International Journal of Mechanical Sciences 81 (2014) 17–25 Contents lists available at ScienceDirect International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci Nonlinear buckling of imperfect eccentrically stiffened metal–ceramic–metal S-FGM thin circular cylindrical shells with temperature-dependent properties in thermal environments Nguyen Dinh Duc n, Pham Toan Thang Vietnam National University, Ha Noi, 144 Xuan Thuy, Cau Giay, Ha Noi, Viet Nam art ic l e i nf o a b s t r a c t Article history: Received 11 November 2013 Received in revised form 10 January 2014 Accepted 20 January 2014 Available online February 2014 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 temperaturedependent 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 Keywords: Nonlinear buckling S-FGM with metal–ceramic–metal layers Eccentrically stiffened cylindrical shells Imperfection Elastic foundation Thermal environment Introduction The material has variable mechanical property with international name Functionally 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 functionally 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 industries Since FGM was researched and developed, the shell calculations 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 n Corresponding author Tel.: ỵ 84 37547978; fax: ỵ84 37547424 E-mail address: ducnd@vnu.edu.vn (N.D Duc) http://dx.doi.org/10.1016/j.ijmecsci.2014.01.016 0020-7403 & 2014 Elsevier Ltd All rights reserved shells A few case studies on the stability of FGM cylindrical shells are introduced below: Lanhe et al [3] have used the uncoupled equation and Shahsiah and Eslami [4] have used couple of 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 cylindrical 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 investigate buckling behavior of functionally graded stiffened cylindrical shells reinforced by rings and stringer subjected to axial compression The buckling analysis of short cylindrical shells surrounded by an elastic medium was carried out by Naili and Oddou [10] Mirzavand and Eslami [11] presented the buckling analysis of 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 18 N.D Duc, P.T Thang / International Journal of Mechanical Sciences 81 (2014) 17–25 Nomenclature h m n N L R EðzÞ; Em ; thickness of shell number of half waves axial direction number of wave in circumferential direction volume-fraction index length of the shell radius of the shell Ec Young's modulus of shell, metal, ceramic respectively stiffeners in the buckling of isotropic cylindrical shells under axial compressive load Matsunaga [13] examined the free vibration and linear buckling of FGM cylindrical shells based on a twodimensional 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 Some researchers have used the first-order and high-order theories for buckling analysis of the perfect and imperfect cylindrical 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 compression Shen and Li [20] studied influence of the local 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 the first-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 nonlinearity is presented by Shen [22] By using higher order shear deformation theory, this author [23] continued to investigate the post-buckling of FGM hybrid cylindrical shells in thermal environments under axial loading Shen [24] studied the postbuckling response of a shear deformable functionally graded cylindrical shell of finite 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 of finite length embedded in a large outer elastic medium [25] C Tx ; C Ty sTx ; sTy ATx ; ATy I Tx ; I Ty zTx ; zTy T T dx ; dy T T hx ; hy μh coupling parameters spacing of the stringer and ring stiffeners, respectively cross-section areas of stiffeners moment of inertia of stiffeners cross section relative to the shell middle surface eccentrically of stiffeners with respect to the middle surface the shell width of the stringer and ring stiffened, respectively height of the stringer and ring stiffeners, respectively known imperfect amplitude For dynamic analysis of FGM cylindrical shells, Ng et al [26] and Darabi et al [27] presented respectively linear and nonlinear parametric 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 unstiffened 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 functionally 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, calculations become complex Duc and Thang [33] studied an analytical approach to investigate the nonlinear static buckling and postbuckling for imperfect eccentrically stiffened functionally graded thin circular cylindrical shells surrounded on elastic foundation with ceramic–metal–ceramic layers (S-FGM) and subjected to axial 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 properties 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 Fig Configuration of an eccentrically stiffened S-FGM circular cylindrical shell N.D Duc, P.T Thang / International Journal of Mechanical Sciences 81 (2014) 17–25 response of the eccentrically stiffened S-FGM shell in thermal environments are analyzed and discussed 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] The volume fractions of metal and ceramic, V m and V c is assumed by the Sigmoid power-law distribution (S-FGM) [34] < 2z ỵ h N ; N Z 0; À h r z r h 1ị V c zị ẳ ; V m zị ỵ V c zị ẳ : 2z ỵ h N ; rz r h h 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 ẵEz; Tị; z; Tị; z; Tị; z; Tị; Kz; Tị ẳ ẵEm Tị; m Tị; m Tị; m Tị; K m Tị ỵ ½Ecm ðTÞ; νcm ðTÞ; ρcm ðTÞ; αcm ðTÞ; K cm Tị < 2z ỵ h N ; N Z 0; À h rz r h  : 2z ỵ h N ; r z r h h ð2Þ where Ecm ðTÞ ¼ Ec ðTÞ À Em ðTÞ; ρcm ðTÞ ¼ ρc Tị m Tị; cm Tị ẳ c Tị m Tị; cm Tị ẳ c Tị m Tị; K cm Tị ẳ K c Tị K m ðTÞ; ð3Þ From Eq (2) we can see that for S-FGM (Fig 1): E ¼ Em at z ¼ À h=2 and z ¼ h=2 (metal) and E ¼ Ec at z ¼ (ceramic) A material coefficient 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 [36–38] Pr ¼ P ðP À T ỵ ỵ P T ỵP T ỵ P T ị; 4ị In which T ẳ T ỵ Tzị and T ¼ 300 K (room temperature); P À ; P ; P ; P ; P are coefficients characterizing of the constituent materials The material properties for the later one have been determined by (4) at room temperature, i.e T ¼ 300 K The shell–foundation interaction is represented by the Pasternak model as q ¼ k1 w À k2 ∇2 w; 5ị where ẳ =x ỵ =y , w is the deflection of the shell, k1 is Winkler foundation modulus and k2 is the shear layer foundation stiffness of the Pasternak model 2 Theoretical formulation The strains at the middle surface relating to the displacement components u; v; w based on the von Karman geometrical nonlinearity assumption are of the form [39,40] 0x ẳ u;x ỵ w;x ị2 ; w R 0y ẳ v;y ỵ w;y ị2 ; 0xy ẳ u;y ỵ v;x ỵ w;x w;y 6ị 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] x ẳ 0x ỵ zkx ; y ẳ 0y ỵzky ; xy ẳ 0xy þ 2zkxy kx ¼ Àw;xx ; ky ¼ À w;yy ; 19 kxy ẳ w;xy ; 7ị In which ε0x ; ε0y are the normal strains and ε0xy 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 dened as sh ssh x ; sy ị ẳ ssh xy ẳ Ez; Tị ẵx ; y ị ỵ y ; x ị ỵ ịTzị1; 1ị; z; Tị Ez; Tị ; 2ẵ1 ỵ νðz; TÞ xy ð8Þ where ΔT is temperature rise from stress free initial state, and more generally, T ẳ Tzị; Eðz; TÞ; νðz; TÞ are the FGM shell's elastic moduli which are determined by (2) For stiffeners in thermal environments with temperaturedependent properties, we have proposed its form adapted from Ref [32] as follows: st ðsst x ; sy Þ ¼ E ð ε x ; ε y Þ À E0 α0 ðTÞΔðTÞð1; 1Þ: À 2ν0 ðTÞ ð9Þ here, E0 ẳ E0 Tị; ẳ Tị; ¼ α0 ðTÞ 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 E0 ¼ 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: ! ET0 ATx Nx ẳ I 10 ỵ T 0x ỵI 20 0y ỵ I 11 ỵ C Tx ịkx ỵ I21 ky ỵ ; sx ! ET0 ATy Ny ẳ I 20 0x ỵ I 10 ỵ T 0y ỵ I21 kx ỵ I11 ỵ C Ty ịky ỵ ; sy Nxy ẳ I 30 0xy ỵ 2I 31 kxy ; ! ET0 I Tx kx ỵ I 22 ky ỵ ; sTx ! ET0 I Ty T M y ẳ I 21 0x ỵ I 11 þ C y Þε0y þ I 22 kx þ I 12 ỵ T ky ỵ ; sy M x ẳ I 11 ỵ C Tx ị0x ỵ I 21 0y ỵ I 12 ỵ M xy ẳ I 31 0xy ỵ 2I 32 kxy ; 10ị The relation (10) is our most important finding, where I ij ði ¼ 1; 2; 3; j ¼ 0; 1; 2Þ: Z h=2 Ezị zj dz; j ẳ 0; I 1j ¼ À h=2 À νðzÞ Z h=2 EðzÞνðzÞ j z dz; j ¼ 0; I 2j ¼ À h=2 À νðzÞ Z h=2 EðzÞ zj dz ẳ I 1j I 2j ị; j ¼ 0; I 3j ¼ À h=2 2½ð1 ỵ zị T I Tx ẳ T T dx hx ị3 ỵ ATx zTx ị2 ; 12 C Tx ẳ E0 ATx zTx ; sTx zTx ẳ hx ỵh ; T T ATx ¼ dx sTx ; C Ty ¼ I Ty ¼ E0 ATy zTy sTy T T zTy ẳ T dy hy ị3 12 ỵ ATy zTy ị2 ; ; T hy ỵ h ; T ATy ẳ dy sTy : Z h=2 Ezịzị ; ị ẳ Tzị1; zịdz: h=2 νðzÞ ð11Þ 20 N.D Duc, P.T Thang / International Journal of Mechanical Sciences 81 (2014) 17–25 where the coupling parameters C x ; C y are negative for outside stiffeners and positive for inside one; I x ; I y are the second moments of cross-section areas; sx ; sy are the spacing of the longitudinal and transversal stiffeners; zx ; zy are 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 crosssection 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 ATx ; ATy After the thermal deformation process, the geometric shapes of stiffeners which can be determined as follows [32]: dx ẳ dx ẵ1 ỵ m Tzị; dy ẳ dy ẵ1 ỵ m Tzị; hx ẳ hx ẵ1 ỵ m Tzị; hy ẳ hy ẵ1 ỵ m Tzị; zTx ẳ zx ẵ1 ỵ m Tzị; zTy ẳ zy ẵ1 ỵ m Tzị; sTx ẳ sx ẵ1 ỵ m Tzị; sTy ẳ sy ẵ1 ỵ m Tzị: T T T 12ị N x;x ỵ Nxy;y ẳ 13aị N xy;x ỵ N y;y ẳ 13bị M x;xx ỵ 2M xy;xy ỵ M y;yy ỵ 13cị 14ị where I 20 J 12 ¼ Δ J 66 ¼ I 30 ; G12 ¼ J 22 I 21 À J 12 I 11 ỵ C Ty ị; G21 ẳ J 11 I 21 J 12 I 11 ỵC Tx ị; I 31 ; I 30 ẳ ET AT I 10 ỵ T x sx 15ị ! I 10 ỵ ET0 ATy sTy D66 ẳ I 32 I 31 G66 ð17Þ ϕðx; yÞ is stress function defined by Nx ¼ ϕ;yy ; Ny ¼ ϕ;xx ; N xy ẳ ;xy 18ị For an imperfect S-FGM circular cylindrical shell Eq (16) is modified into form as G21 ;xxxx ỵ G11 ỵ G22 2G66 ị;xxyy ỵ G12 ;yyyy D11 w;xxxx D22 w;yyyy ;xx R ỵ ;yy w;xx ỵ wn;xx ị 2;xy w;xy ỵ wn;xy ị þ ϕ;xx ðw;yy þ wn;yy Þ þ q À k1 w ỵk2 w ẳ 19ị In which wn ðx; yÞ is a known function representing initial small imperfection of the shell The geometrical compatibility equation for imperfect cylindrical shells written as R 0x;yy ỵ 0y;xx 0xy;xy ẳ w;xx ỵ w2;xy w;xx w;yy ỵ2w;xy wn;xy À w;xx wn;yy À w;yy wn;xx ð20Þ From the constitutive relations Eq (14) in conjunction with Eq (18) one can write ε0x ¼ J 22 ϕ;yy À J 12 ϕ;xx þ G11 w;xx þ G12 w;yy ÀðJ 22 À J 12 ị1 0y ẳ J 12 ;yy ỵ J 11 ;xx ỵ G21 w;xx ỵ G22 w;yy J 11 À J 12 ÞΦ1 ð21Þ Setting Eq (21) into Eq (20) gives the compatibility equation of an imperfect S-FGM shell as J 11 ;xxxx ỵJ 66 2J 12 ị;xxyy þ J 22 ϕ;yyyy þ G21 w;xxxx þ G12 w;yyyy 22ị G22 ẳ J 11 I 11 ỵ C Ty Þ À J 12 I 21 ; and D21 ẳ I 22 G11 I 21 I 11 ỵ C Ty ịG21 ; ỵ G11 ỵ G22 2G66 ịw;xxyy wxx ẳ0 w2;xy w;xx w;yy ỵ 2w;xy wn;xy Àw;xx wn;yy À w;yy wn;xx À R ; G11 ẳ J 22 I 11 ỵ C Tx ị À J 12 I 21 ; G66 ¼ À G12 I 12 I 22 ỵ C Ty ịG22 ; 0xy ẳ J 66 ;xy ỵ 2G66 w;xy Calculated from Eq (10) ε0x ¼ J 22 Nx À J 12 Ny ỵ G11 w;xx ỵ G12 w;yy J 22 J 12 ị1 ; 0y ẳ J 12 N x ỵ J 11 N y ỵ G21 w;xx ỵ G22 w;yy À ðJ 11 À J 12 ÞΦ1 ; 0xy ẳ J 66 N xy ỵ2G66 w;xy ; ! ET0 ATx I ỵ J 11 ẳ ; 10 sTx ! ET0 ATy I 10 ỵ T J 22 ¼ ; Δ sy sTy D12 ¼ I 22 G22 I 21 I 11 ỵ C Tx ịG12 ; ỵ 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] ET0 ATy À ðD12 þ D21 þ 4D66 Þw;xxyy T Ny þ N x w;xx ỵ 2Nxy w;xy ỵ Ny w;yy R ỵ q k1 w ỵ k2 w ẳ D22 ẳ I 12 ỵ Eqs (19) and (22) are nonlinear equations in terms of variables w and ϕ and used to investigate the nonlinear buckling of imperfect eccentrically stiffened functionally graded thin circular cylindrical shells 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 the following approximate solutions [41,42] w; wn ị ẳ W; μhÞ sin λm x sin δn y ! ÀI 220 : f ẳ A1 cos 2m x ỵ A2 cos 2n yỵ A3 sin m x sin n yỵ 1=2ịN x0 y2 Substituting once again Eq (14) into the expression of M ij in (10), then M ij into Eq (13c) leads to N x;x ỵ Nxy;y ẳ 0; N xy;x ỵ N y;y ẳ 0; G21 ;xxxx ỵ G11 þ G22 À2G66 Þϕ;xxyy þ G12 ϕ;yyyy where E T AT D11 ẳ I 12 ỵ T x G21 I 21 I 11 ỵ C Tx ịG11 ; sx 24ị m ẳ m =L; n ẳ n=R, W are amplitude of the deflection and μ is imperfection parameter The coefcients Ai i ẳ 1=3ị are determined by substitution of Eqs (23) and (24) into Eq (22) as A1 ẳ D11 w;xxxx D22 w;yyyy D12 ỵ D21 ỵ 4D66 ịw;xxyy ỵ Ny ỵ N x w;xx ỵ 2Nxy w;xy ỵ Ny w;yy ỵ q k1 w ỵ k2 w ẳ 0; R 23ị 16ị A3 ẳ 2n 32J 11 m WW ỵ 2hị; A2 ẳ 2m 32J 22 n 2 m 2 RẵJ 11 m ỵ J 22 n ỵ J 66 2J 12 ị m n W ẵG21 m ỵ G12 n ỵG11 ỵ G22 2G66 ịm n WW ỵ 2hị; 2 ẵJ 11 m ỵ J 22 n ỵ J 66 2J 12 ị m n W ð25Þ N.D Duc, P.T Thang / International Journal of Mechanical Sciences 81 (2014) 17–25 Substitution of Eqs (23) and (24) into (19) and applying the Galerkin procedure for the resulting equation yield 2 2m ẵG21 m ỵ G12 n þ ðG11 þ G22 À 2G66 Þλm δn À 2 R ½J 11 m ỵ J 22 n ỵ J 66 2J 12 Þλm δn 7 ẵG ỵ G ỵ G ỵ G 2G Þλ2 δ2 2 21 12 11 22 66 6À n m m n À 2 7W ẵJ 11 m ỵ J 22 n ỵ J 66 2J 12 ịm δn λm δ n λm 6À À R2 2 ẵJ 11 m ỵ J 22 n ỵ J 66 2J 12 ịm n 2 D11 m D22 n D12 ỵ D21 þ 4D66 Þλm δn À k1 À ðλm þ δn ịk2 ! 4m 4n m ỵ N W ỵ hị ẳ 0; W ỵ hịWW ỵ 2hị δn x0 16λm δn J 22 J 11 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 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 u ¼ at x ¼ 0; L is justified in an average sense as Z 2π R Z L ∂u dx dy ẳ x 27ị From Eqs (6) and (14) one can obtain the following expression in which Eq (18) and imperfect have been included ∂u ¼ J 22 ;yy J 12 ;xx ỵ G11 w;xx ỵ G12 w;yy À w2;x À w;x wnx ÀðJ 22 À J 12 ÞΦ1 ∂x ð28Þ 21 where ẵB21 m4 ỵ B12 n4 L4R ỵ B11 þ B22 À 2B66 Þm2 n2 π L2R 2 þ 6À 2 2 n m π L h Rh ½A11 m4 π ỵ A22 n4 LR ỵ A66 2A12 ịm2 n2 π LR 7 4 4 2 2 ½B21 m ỵ B12 n LR ỵ B11 ỵ B22 2B66 ịm n LR 6ỵ Rh ẵA m4 ỵ A n4 L4 ỵ A 2An ịm2 n2 L2 11 22 66 12 R R b1 ¼ 2 m L R ỵ n ẵA11 m4 ỵ A22 n4 LR þ ðA66 À 2A12 Þm2 n2 π LR 7 n n n n n 2 2 2 D11 m2 D22 n4 LR D ỵ D ỵ 4D ị k L R m ỵ n L R h R 12 21 66 À À À n À À k 2 2 2π2 2π2 2 m m L R Rh m π Rh Rh " ! # A22 m4 π n4 L4R m2 ỵ b2 ẳ À 2 2 PðA22 À A12 Þ 16m2 π LR Rh 8LR Rh An22 An11 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 shell μ ¼ 0) Using Eq (32), we have derived the temperah i A22 b1 , which sets them into the buckture change, ΔT b ¼ 1P ðA22 À A12 Þ ling state under the condition W ¼ 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 parameter ðLR ; Rh Þ and the value of W=h, we can use these to determine ΔT in (32) as follows: we choose an initial step for ΔT on the right hand side in Eq (32) with ΔT ¼ (since T ¼ T ¼ 300 K, the initial room temperature) In the next iterative step, we replace the known value of ΔT found in the previous step to determine the right hand side of Eq (32), ΔT This iterative procedure will stop at the kth-steps if ΔT k satisfies the conditionjΔT À ΔT k jr ε Here, ΔT is a desired solution for the temperature and ε is a tolerance used in the iterative steps 4.2 Mechanical buckling analysis By using Eq (11), the thermal parameter Φ1 can be expressed in terms of ΔT: To clarify the effects of buckling load of the S-FGM shell with metal–ceramic–metal layers compared with the P-FGM 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 Nx0 ¼ À P x h, where P x is the average axial stress on the shell's end sections, positive when the shells subjected to axial compression The boundary conditions considered in this paper are Φ1 ¼ PhΔT w ¼ M x ¼ 0; N x ¼ N x0 ; Nx0 ¼ À P x h at x ¼ 0; L Substitution of Eqs (23) and (24) into (28) and then the result into Eq (27) give fictitious edge compressive loads as J ÀJ N x0 ¼ 22 12 ỵ WW ỵ 2hị 29ị 8J 22 m J 22 in which "Z Pẳ ỵ 1 ð34Þ And Eq (26) leads to Z 30ị E c c dt ỵ vc ỵvmc t N Þ Emc αmc t 2N dt À vc ỵ vmc t N ị Z Ec mc ỵ Emc c ịt N dt vc ỵ vmc t N ị P x ẳ a1 W W þ μÞ þa2 W ðW þ 2μÞ ð35Þ where # ð31Þ Although ΔT is included in the expression for L due to the temperature dependence of material properties ðT ¼ T ỵ Tị, one may formally express T from Eqs (26) and (30) as follows: " # A22 W T ẳ 32ị ỵ b2 W W ỵ 2ị b1 P A22 A12 ị W ỵ ị ẵG21 m4 ỵG12 n4 L4R ỵG11 ỵ G22 À2G66 Þm2 n2 π L2R 2 2 2 ẵJ 11 m4 ỵJ 22 n4 L4R ỵJ 66 2J 12 ịm2 n2 L2R m π LR Rh 6 ẵG m4 ỵG n4 L4 ỵG ỵ G À 2G Þm2 n2 π L2 21 12 11 22 66 R R 6À Rh ẵJ 11 m4 ỵJ 22 n4 L4R ỵJ 66 À 2J 12 Þm2 n2 π L2R a1 ẳ 6 m2 L2R 6ỵ ẵJ m4 ỵJ n4 L4 ỵJ 2J Þm2 n2 π L2 11 22 66 12 R R n n n Dn m2 π Dn n4 L2 k1 L2 R2 m2 ỵ n2 L2R ỵ 11 22 R D12 ỵD21 ỵ4D66 ị ỵ ỵn ỵ 2R 2h þ k2 2 2 2 m π m2 π LR Rh m π Rh Rh 7 7 7 7 7 7 7 22 N.D Duc, P.T Thang / International Journal of Mechanical Sciences 81 (2014) 17–25 a2 ¼ 16m2 π L2R R2h m4 π J 22 ỵ n4 L4R ! J 11 L R W k2 G21 ; Rh ¼ ; W ¼ ; k2 ¼ ; k1 ¼ k1 h; G21 ¼ ; R h h h h G12 G11 G22 G66 ; G11 ¼ ; G22 ¼ ; G66 ¼ ; J 11 ¼ J 11 h; G12 ¼ h h h h D11 D22 J 22 ¼ J 22 h; J 12 ¼ J 12 h; J 66 ¼ J 66 h; D11 ¼ ; D22 ¼ ; h h D12 D21 D66 D12 ¼ ; D21 ¼ ; D66 ¼ : h h h LR ẳ 36ị For a perfect cylindrical shells ẳ Eq (35) leads to P x0 ẳ a1 37ị Numerical result and discussion To illustrate, we consider a symmetric S-FGM circular cylindrical shell with the parameters as follows: L ¼ 0:75 m; R ¼ 0:5 m; h ¼ R=80; 2π R L sTx ¼ ; sTy ¼ ; ns ¼ 20; nr ¼ 70; ns nr Fig Effects of N index on the nonlinear response of the S-FGM circular cylindrical shells under mechanical load Fig Nonlinear response of the un-stiffened imperfect S-FGM and P-FGM circular cylindrical shells (without elastic foundations) Fig Effects of N index on the nonlinear response of the S-FGM circular cylindrical shells under thermal load Fig Nonlinear response of the stiffened S-FGM and P-FGM circular cylindrical shells (without elastic foundations) Fig Effect of imperfection on buckling of eccentrically stiffened S-FGM circular cylindrical shells under mechanical load N.D Duc, P.T Thang / International Journal of Mechanical Sciences 81 (2014) 17–25 Fig Effects of the stiffeners on the nonlinear response of the S-FGM circular cylindrical shells under mechanical load 23 Fig 10 Effects of imperfection and elastic foundation on the nonlinear response of S-FGM circular cylindrical shells under mechanical load Fig Effects of R/h index 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 Effects of R/h index on the nonlinear response of S-FGM circular cylindrical shells under thermal load Fig 12 Effects of ratio L/R on the nonlinear response of S-FGM circular cylindrical shells under thermal load 24 N.D Duc, P.T Thang / International Journal of Mechanical Sciences 81 (2014) 17–25 Table Material properties of the constituent materials of the considered FGM shells [36–38] Material Properties P0 PÀ1 P1 P2 P3 Si3N4 (ceramic) E (Pa) ρ (kg/m3) α (K À 1) K (W/m K) ν 348.43e9 2370 5.8723e À 13.723 0.24 0 0 À 3.70e À 9.095e À 0 2.160e À 0 0 À 8.946e À 11 0 0 SUS304 (Metal) E (Pa) ρ (kg/m3) α (K À 1) K (W/m K) ν 201.04e9 8166 12.330eÀ 15.379 0.3177 0 0 3.079eÀ 8.086e À 0 À 6.534e À 0 0 0 0 T T hx ¼ hy ¼ 0:01 m; k1 ¼ 100; m ¼ n ¼ 1; T T dx ¼ dy ¼ 0:0025 m; k2 ¼ 30; ð38Þ where ns and ns are the number of strings, rings of the shells, respectively Figs and show a comparison between the present results for the S-FGM shell and Bich's results [31] for the P-FGM shell with the same geometrical parameters In Fig 2, we consider the imperfect shell without stiffeners and elastic foundation with N ¼ 2, we can see that the solid line is much higher than the dash line, revealing the loading capacity of the S-FGM circular cylindrical shell with metal–ceramic–metal layers is higher than P-FGM shell with metal–ceramic layers In Fig 3, we consider the shells with stiffeners, imperfection but without the elastic foundation and realized that the solid line (μ ¼ – perfect shell) is always higher than dash line (μ ¼ 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 and Fig 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 (Em o Ec ) 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 shows 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 μ is increased Fig shows the comparison of S-FGM axial compressive cylinder shells with stiffeners and without stiffeners From the figure, we can see that in both cases, the perfect ẳ 0ị and imperfect ẳ 0:1ị cylindrical shells with stiffeners can withstand higher compression than the ones without the stiffeners This clearly shows the better effectiveness of stiffeners Figs and show the influence of radius ratio on the thickness R=h ẳ 100; 150; 200ị on buckling behavior of S-FGM cylindrical shell 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 cylindrical shell becomes thinner and the load capacity is decreased Fig 10 presents the effects of the elastic foundations on buckling behavior of perfect (μ ¼ 0) and imperfect (μ ¼ 0:1) S-FGM circular cylindrical shells under mechanical load Obviously, buckling load is enhanced due to the presence of elastic foundations and the effect of Pasternak foundation k2 on the loading capacity is higher than the Winkler foundation k1 Figs 11 and 12 show 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 in Fig 11, the mechanical loading capacity of the shell is increased when the ratio of L=R is increased Fig 12 shows that the thermal loading capacity of the shell is decreased when the ratio of L=R is increased 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 surrounded 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 formulations are based on the Donnell shell 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, effects of material and geometrical properties, temperature, elastic foundation 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 Reference [1] Yamanouchi M, Koizumi M, Hirai T, Shiota I., In: Proceedings of the first international symposium on functionally gradient materials, Japan; 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(22) are nonlinear equations in terms of variables w and ϕ and used to investigate the nonlinear buckling of imperfect eccentrically stiffened functionally graded thin circular cylindrical shells. .. approach to investigate the nonlinear static buckling and postbuckling for imperfect eccentrically stiffened functionally graded thin circular cylindrical shells surrounded on elastic foundation with. .. of N index on the nonlinear response of the S-FGM circular cylindrical shells under mechanical load Fig Nonlinear response of the un -stiffened imperfect S-FGM and P-FGM circular cylindrical shells