DSpace at VNU: Nonlinear response of imperfect eccentrically stiffened FGM cylindrical panels on elastic foundation subjected to mechanical loads

European Journal of Mechanics A/Solids 46 (2014) 60e71 Contents lists available at ScienceDirect European Journal of Mechanics A/Solids journal homepage: www.elsevier.com/locate/ejmsol Nonlinear response of imperfect eccentrically stiffened FGM cylindrical panels on elastic foundation subjected to mechanical loads Nguyen Dinh Duc*, Tran Quoc Quan University of Engineering and Technology, Vietnam National University, Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Viet Nam a r t i c l e i n f o a b s t r a c t Article history: Received 23 November 2013 Accepted February 2014 Available online 18 February 2014 In the present paper, the nonlinear response of eccentrically stiffened FGM cylindrical panels on elastic foundation subjected to mechanical loads is presented Material properties are graded in the thickness direction of the FGM panel according to a simple power law distribution By applying Bubnov-Galerkin method, the Lekhnitsky smeared stiffeners technique with Pasternak type elastic foundation and stress function, explicit relations of load-deflection curves for simply supported eccentrically stiffened FGM panels are determined Numerical results are given for evaluate effects of material and geometrical properties, elastic foundation and eccentrically outside stiffeners on the buckling and postbuckling of the FGM panels The obtained results are validated by comparing with those in the literature Ó 2014 Elsevier Masson SAS All rights reserved Keywords: Nonlinear response Eccentrically stiffened cylindrical panels Elastic foundations Introduction Flat and curved panels are important components of space structures and nuclear reactors The static behavior of panels attracts special attention of a lot of authors in the world Kabir and Chaudhur (1993) presented a direct Fourier approach for the analysis of thin finite-dimensional cylindrical shells Alijani and Aghdam (2009), by applying the extended Kantorovich method, given a semi-analytical solution for stress analysis of moderately thick laminated cylindrical panels with various boundary conditions Dennis et al (1994), Yamada and Croll (1989) investigated instability, buckling behaviors of pressure loaded cylindrical panels Shahraki et al (2013) obtained nonlinear buckling analysis of laminated composite curved panels constrained by Winkler tensionless foundation The study involving postbuckling of laminated cylindrical panels loaded by improved arc-length method can be found in the paper of Kweon and Hong (1994) Functionally Graded Materials (FGMs) are microscopically inhomogeneous made from a mixture of metal and ceramic, and its mechanical properties vary smoothly and continuously from one surface to the other Functionally graded structures such as cylindrical panels and cylindrical shells in recent years, play the important part in the modern industries Therefore, researchers on stability problems of FGMs structures have received considerable attention Regarding to the static stability of FGM shells, Yang et al * Corresponding author E-mail address: ducnd@vnu.edu.vn (N.D Duc) http://dx.doi.org/10.1016/j.euromechsol.2014.02.005 0997-7538/Ó 2014 Elsevier Masson SAS All rights reserved (2006) presented investigations on thermo-mechanical postbuckling of FGM cylindrical panels with temperature-dependent properties Sofiyev (2005) studied the stability of compositionally graded ceramic-metal cylindrical shells under periodic axial impulsive loading Closed-form solutions of free-vibration problems of simply supported multilayered shells made of functionally graded material have been examined by Cinefra et al (2010a) Cinefra et al (2010b) also considered the thermo-mechanical analysis of a simply supported functionally graded shell Using the first-order shear deformation shell theory and von Karman strains, Liew et al (2012) investigated post-buckling responses of functionally graded cylindrical shells under axial compression and thermal loads Li and Batra (2006) determined critical compressive loads for hybrid cylindrical shells with FGM between layer Elasticity solution is presented in Javanbakht et al (2011) for finitely long, simply-supported, functionally graded shallow and nonshallow shell panel with two piezoelectric layers under pressure and electrostatic excitation Some investigations on postbuckling of FGM cylindrical panels and cylindrical shells subjected to axial loading or pressure loading in thermal environments are presented by Shen (2002), Shen and Noda (2005) Recently, Shen and Wang (2014) focused on the large amplitude vibration behavior of a shear deformable FGM cylindrical panel resting on elastic foundations in thermal environments Alibeigloo (2014) studied free vibration behavior of functionally graded carbon nanotubereinforced composite cylindrical panel embedded in piezoelectric layers with simply supported boundary conditions is investigated by using three-dimensional theory of elasticity He et al (2010), Alibeigloo and Chen, 2010 published the results on the three- N.D Duc, T.Q Quan / European Journal of Mechanics A/Solids 46 (2014) 60e71 dimensional elasticity solution for static analysis of a functionally graded material cylindrical panel with simply supported edges Duc and Tung (2010a, 2010b) investigated the nonlinear response of thin and moderately thick FGM cylindrical panels subjected to mechanical and thermo-mechanical loads In recent years, the investigation of nonlinear stability of FGM structures on elastic foundations takes much attention In spite of practical importance and increasing use of FGM structures, investigations on the effects of elastic media on the response of FGM plates and shells are comparatively scarce Tung (2013) has studied the postbuckling behavior of FGM cylindrical panels with tangential edge constraints and resting on elastic foundations (without stiffeners) Duc and Quan (2013) researched the nonlinear postbuckling of imperfect eccentrically stiffened P-FGM double curved thin shallow shells on elastic foundation in thermal environments For FGM structures, many researches focused on nonlinear stability of un-stiffened plates and shells However, the investigation on static and dynamic of reinforced functionally graded plates and shells by outside stiffeners have received comparatively little attention, this may be because of their inherent complexity Najafizadeh et al (2009) used both analytical and finite element methods to obtain the critical loads of FGM stiffened cylindrical shells under axial compression Bich et al (2011) studied the nonlinear static postbuckling of eccentrically stiffened thin ESFGM plates and shallow shells The nonlinear dynamical analysis of eccentrically stiffened functionally graded cylindrical panels based on classical theory is investigated by Bich et al (2012) Duc (2013) investigated nonlinear dynamic response of imperfect eccentrically stiffened doubly curved ESP-FGM shallow shells on elastic foundations There is no investigation on the nonlinear static stability of FGM cylindrical panel with stiffeners on elastic foundation This paper considered the nonlinear postbuckling for imperfect eccentrically stiffened FGM cylindrical panels on elastic foundations using a simple power-law distribution under mechanical loads (ESP-FGM) Using Bubnov-Galerkin method and using stress function, the effects of geometrical and material properties, elastic foundation and eccentrically outside stiffeners on the nonlinear response of the imperfect eccentrically stiffened ESP-FGM cylindrical panels are analyzed and discussed 61 Eccentrically stiffened FGM cylindrical panels on elastic foundations Consider an eccentrically stiffened functionally graded cylindrical panel as shown in Fig The radii of curvature, thickness, axial length and length of the panel are R, h, a and b, respectively For ES-FGM panel, the volume fractions of constituents are assumed to vary through the thickness according to the following power law distribution (ESP-FGM)  Vm zị ẳ  2z ỵ h N ; Vc zị ẳ Vm ðzÞ; 2h (1) where N is volume fraction index (0 N < N) Material coefficient E(z) is obtained as  Ezị ẳ Ec ỵ Emc  2z ỵ h N ; 2h (2) where Emc ¼ Em À Ec : (3) The values with subscripts m and c belong to metal and ceramic, respectively It is evident from Eqs (2) and (3) that the upper surface of the panel (z ¼ Àh/2) is ceramic-rich, while the lower surface (z ¼ h/2) is metal-rich, and the percentage of ceramic constituent in the panel is enhanced when N increases The Poisson ratio n is assumed to be constant (n ¼ const) Assume that the panel is reinforced by eccentrically longitudinal and transversal homogeneous stiffeners with the elastic modulus E0 In order to provide the continuity between the shell and stiffeners, suppose that stiffeners are made of full metal (E0 ¼ Em) if putting them at the metal-rich side of the panel, and conversely full ceramic ones (E0 ¼ Ec) at the ceramic-rich side of the panel (Duc, 2013; Duc and Quan, 2013; Bich et al., 2011, 2012) The panelefoundation interaction is represented by Pasternak model as Fig Configuration of an eccentrically stiffened cylindrical panel (ES-FGM panel) on elastic foundations 62 N.D Duc, T.Q Quan / European Journal of Mechanics A/Solids 46 (2014) 60e71 qe ¼ k1 w À k2 V2 w; (4) whereV2 ẳ v2/vx2 ỵ v2/vy2, w is the deection of the panel, k1 is Winkler foundation modulus and k2 is the shear layer foundation stiffness of Pasternak model where A66 B12 ¼ Theoretical formulation According to the classical shell theory, for ESP-FGM panel, the strain at the middle surface and curvatures are related to the displacement components u, v, w in the x, y, z coordinate as @ 3x 3y gxy A ¼ B @ 1 kx C A ỵ z@ ky A; 2kxy g0xy 3x 3y B @ 3x 3y gxy 1 1 u;x ỵ w2;x =2 wx;x kx C A ẳ @ v;y w=R ỵ w2;y =2 A; @ ky A ẳ @ wy;y A; w;xy kxy u;y ỵ v;x þ w;x w;y  ssh xy ¼ Á À ÁÃ Ezị 3x; 3y ỵ n 3y; 3x ; n2 (7) Ezị g ; 21 ỵ nị xy and for stiffeners, modified from Duc (2013); Duc and Quan (2013); Bich et al (2011), (2012) as  st sst x ; sy  ¼ E0 À 3x; 3y Á : (8) Taking into account the contribution of stiffeners by the smeared stiffeners technique and omitting the twist of stiffeners and integrating the stressestrain equations and their moments through the thickness of panel, the expressions for force and moment resultants of an ESP-FGM cylindrical shell are of the form (Duc, 2013; Duc and Quan, 2013; Bich et al., 2011, 2012)  E0 A1 0 x ỵ A12 y ỵ B11 ỵ C1 ịkx ỵ B12 ky ; s1   E A Ny ¼ A12 0x þ A22 þ 0y þ B12 kx ỵ B22 ỵ C2 ịky ; s2 Nx ẳ    E I Mx ẳ B11 ỵ C1 ị3 0x ỵ B12 0y ỵ D11 ỵ kx ỵ D12 ky ; s1   E I Mx ẳ B12 0x ỵ B22 ỵ C2 ị3 0y ỵ D12 kx ỵ D22 ỵ ky ; s2 B66 g0xy ỵ 2D66 kxy ; E3 ; 21 ỵ nị d1 h1 ị3 d h ị3 ỵ A1 z1 ị2 ; I2 ẳ 2 ỵ A2 ðz2 Þ2 ; 12 12 E0 A1 z1 E0 A2 z2 ; C2 ¼ ; C1 ¼ s1 s2 I1 ẳ h1 ỵ h h ỵh ; z2 ẳ ; 2   Em À Ec E1 ¼ Ec ỵ h; kỵ1 (11) Em Ec ịNh2 ; 2N ỵ 1ịN ỵ 2ị !  Ec 1 ỵ h3 ; E3 ẳ ỵ Em Ec ị N ỵ N ỵ 4N ỵ 12 E2 ¼ where s1,s2 are 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 panel; 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 The nonlinear equilibrium equations of perfect ESP-FGM cylindrical panels based on the classical shell theory are (Duc and Tung, 2010a, 2010b; Tung, 2013) Nx;x ỵ Nxy;y ẳ 0; (12a) Nxy;x ỵ Ny;y ẳ 0; (12b) A11 ỵ Nxy ẳ A66 g0xy ỵ 2B66 kxy ; Mxy ẳ D66 ẳ E3 E3 n ; D12 ¼ ; À n2 À n2 z1 ¼ in which u, v are the displacement components along the x, y directions, respectively Hooke law for an ESP-FGM panel is defined as sh ssh ¼ x ; sy D11 ¼ D22 ¼ (10) and (6)  E2 n E2 ; ; B66 ¼ 2ð1 þ nÞ À n2 (5) in which 0x ; 0y are normal strains, 0xy is shear strain at the middle surface; kx, ky, kxy and R are curvatures and radii of the panel, respectively The nonlinear strainedisplacement relationship of the FGM cylindrical panel are (Tung, 2013; Bich et al., 2012) E1 E1 n ; A12 ¼ ; À n2 À n2 E1 E2 ; B11 ẳ B22 ẳ ẳ ; 21 ỵ nị n2 A11 ẳ A22 ẳ Ny ỵ Nx w;xx ỵ 2Nxy w;xy ỵ Ny w;yy R ỵ q k1 w ỵ k2 V2 w ẳ 0: Mx;xx þ 2Mxy;xy þ My;yy þ (12c) (9) Calculated from Eq (9) 3x ẳ A*22 Nx A*12 Ny ỵ B*11 w;xx ỵ B*12 w;yy ; ẳ A*11 Ny A*12 Nx ỵ B*21 w;xx ỵ B*22 w;yy ; 3y g0xy ẳ A*66 Nxy ỵ 2B*66 w;xy ; (13) N.D Duc, T.Q Quan / European Journal of Mechanics A/Solids 46 (2014) 60e71 where x;yy  E0 A1 A ; A*12 ¼ 12 ; D D s1   E A A*22 ẳ A ỵ ; A*66 ¼ ; D 22 A66 s2    E A E A D ẳ A11 ỵ A22 ỵ A212 ; s1 s2 A*11 ẳ  A11 ỵ B*11 ẳ A*22 B11 ỵ C1 ị A*12 B12 ; 63 ỵ 0y;xx g0xy;xy ẳ w2;xy w;xx w;yy ỵ 2w;xy w*;xy À w;xx w*;yy w;xx : À w;yy w*;xx À R (19) From the constitutive relations (13) in conjunction with Eq (17) one can write 3x (14) 3y   ẳ A*22 f;yy A*12 f;xx ỵ B*11 w;xx ỵ B*12 w;yy A*22 A*12 F1 ;   ẳ A*11 f;xx A*12 f;yy ỵ B*21 w;xx ỵ B*22 w;yy A*11 A*12 F1 ; g0xy ẳ A*66 f;xy ỵ 2B*66 w;xy B*22 ẳ A*11 B22 ỵ C2 ị A*12 B12 ; (20) B*12 ẳ A*22 B12 A*12 B22 ỵ C2 ị; B*21 ẳ A*11 B12 A*12 B11 ỵ C1 ị; B*66 ¼ Setting Eq (20) into Eq (19) gives the compatibility equation of an imperfect FGM cylindrical panel as B66 A66 Substituting once again Eq (13) into the expression of Mij in (9), then Mij into the Eq (12c) leads to Nx;x ỵ Nxy;y ẳ 0; Nxy;x ỵ Ny;y ẳ 0; (15)   B*21 f;xxxx ỵ B*12 f;yyyy ỵ B*11 ỵ B*22 2B*66 f;xxyy D*11 w;xxxx   D*22 w;yyyy D*12 ỵ D*21 ỵ 4D*66 w;xxyy ỵ Nx w;xx ỵ 2Nxy w;xy ỵ Ny w;yy þ Ny þ q À k1 w þ k2 V2 w ẳ 0; R   A*11 f;xxxx ỵ A*22 f;yyyy ỵ A*66 2A*12 f;xxyy ỵ B*21 w;xxxx ỵ B*12 w;yyyy    ỵ B*11 ỵ B*22 2B*66 w;xxyy w2;xy w;xx w;yy ỵ 2w;xy w*;xy w;xx  ¼ 0: À w;xx w*;yy À w;yy w*;xx À R (21) Eqs (18) and (21) are nonlinear equations in terms of variables w and f and used to investigate the stability of the FGM cylindrical panels on elastic foundations subjected to mechanical loads In the present study, the edges of cylindrical panels are assumed to be simply supported Depending on an in-plane restrain at the edges, the boundary conditions are w ¼ u ¼ Mx ¼ 0; Nx ¼ Nx0 at x ¼ 0; a w ¼ v ¼ My ¼ 0; Ny ¼ Ny0 at y ¼ 0; b; where D*11 ẳ D11 ỵ E0 I1 B11 ỵ C1 ịB*11 B12 B*21 ; s1 D*22 ẳ D22 ỵ E0 I2 B12 B*12 B22 ỵ C2 ịB*22 ; s2 D*12 ẳ D12 B11 þ C1 ÞB*12 À B12 B*22 ; (16) where Nx0, Ny0 are fictitious compressive edge loads at immovable edges The mentioned conditions (22) can be satisfied identically if the buckling mode shape is chosen by Duc (2013); Duc and Quan (2013); Tung (2013); Duc and Tung (2010a), (2010b) w ¼ Wsin lm x sin dn y; D*21 ¼ D12 À B12 B*11 B22 ỵ C2 ịB*21 ; D*66 ẳ D66 À B66 B*66 f(x,y) is stress function defined by Nx ¼ f;yy ; Ny ¼ f;xx ; Nxy ¼ Àf;xy : (22) (17) For an imperfect FGM cylindrical panel, Eq (15) are modified into form as   B*21 f;xxxx ỵ B*12 f;yyyy ỵ B*11 ỵ B*22 2B*66 f;xxyy À D*11 w;xxxx     À D*22 w;yyyy D*12 ỵ D*21 ỵ 4D*66 w;xxyy ỵ f;yy w;xx ỵ w*;xx     f ;xx ỵ q k1 w 2f;xy w;xy ỵ w*;xy ỵ f;xx w;yy ỵ w*;yy ỵ R ỵ k2 V2 w ¼ 0; (18) in which w*(x,y) is a known function representing initial small imperfection of the panel The geometrical compatibility equation for imperfect cylindrical panels written as (23) where lm ¼ mp/a, dn ¼ np/b, m,n ¼ 1,2, are the natural numbers of half waves in the corresponding direction x,y; W is amplitude of the deflection The initial imperfection w* is assumed to have the same form of the panel deection w, i.e w* x; yị ẳ W0 sin lm x sin dn y; (24) in which W0 ¼ const is a known initial amplitude Substituting Eqs (23) and (24) into the compatibility Equation (21), we define the stress function as f ẳ A1 cos 2lm x ỵ A2 cos 2dn y ỵ A3 sin lm x sin dn y 1 ỵ Nx0 y2 ỵ Ny0 x2 ; 2 with A1 ẳ d2n 32A*11 lm WW ỵ 2mhị; A2 ẳ l2m 32A*22 dn WW ỵ 2mhị; (25) 64 N.D Duc, T.Q Quan / European Journal of Mechanics A/Solids 46 (2014) 60e71     4 2 B*21 lm ỵ B*12 dn ỵ B*11 þ B*22 À 2B*66 lm dn   W À     W: A3 ¼  4 2 4 2 A*11 lm ỵ A*22 dn ỵ A*66 2A*12 lm dn R A*11 lm þ A*22 dn þ A*66 À 2A*12 lm dn l2m Subsequently, substitution of Eqs 23e25 into Eq (18) and applying the Galerkin procedure for the resulting equation yield (26) 3.1 ESP-FGM cylindrical panel under uniform external pressure We consider an ESP-FGM cylindrical panel only subjected to uniform external pressure in the absence of thermal and edge h  i h  i   4 2 > B* l4 ỵ B* d4 ỵ B* ỵ B* 2B* > l2m d2n B*21 lm ỵ B*12 dn ỵ B*11 ỵ B*22 2B*66 lm dn > m n l 21 12 11 22 66 > > h  i h  i   À À m > > 4 2 4 2 > R A* l ỵ A* d ỵ A* 2A* l d A* l ỵ A* d þ A* À 2A* l d 2< > > > > > > > > = m n m n 11 m 22 n 66 12 W > > >     > > i À D*11 l4m D*22 d4n D*12 ỵ D*21 ỵ 4D*66 l2m d2n k2 l2m ỵ d2n k1 > > > ; mnp 11 m 22 n 66 12 4lm dn > > > > l4 > >   > À m2 h > : R A* l4 ỵ A* d4 ỵ A* 2A* l2 d m n 11 m 22 n 66 12 l2m À 2 * * * * R " ! # A11 lm ỵ A22 dn ỵ A66 2A12 lm dn dn 8lm dn B*21 B*12 7WW ỵ mhị ỵ  l d ỵ ỵ    m n WW ỵ 2mhị 4 2 6A*11 Rlm A*11 A*22 B*21 lm ỵ B*12 dn ỵ B*11 ỵ B*22 2B*66 lm dn    À  4 2 A*11 lm ỵ A*22 dn ỵ A*66 2A*12 lm dn !  l4m d4n mnp2 mnp2  Ny0 4q 2 Nx0 lm ỵ Ny0 dn W ỵ mhị ỵ ỵ ỵ * WW ỵ mhịW ỵ 2mhị ẳ 0; * lm dn R lm dn 64lm dn A22 A11 4lm dn 2    (27) where m,n are odd numbers This is basic equation governing the nonlinear response of ESP-FGM cylindrical panels under mechanical loads In this paper, we consider a simply supported ESP-FGM cylindrical panel with all movable edges and is rested on elastic foundations Two cases of mechanical loads will be analyzed compressive loads In this case, Nx0 ¼ Ny0 ¼ 0, and Eq (27) is reduced to À À Á q ẳ b11 W ỵ b12 W W ỵ m ỵ b13 W W ỵ 2m þ b14 W W þ m W þ 2m ; (28) where b11 ¼ mnp2 B4a D*11 K1 16B4h À þ m3 np4 B2a Rb 8B3h mnp6 16B4h þ mnp4 B2a D*11 K2  16B4h  m2 B2a ỵ n2 þ   m5 np6 B4a D*11 þ mn5 p6 D*22 ỵ m3 n3 p6 B2a D*12 ỵ D*21 ỵ 4D*66 16B4h  ! B*21 m4 B4a ỵ B*11 ỵ B*22 2B*66 m2 n2 B2a ỵ B*12 n4 m5 np2 B4a R2b  ! ỵ  !   A*11 m4 B4a ỵ A*66 2A*12 m2 n2 B2a þ A*22 n4 16B2h A*11 m4 B4a þ A*66 À 2A*12 m2 n2 B2a ỵ A*22 n4   !2  B*21 m4 B4a ỵ B*11 ỵ B*22 2B*66 m2 n2 B2a ỵ B*12 n4  ! ;  A*11 m4 B4a ỵ A*66 2A*12 m2 n2 B2a þ A*22 n4 (29) N.D Duc, T.Q Quan / European Journal of Mechanics A/Solids 46 (2014) 60e71 b12 ¼ À 3B3h  !  n2 p4 B2 B* m4 B4 ỵ B* ỵ B* 2B* n2 B2 þ B* n4 m 2m a a a 21 11 22 66 12 2m4 n2 p2 B4a Rb  !ỵ  ! ;   A*11 m4 B4a ỵ A*66 2A*12 m2 n2 B2a ỵ A*22 n4 3B4h A*11 m4 B4a ỵ A*66 2A*12 m2 n2 B2a ỵ A*22 n4 b13 ¼ 65 p2 n2 R b 24A*11 B3h ỵ m2 n2 p4 B2a 12B4h * BB21 @ A*11 ỵ B*12 A*22 (29) For a perfect cylindrical panel (m ¼ 0), Eq (28) leads to C A;   q ¼ b11 W þ b12 þ b13 W þ b14 W : (31) 3.2 ESP-FGM cylindrical panels under axial compressive loads b14 ¼ An ESP-FGM cylindrical panel supported by elastic foundations and subjected to axial compressive loads Fx uniformly distributed at two curved edges x¼0,a in the absence of external pressure and thermal loads is considered In this case, q ¼ 0, Ny0 ¼ 0, Nx0 ¼ ÀFxh, and Eq (27) leads to mnp6 Bm4 B4a n4 C ỵ @ A; * 256B4h A A* 22 11 where K1 ¼ Fx ¼ b21 k1 a4 k a2 ; K2 ¼ 2* ; * D11 D11 m2 p2 B2a D*11 b21 ¼ B2h ỵ W W ỵ 2m W ỵ b22 W ỵ b23 ỵ b24 W W þ 2m ; W þm W þm (32) where n4 p2 D*22 m2 B2a B2h ỵ   n2 p2 D*12 ỵ D*21 ỵ 4D*66 B2h ỵ p2 m2 B2a R2b  !  A*11 m4 B4a ỵ A*66 2A*12 m2 n2 B2a ỵ A*22 n4  !2  !   n2 B2 ỵ B* n4 n2 B2 ỵ B* n4 * m4 B4 ỵ B* þ B* À 2B* * m4 B4 þ B* þ B* À 2B* m m B B a a a a 21 11 22 66 12 21 11 22 66 12 p2 2R b ỵ    !  ! 2 B m Ba Bh h A*11 m4 B4a ỵ A*66 2A*12 m2 n2 B2a þ A*22 n4 A*11 m4 B4a þ A*66 À 2A*12 m2 n2 B2a ỵ A*22 n4   D*11 m2 B2a ỵ n2 K2 B2 D* K1 ỵ a2 11 ; ỵ m2 B2h m p2 B2h 32mnB2a Rb 32n b22 ẳ  !ỵ  3mB h 3p2 Bh A*11 m4 B4a ỵ A*66 2A*12 m2 n2 B2a ỵ A*22 n4 A*11 ẳ hA*11 ; A*22 ẳ hA*22 ; A*12 ¼ hA*12 ; A*66 ¼ hA*66 ; B*11  !  B*21 m4 B4a ỵ B*11 ỵ B*22 2B*66 m2 n2 B2a ỵ B*12 n4  !  A*11 m4 B4a ỵ A*66 2A*12 m2 n2 B2a ỵ A*22 n4 ; 2n Rb 8n b23 ẳ ỵ 2 * 3mB 3m p Ba Bh A11 h B* B* B* B* B* ¼ 11 ; B*22 ¼ 22 ; B*12 ¼ 12 ; B*21 ¼ 21 ; B*66 ¼ 66 ; h h h h h (30) D*11 ¼ D*11 * D* D* D* D* ; D22 ¼ 22 ; D*12 ¼ 12 ; D*21 ¼ 21 ; D*66 ¼ 66 ; 3 3 h h h h h3 b24 ¼ p2 4 Bm Ba @ 2 16m Ba Bh A*22 B*21 A*11 ỵ B*12 A*22 (33) ! ; ỵ n4 A*11 C A: Bh ¼ b=h; Ba ¼ b=a; W ¼ W=h; Rb ¼ b=R: For a perfect cylindrical panel (m¼0) only subjected to axial compressive load Fx, Eq (32) leads to Eq (28) expresses explicit relation of pressure-deflection curves for ESP-FGM cylindrical panels rested on elastic foundations and under uniform external pressure   Fx ẳ b21 ỵ b22 ỵ b23 W þ b24 W : (34) 66 N.D Duc, T.Q Quan / European Journal of Mechanics A/Solids 46 (2014) 60e71 From which upper buckling compressive load may be obtained with W/0 as Fx ¼ m2 p2 B2a D*11 b21 ¼ B2h þ n4 p2 D*22 m2 B2a B2h þ   n2 p2 D*12 ỵ D*21 ỵ 4D*66 B2h ỵ m ¼ n ¼ 1, and unless otherwise stated, the FGM panelefoundation interaction is ignored  m2 B2a R2b  p2 A*11 m4 B4a ỵ A*66 2A*12 m2 n2 B2a þ A*22 n4 !  !2  !   n2 B2 ỵ B* n4 n2 B2 ỵ B* n4 * m4 B4 ỵ B* ỵ B* 2B* * m4 B4 ỵ B* ỵ B* 2B* m m B B a a a a 21 11 22 66 12 21 11 22 66 12 p2 2R b ỵ  !  !   2 Bh m Ba Bh A*11 m4 B4a ỵ A*66 2A*12 m2 n2 B2a ỵ A*22 n4 A*11 m4 B4a ỵ A*66 2A*12 m2 n2 B2a ỵ A*22 n4   D*11 m2 B2a ỵ n2 K2 B2 D* K1 ỵ a2 11 : ỵ m2 B2h m p2 B2h (35) The height of stiffeners is equal to 30 mm, its width is mm, the spacing of stiffeners s1 ¼ s2 ¼ 0.4(m),z1 ¼ z2 ¼ 0.0225(m) Numerical results and discussion This section presents illustrative results for ceramic-metal cylindrical panels made of aluminum and alumina with properties (Duc, 2013; Duc and Quan, 2013; Tung, 2013; Duc and Tung, 2010a, 2010b) Em ¼ 70 GPa; Ec ¼ 380 GPa: (36) Poisson’s ratio is chosen to be n ¼ 0.3 and have a square plane form (a ¼ b) In addition, the results presented in this section correspond to deformation mode with half-wave numbers Fig Effects of N index on the nonlinear response of ESP-FGM cylindrical panels 4.1 ESP-FGM cylindrical panel under uniform external pressure Fig shows effect of volume fraction index N on the nonlinear response of ESP-FGM cylindrical panels under uniform external pressure Obviously, the mechanical load get better if the volume N decreases Fig Effects of elastic foundations on the nonlinear response of ESP-FGM cylindrical panels N.D Duc, T.Q Quan / European Journal of Mechanics A/Solids 46 (2014) 60e71 Fig Effects of initial imperfection on the nonlinear response of ESP-FGM cylindrical panels under uniform external pressure Fig indicates effects of elastic foundations on the nonlinear response of ESP-FGM cylindrical panels under uniform external pressure When K1 ¼ K2 ¼ (without elastic foundation), the obtained result coincides with the result presented by Bich et al Fig Nonlinear response of ESP-FGM cylindrical panels without elastic foundations under uniform external pressure 67 Fig Effects of ratio b/a on the nonlinear response of ESP-FGM cylindrical panels under uniform external pressure (2011) We can see that the load-carrying capacity of panels increases when increasing modulus K1,K2 In addition, the stiffness K2 of the shear layer of the Pasternak foundation model has more pronounced effect on the postbuckling response of ESP-FGM Fig Effects of ratio b/h on the nonlinear response ESP- FGM cylindrical panels under uniform external pressure 68 N.D Duc, T.Q Quan / European Journal of Mechanics A/Solids 46 (2014) 60e71 Fig Effects of ratio b/R on the nonlinear response of ESP-FGM cylindrical panels under uniform external pressure Fig 10 Effects of imperfection on the nonlinear postbuckling response of ESP- FGM cylindrical panels cylindrical panels under uniform external pressure than the modulus K1 of the Winkler model The effects of initial imperfection (four values of m: 0, 0.1, 0.2 and 0.5) on the nonlinear postbuckling response of ESP - FGM cylindrical panels are shown in Fig The nonlinear response of ESP- FGM cylindrical panels without elastic foundations is presented in Fig 5, which has a good agreement with those reported by Bich et al (2011) Figs 6e8 analyze the effects of geometrical parameters on the postbuckling behavior of ESP-FGM cylindrical panels when Fig Effect of stiffeners on the nonlinear response of ESP-FGM cylindrical panels under uniform external pressure Fig 11 Effects of elastic foundations on the nonlinear postbuckling response of ESPFGM cylindrical panels N.D Duc, T.Q Quan / European Journal of Mechanics A/Solids 46 (2014) 60e71 69 Fig 12 Effects of N index on the nonlinear response of ESP-FGM cylindrical panels under axial compression n ¼ 0.3 Fig shows effect of stiffeners on the nonlinear response of FGM cylindrical panels which not rest on elastic foundations 4.2 ESP-FGM cylindrical panels under axial compressive loads In Figs 10e17, we consider the ESP-FGM panels under axial compressive loads The effects of imperfection on the postbuckling behavior of cylindrical FGM panels are shown in Fig 10 The curves in Fig 10 show us that in postbuckling periods, the imperfect cylindrical panel has a better mechanical and thermal loading capacity than those of the perfect panel Fig 11 shows the effects of elastic foundations on nonlinear response of ESP-FGM cylindrical Fig 13 Effects of elastic foundations on the nonlinear response of ESP-FGM cylindrical panels under axial compression Fig 14 Effects of ratio b/a on the nonlinear response of cylindrical ESP-FGM panels under axial compressive loads panels and comparison with results reported by Tung (2013) (without stiffeners) Fig 12 shows the effects of volume fraction index N on the nonlinear response of ESP- FGM cylindrical panels with movable edges It is obviously that the loading capacity of the panels decreases along with the increase of N Fig 13 shows the effects of elastic foundations on the postbuckling behavior of ESP-FGM cylindrical panels with movable Fig 15 Effects of ratio b/h on the nonlinear response of ESP-FGM cylindrical panels 70 N.D Duc, T.Q Quan / European Journal of Mechanics A/Solids 46 (2014) 60e71 stiffened (P-FGM) cylindrical panels, we obtain the result as the result in Tung (2013) It is clear that the stiffeners can enhance the mechanical loading capacity for the cylindrical FGM panels Conclusions This paper presents an analytical investigation on the nonlinear postbuckling response for imperfect eccentrically stiffened FGM cylindrical panels on elastic foundation using a simple power-law distribution (ESP-FGM) under mechanical loads Applying Bubnov-Galerkin method, explicit expressions of critical buckling loads and nonlinear load-deflection curves for simply supported eccentrically stiffened FGM cylindrical panels under two cases of mechanical loads are determined From the obtained results in this paper we can conclude that: Fig 16 Effects of ratio b/R on the nonlinear response of ESP- FGM cylindrical panels edges and subjected to axial compression Obviously, both buckling loads and postbuckling equilibrium paths of ESP-FGM cylindrical panels become considerably higher due to the support of elastic media, especially Pasternak type foundations However, the severity of snap-through instability is almost unchanged with different values of foundation parameters Figs 14e16 analyze the effects of geometrical parameters on the nonlinear response of cylindrical ESP-FGM panels when n ¼ 0.3 We can see that the geometrical parameters considerably impact on the nonlinear buckling and postbuckling of the FGM panels However, imperfect panels have a little better loading capacity than perfect ones, especially, in postbuckling periods Fig 17 shows effects of stiffeners on the nonlinear response of ESP-FGM cylindrical panels under axial compression For un- Fig 17 Effects of stiffeners on nonlinear response of FGM cylindrical panels under axial compressive loads - The critical bucking loads for FGM cylindrical panels decreases when increasing the power law index - The geometrical parameters considerably impact on the nonlinear buckling and postbuckling of the FGM cylindrical panels - The initial imperfection has a significant influence on the nonlinear buckling and postbuckling of the FGM cylindrical panels - The stiffener system strongly enhances the load-carrying capacity of the FGM cylindrical panels - The elastic foundations have a strong effect on the nonlinear response of the FGM cylindrical panels and the beneficial effect of the Pasternak foundation is higher than the Winkler one Acknowledgment This stability shapes” grateful paper was supported by Project “Nonlinear analysis on and dynamics of functionally graded shells with special of Vietnam National University, Hanoi The authors are for this support References Alibeigloo, A., 2014 Free vibration analysis of functionally graded carbon nanotubereinforced composite cylindrical panel embedded in piezoelectric layers by using theory of 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with tangential edge constraints and resting on elastic foundations Compos Struct 100, 532e541 Yamada, S., Croll, J.G.A., 1989 Buckling behavior of pressure loaded cylindrical panels J Eng Mech 115 (2), 327e344 Yang, J., Liew, K.M., Wu, Y.F., Kitipornchai, S., 2006 Thermo-mechanical postbuckling of FGM cylindrical panels with temperature-dependent properties Int J Solids Struct 43, 307e324  ... those of the perfect panel Fig 11 shows the effects of elastic foundations on nonlinear response of ESP -FGM cylindrical Fig 13 Effects of elastic foundations on the nonlinear response of ESP -FGM cylindrical. .. Fig Nonlinear response of ESP -FGM cylindrical panels without elastic foundations under uniform external pressure 67 Fig Effects of ratio b/a on the nonlinear response of ESP -FGM cylindrical panels. .. index on the nonlinear response of ESP -FGM cylindrical panels under axial compression n ¼ 0.3 Fig shows effect of stiffeners on the nonlinear response of FGM cylindrical panels which not rest on elastic