretical formulation

The strains at the middle surface relating to the displacementcomponents u; v;w based on the von Karman geometrical non-linearity assumption are of the form [39,40]

ε0x ¼ u;xþ12ðw;xÞ2; ε0y ¼ v;y�w

Rþ12ðw;yÞ2; γ0xy ¼ u;yþv;xþw;xw;y

ð6ÞAccording to the Donnell shell theory, the nonlinear strain–

displacement relations from the middle surface for a thin circularcylindrical shell have the form [39,40]

εx ¼ ε0xþzkx; εy ¼ ε0yþzky; γxy ¼ γ0xyþ2zkxy

kx ¼ �w;xx; ky ¼ �w;yy; kxy ¼ �w;xy; ð7Þ

In which ε0x; ε0y are the normal strains and ε0xy is the shearstrain at the middle surface of the shell and kx; ky, kxy are thecurvatures and twist.

Hooke law for an FGM shell with temperature-dependentproperties is defined as

ðsshx ;ssh

y Þ ¼ Eðz; TÞ1�ν2ðz; TÞ½ðεx; εyÞþνðεy; εxÞ�ð1þνÞαΔTðzÞð1;1Þ�;

sshxy ¼

Eðz; TÞ2½1þνðz; TÞ� γxy; ð8Þ

where ΔT is temperature rise from stress free initial state, andmore generally, ΔT ¼ΔTðzÞ; Eðz; TÞ;νðz; TÞ are the FGM shell'selastic moduli which are determined by (2).

For stiffeners in thermal environments with temperature-dependent properties, we have proposed its form adapted fromRef. [32] as follows:

ðsstx ;sst

y Þ ¼ E0ðεx; εyÞ�E0

1�2ν0ðTÞα0ðTÞΔðTÞð1;1Þ: ð9Þ

here, E0 ¼ E0ðTÞ; ν0 ¼ ν0ðTÞ; α0 ¼ α0ðTÞ are Young's modulus,Poisson ratio and thermal expansion coefficient of the stiffeners,respectively. Where E0 is Young's modulus of stringers and ringsstiffeners with E0 ¼ Em.

We have assumed that the thermal stress of stiffeners is subtlewhich distributes uniformly through the whole shell structure.Therefore, we can ignore it and Lekhnitsky smeared stiffenerstechnique can be adapted from Ref. [41–44] as follows:

Nx ¼ I10þET0A

Tx

sTx

!ε0xþ I20ε0yþðI11þCT

x Þkxþ I21kyþΦ1;

Ny ¼ I20ε0xþ I10þET0A

Ty

sTy

!ε0yþ I21kxþðI11þCT

y ÞkyþΦ1;

Nxy ¼ I30γ0xyþ2I31kxy;

Mx ¼ ðI11þCTx Þε0xþ I21ε0yþ I12þ

ET0ITx

sTx

!kxþ I22kyþΦ2;

My ¼ I21ε0xþðI11þCTy Þε0yþ I22kxþ I12þ

ET0ITy

sTy

!kyþΦ2;

Mxy ¼ I31γ0xyþ2I32kxy; ð10Þ

The relation (10) is our most important finding, whereIijði¼ 1;2;3; j¼ 0;1;2Þ:

I1j ¼Z h=2

�h=2

EðzÞ1�νðzÞ2

zj dz; j¼ 0;2

I2j ¼Z h=2

�h=2

EðzÞνðzÞ1�νðzÞ2

zj dz; j¼ 0;2

I3j ¼Z h=2

�h=2

EðzÞ2½ð1þνðzÞ�z

j dz¼ 12ðI1j� I2jÞ; j¼ 0;2

ITx ¼dTx ðhT

x Þ312

þATx ðzTx Þ2; ITy ¼

dTy ðhTy Þ3

12þAT

y ðzTy Þ2;

CTx ¼

E0ATx z

Tx

sTx; CT

y ¼E0A

Tyz

Ty

sTy;

zTx ¼hTx þhT

2; zTy ¼

hTyþhT

2;

ATx ¼ dTx s

Tx ; AT

y ¼ dTysTy :

ðΦ1;Φ2Þ ¼ �Z h=2

�h=2

EðzÞαðzÞ1�νðzÞΔTðzÞð1; zÞdz: ð11Þ

N.D. Duc, P.T. Thang / International Journal of Mechanical Sciences 81 (2014) 17–25 19

where the coupling parameters Cx;Cy are negative for outsidestiffeners and positive for inside one; Ix; Iy are the secondmoments of cross-section areas; sx; sy are the spacing of thelongitudinal and transversal stiffeners; zx; zy are the eccentricitiesof stiffeners with respect to the middle surface of shell; and thewidth and thickness of longitudinal and transversal stiffeners aredenoted by dx;hx and dy;hy respectively. Ax;Ay are the cross-section areas of stiffeners. Although the stiffeners are deformedby temperature, we, however, have assumed that the stiffenerkeep its rectangular shape of the cross section. Therefore, it isstraightforward to calculate AT

x ;ATy .

After the thermal deformation process, the geometric shapes ofstiffeners which can be determined as follows [32]:

dTx ¼ dx½1þαmTðzÞ�; dTy ¼ dy½1þαmTðzÞ�;hTx ¼ hx½1þαmTðzÞ�; hTy ¼ hy½1þαmTðzÞ�;zTx ¼ zx½1þαmTðzÞ�; zTy ¼ zy½1þαmTðzÞ�;sTx ¼ sx½1þαmTðzÞ�; sTy ¼ sy½1þαmTðzÞ�: ð12Þ

Interestingly, in this paper, from Eqs. (9) and (12), we can seethat the material properties of eccentrically outside stiffeners alsodepend on temperature.

The nonlinear equilibrium equations of the perfect S-FGMcylindrical shells based on the classical shell theory are [39,40]

Nx;xþNxy;y ¼ 0 ð13aÞ

Nxy;xþNy;y ¼ 0 ð13bÞ

Mx;xxþ2Mxy;xyþMy;yyþNy

RþNxw;xxþ2Nxyw;xyþNyw;yy

þq�k1wþk2∇2w¼ 0 ð13cÞCalculated from Eq. (10)

ε0x ¼ J22Nx� J12NyþG11w;xxþG12w;yy�ðJ22� J12ÞΦ1;

ε0y ¼ � J12Nxþ J11NyþG21w;xxþG22w;yy�ðJ11� J12ÞΦ1;

γ0xy ¼ J66Nxyþ2G66w;xy; ð14Þ

where

J11 ¼1Δ

I10þET0A

Tx

sTx

!; J12 ¼

I20Δ;

J22 ¼1Δ

I10þET0A

Ty

sTy

!; J66 ¼

1I30

;

G11 ¼ J22ðI11þCTx Þ� J12I21; G22 ¼ J11ðI11þCT

y Þ� J12I21;

G12 ¼ J22I21� J12ðI11þCTy Þ; G21 ¼ J11I21� J12ðI11þCT

x Þ;

G66 ¼I31I30

; ð15Þ

and

Δ¼ I10þET0A

Tx

sTx

!I10þ

ET0ATy

sTy

!� I220:

Substituting once again Eq. (14) into the expression of Mij in(10), then Mij into Eq. (13c) leads to

Nx;xþNxy;y ¼ 0;Nxy;xþNy;y ¼ 0;G21ϕ;xxxxþðG11þG22�2G66Þϕ;xxyyþG12ϕ;yyyy

�D11w;xxxx�D22w;yyyy�ðD12þD21þ4D66Þw;xxyyþ⋯Ny

RþNxw;xxþ2Nxyw;xyþNyw;yyþq�k1wþk2∇2w¼ 0; ð16Þ

where

D11 ¼ I12þET0A

Tx

sTx�G21I21�ðI11þCT

x ÞG11;

D22 ¼ I12þET0A

Ty

sTy�G12I12�ðI22þCT

y ÞG22;

D12 ¼ I22�G22I21�ðI11þCTx ÞG12;

D21 ¼ I22�G11I21�ðI11þCTy ÞG21;

D66 ¼ I32� I31G66 ð17Þϕðx; yÞ is stress function defined by

Nx ¼ϕ;yy;Ny ¼ϕ;xx;Nxy ¼ �ϕ;xy ð18Þ

For an imperfect S-FGM circular cylindrical shell Eq. (16) ismodified into form as

G21ϕ;xxxxþðG11þG22�2G66Þϕ;xxyyþG12ϕ;yyyy�D11w;xxxx�D22w;yyyy

�ðD12þD21þ4D66Þw;xxyy

þϕ;xx

Rþϕ;yyðw;xxþwn

;xxÞ�2ϕ;xyðw;xyþwn

;xyÞþϕ;xxðw;yyþwn

;yyÞ

þq�k1wþk2∇2w¼ 0 ð19ÞIn which wnðx; yÞ is a known function representing initial smallimperfection of the shell. The geometrical compatibility equationfor imperfect cylindrical shells written as

ε0x;yyþε0y;xx�γ0xy;xy ¼ �1Rw;xxþw2

;xy�w;xxw;yyþ2w;xywn

;xy

�w;xxwn

;yy�w;yywn

;xx ð20Þ

From the constitutive relations Eq. (14) in conjunction with Eq.(18) one can write

ε0x ¼ J22ϕ;yy� J12ϕ;xxþG11w;xxþG12w;yy�ðJ22� J12ÞΦ1

ε0y ¼ � J12ϕ;yyþ J11ϕ;xxþG21w;xxþG22w;yy�ðJ11� J12ÞΦ1

γ0xy ¼ � J66ϕ;xyþ2G66w;xy ð21Þ

Setting Eq. (21) into Eq. (20) gives the compatibility equation of animperfect S-FGM shell as

J11ϕ;xxxxþðJ66�2J12Þϕ;xxyyþ J22ϕ;yyyyþG21w;xxxxþG12w;yyyy

þðG11þG22�2G66Þw;xxyy

� w2;xy�w;xxw;yyþ2w;xywn

;xy�w;xxwn

;yy�w;yywn

;xx�wxx

R

� �¼ 0

ð22ÞEqs. (19) and (22) are nonlinear equations in terms of variables wand ϕ and used to investigate the nonlinear buckling of imperfecteccentrically stiffened functionally graded thin circular cylindricalshells surrounded on elastic foundation with metal–ceramic–metal layers (S-FGM) and subjected mechanical and thermal loads.

The approximate solutions of w,w* and f, we assumed thefollowing approximate solutions [41,42]

ðw;wnÞ ¼ ðW ;μhÞ sin λmx sin δny ð23Þ

f ¼ A1 cos 2λmxþA2 cos 2δnyþA3 sin λmx sin δnyþð1=2ÞNx0y2

ð24Þλm ¼mπ=L; δn ¼ n=R, W are amplitude of the deflection and μ isimperfection parameter. The coefficients Aiði¼ 1=3Þ are deter-mined by substitution of Eqs. (23) and (24) into Eq. (22) as

A1 ¼δ2n

32J11λ2m

WðWþ2μhÞ; A2 ¼λ2m

32J22δ2n

WðWþ2μhÞ;

A3 ¼λ2m

R½J11λ4mþ J22δ4nþðJ66�2J12Þλ2mδ2n�

W

�½G21λ4mþG12δ

4nþðG11þG22�2G66Þλ2mδ2n�

½J11λ4mþ J22δ4nþðJ66�2J12Þλ2mδ2n�

W ð25Þ

N.D. Duc, P.T. Thang / International Journal of Mechanical Sciences 81 (2014) 17–2520

Substitution of Eqs. (23) and (24) into (19) and applying theGalerkin procedure for the resulting equation yield

1λmδn

2λ2mR

½G21λ4mþG12δ

4nþðG11þG22�2G66Þλ2mδ2n�

½J11λ4mþ J22δ4nþðJ66�2J12Þλ2mδ2n�

�

�½G21λ4mþG12δ

4nþðG11þG22�2G66Þλ2mδ2n�2

½J11λ4mþ J22δ4nþðJ66�2J12Þλ2mδ2n�

�

�λ4mR2

1

½J11λ4mþ J22δ4nþðJ66�2J12Þλ2mδ2n�

�

�D11λ4m�D22δ

4n�ðD12þD21þ4D66Þλ2mδ2n�k1�ðλ2mþδ2nÞk2

2666666666666664

3777777777777775

W

� 116λmδn

λ4mJ22

þ δ4nJ11

!ðWþμhÞWðWþ2μhÞ�λm

δnNx0ðWþμhÞ ¼ 0;

ð26Þ

where m;n are odd numbers. This equation will be used to analyzethe buckling behaviors of eccentrically stiffened S-FGM shellsunder mechanical and thermal loads.

4. Nonlinear buckling analysis

4.1. Thermal buckling analysis

A simply supported S-FGM circular cylindrical shell on twoimmovable edges and under steadily increasing temperature isconsidered (Table 1). The condition expressing the immovabilityon the boundary edges of the shell, i.e. u¼ 0 at x¼ 0; L is justifiedin an average sense as

Z 2πR

0

Z L

0

∂u∂x

dx dy¼ 0 ð27Þ

From Eqs. (6) and (14) one can obtain the following expressionin which Eq. (18) and imperfect have been included

∂u∂x

¼ J22ϕ;yy� J12ϕ;xxþG11w;xxþG12w;yy�12w2

;x�w;xwn

x�ðJ22� J12ÞΦ1

ð28Þ

Substitution of Eqs. (23) and (24) into (28) and then the result intoEq. (27) give fictitious edge compressive loads as

Nx0 ¼J22� J12

J22

� �Φ1þ

18J22

λ2mWðWþ2μhÞ ð29Þ

By using Eq. (11), the thermal parameter Φ1 can be expressedin terms of ΔT:

Φ1 ¼ PhΔT ð30Þ

in which

P ¼ �Z 1

0

Ecαc

1�ðvcþvmctNÞdtþ

Z 1

0

ðEcαmcþEmcαcÞtN1�ðvcþvmctNÞ

dt

"

þZ 1

0

Emcαmct2N

1�ðvcþvmctNÞdt

#ð31Þ

Although ΔT is included in the expression for L due to thetemperature dependence of material properties ðT ¼ T0þΔTÞ, onemay formally express ΔT from Eqs. (26) and (30) as follows:

ΔT ¼ 1P

A22

ðA22 �A12 Þ

" #b1

W

ðW þμÞþb2W ðW þ2μÞ ð32Þ

where

b1 ¼

� 1

m2π2L2hR2h

½B21m4π4þB12n4L4RþðB11 þB22 �2B66 Þm2n2π2L2R�2

½A11m4π4þA22n4L4RþðA66 �2An

12 Þm2n2π2L2R�þ

þ 2Rh

½B21m4π4þB12n4L4RþðB11 þB22 �2B66 Þm2n2π2L2R�½A11m4π4þA22n4L4RþðA66 �2An

12 Þm2n2π2L2R��

� m2π2L2R½A11m4π4þA22n4L4RþðA66 �2An

12 Þm2n2π2L2R�þ

�Dn

11m2π2

L2RR2h

�Dn

22n4L2R

m2π2R2h

�n2ðDn

12 þDn

21 þ4Dn

66 ÞR2h

�k1L2RR

2h

m2π2 �m2π2þn2L2Rm2π2 k2

26666666666666664

37777777777777775

b2 ¼ � 1PðA22 �A12 Þ

A22

16m2π2L2RR2h

m4π4

An

22

þn4L4RAn

11

!�m2π2

8L2RR2h

" #

ð33Þ

Eq. (32) is the analytical form to determine the non-linearrelation between the bending deflection and temperature for bothof the perfect and imperfect shells under the thermal loads (forperfect shell μ¼ 0). Using Eq. (32), we have derived the tempera-

ture change, ΔTb ¼ 1P

A22

ðA22 �A12 Þ

h ib1, which sets them into the buck-

ling state under the condition W ¼ 0.Eq. (32) is temperature dependence which makes it very difficult

to solve. Fortunately, we have applied a numerical technique usingthe iterative algorithm to determine the buckling loads as well as todetermine the deflection – load relations in the buckling period ofthe S-FGM shells. To be more specific, given the material parameterN, the geometrical parameter ðLR;RhÞ and the value of W=h, we canuse these to determine ΔT in (32) as follows: we choose an initialstep for ΔT1 on the right hand side in Eq. (32) with ΔT ¼ 0 (sinceT ¼ T0 ¼ 300 K, the initial room temperature). In the next iterativestep, we replace the known value of ΔT found in the previous stepto determine the right hand side of Eq. (32), ΔT2. This iterativeprocedure will stop at the kth-steps if ΔTk satisfies the con-ditionjΔT�ΔTkjrε. Here, ΔT is a desired solution for the tem-perature and ε is a tolerance used in the iterative steps.

4.2. Mechanical buckling analysis

To clarify the effects of buckling load of the S-FGM shell withmetal–ceramic–metal layers compared with the P-FGM withmetal–ceramic layers, in this section, we consider the effects ofshell under axial compression without temperature, and afterwardcompare the results with those of Bich et al. [31].

Suppose that an imperfect S-FGM circular cylindrical shell issimply supported and subjected to axial compressive loadNx0 ¼ �Pxh, where Px is the average axial stress on the shell'send sections, positive when the shells subjected to axial compres-sion. The boundary conditions considered in this paper are

w¼Mx ¼ 0; Nx ¼Nx0; Nx0 ¼ �Pxh at x¼ 0; L ð34Þ

And Eq. (26) leads to

Px ¼ a1W

ðW þμÞþa2W ðW þ2μÞ ð35Þ

where

a1 ¼

1

m2π2L2RR2h

½G21m4π4þG12n4L4RþðG11 þG22 �2G66 Þm2n2π2L2R�2½J11m4π4þ J22n4L4RþðJ66 �2J12 Þm2n2π2L2R�

� 2Rh

½G21m4π4þG12n4L4RþðG11 þG22 �2G66 Þm2n2π2L2R�½J11m4π4þ J22n4L4RþðJ66 �2J12 Þm2n2π2L2R�

þ m2π2L2R½J11m4π4þ J22n4L4RþðJ66 �2J12 Þm2n2π2L2R�

þDn

11m2π2

L2RR2h

þDn

22n4L2R

m2π2R2h

þn2ðDn

12 þDn

21 þ4Dn

66 ÞR2h

þk1L2RR

2h

m2π2 þm2π2þn2L2Rm2π2 k2

26666666666666664

37777777777777775

N.D. Duc, P.T. Thang / International Journal of Mechanical Sciences 81 (2014) 17–25 21

a2 ¼1

16m2π2L2RR2h

m4π4

J22þn4L4R

J11

!

LR ¼LR; Rh ¼

Rh; W ¼W

h; k2 ¼ k2

h; k1 ¼ k1h; G21 ¼ G21

h;

G12 ¼ G12

h; G11 ¼ G11

h; G22 ¼ G22

h; G66 ¼ G66

h; J11 ¼ J11h;

J22 ¼ J22h; J12 ¼ J12h; J66 ¼ J66h;D11 ¼D11

h3; D22 ¼D22

h3;

D12 ¼D12

h3 ; D21 ¼D21

h3; D66 ¼D66

h3: ð36Þ

For a perfect cylindrical shells μ¼ 0 Eq. (35) leads to

Px0 ¼ a1 ð37Þ5. 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; R¼ 0:5 m; h¼ R=80;

sTx ¼2πRns

; sTy ¼Lnr; ns ¼ 20; nr ¼ 70;

Fig. 2. Nonlinear response of the un-stiffened imperfect S-FGM and P-FGM circularcylindrical shells (without elastic foundations).

Fig. 3. Nonlinear response of the stiffened S-FGM and P-FGM circular cylindricalshells (without elastic foundations).

Fig. 4. Effects of N index on the nonlinear response of the S-FGM circularcylindrical shells under mechanical load.

Fig. 5. Effects of N index on the nonlinear response of the S-FGM circularcylindrical shells under thermal load.

Fig. 6. Effect of imperfection on buckling of eccentrically stiffened S-FGM circularcylindrical shells under mechanical load.

N.D. Duc, P.T. Thang / International Journal of Mechanical Sciences 81 (2014) 17–2522

Fig. 7. Effects of the stiffeners on the nonlinear response of the S-FGM circularcylindrical shells under mechanical load.

Fig. 8. Effects of R/h index on the nonlinear response of S-FGM circular cylindricalshells under mechanical load.

Fig. 9. Effects of R/h index on the nonlinear response of S-FGM circular cylindricalshells under thermal load.

Fig. 10. Effects of imperfection and elastic foundation on the nonlinear response ofS-FGM circular cylindrical shells under mechanical load.

Fig. 11. Effects of ratio L/R on the nonlinear response of S-FGM circular cylindricalshells under mechanical load.

Fig. 12. Effects of ratio L/R on the nonlinear response of S-FGM circular cylindricalshells under thermal load.

N.D. Duc, P.T. Thang / International Journal of Mechanical Sciences 81 (2014) 17–25 23

hTx ¼ hTy ¼ 0:01 m; dTx ¼ dTy ¼ 0:0025 m;

k1 ¼ 100; k2 ¼ 30;

m¼ n¼ 1; ð38Þ

where ns and ns are the number of strings, rings of the shells,respectively.

Figs. 2 and 3 show a comparison between the present resultsfor the S-FGM shell and Bich's results [31] for the P-FGM shell withthe same geometrical parameters. In Fig. 2, we consider theimperfect shell without stiffeners and elastic foundation withN¼ 2, we can see that the solid line is much higher than the dashline, revealing the loading capacity of the S-FGM circular cylind-rical shell with metal–ceramic–metal layers is higher than P-FGMshell with metal–ceramic layers. In Fig. 3, we consider the shellswith stiffeners, imperfection but without the elastic foundationand realized that the solid line (μ¼ 0 – perfect shell) is alwayshigher than dash line (μ¼ 0:1 – imperfect shell), and the loadingcapacity of the shell with stiffeners (Fig. 3) is better than the shellwithout stiffeners (Fig. 2).

Fig. 4 and Fig. 5 show the influence of the volume ratio andimperfection on buckling behavior of S-FGM cylindrical shellunder mechanical and thermal loads, respectively. From twofigures, we can see that when N is increased, the curve becomeslower; this means the weaker loading capacity of the shells. This isright 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 loadingcapacity of the perfect shell is a little better than imperfect one.

Fig. 6 shows the influence of imperfection of initial shape onbuckling behavior of S-FGM shell under mechanical load. Itindicates that the loading capacity of the shell is decreased whenμ is increased. Fig. 7 shows the comparison of S-FGM axialcompressive cylinder shells with stiffeners and without stiffeners.From the figure, we can see that in both cases, the perfect ðμ¼ 0Þand imperfect ðμ¼ 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 9 show the influence of radius ratio on the thicknessR=h¼ ð100;150;200Þ on buckling behavior of S-FGM cylindricalshell under mechanical and thermal loads. From these two figures,we can see that when R=h is increased, the curve becomes lower.This is right because when R=h is increased, the circular cylindricalshell becomes thinner and the load capacity is decreased.

Fig. 10 presents the effects of the elastic foundations onbuckling 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 loadingcapacity is higher than the Winkler foundation k1.

Figs. 11 and 12 show the influence of the ratio of the length onradius L=R on buckling behavior of S-FGM cylindrical shell undermechanical and thermal loads. As shown in Fig. 11, the mechan-ical loading capacity of the shell is increased when the ratio ofL=R is increased. Fig. 12 shows that the thermal loading capacityof the shell is decreased when the ratio of L=R is increased.

6. Concluding remarks

This paper presents an analytical investigation on the nonlinearbuckling response for imperfect eccentrically stiffened S-FGM thincircular cylindrical shells with metal–ceramic–metal layers sur-rounded on elastic foundation in thermal environment. The shellsubjected to axial compression and thermal loads. Both S-FGMshell and stiffeners are deformed by temperature. The formula-tions are based on the Donnell shell theory taking into accountgeometrical nonlinearity, initial geometrical imperfection,temperature-dependent properties and the Lekhnitsky smearedstiffeners technique with Pasternak type elastic foundation.

Using the Galerkin method and stress function, effects ofmaterial and geometrical properties, temperature, elastic founda-tion and eccentrically outside stiffeners on the buckling loadingcapacity of the imperfect eccentrically stiffened S-FGM shell inthermal environment are analyzed and discussed. The resultsshowed that the addition of stiffeners increases the mechanicaland thermal loading capacity of the FGM shell, and the loadingcapacity of the S-FGM shell with metal–ceramic–metal layers ishigher than P-FGM shell with metal–ceramic layers with the samegeometrical parameters. Some results were compared with theones of the other authors.

Acknowledgments

This paper was supported by Project “Nonlinear analysis onstability and dynamics of functionally graded shells with specialshapes”of Vietnam National University, Hanoi. The authors aregrateful for this support.

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