DSpace at VNU: Nonlinear mechanical, thermal and thermo-mechanical postbuckling of imperfect eccentrically stiffened thin FGM cylindrical panels on elastic foundations

Thin-Walled Structures 96 (2015) 155–168 Contents lists available at ScienceDirect Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws Nonlinear mechanical, thermal and thermo-mechanical postbuckling of imperfect eccentrically stiffened thin FGM cylindrical panels on elastic foundations Nguyen Dinh Duc a,n, Ngo Duc Tuan b, Tran Quoc Quan a, Nguyen Van Quyen a, Tran Van Anh a a b Vietnam National University, Hanoi, 144 Xuan Thuy-Cau Giay, Hanoi, Vietnam The University of Melbourne, Parkville, VIC 3010, Australia art ic l e i nf o a b s t r a c t Article history: Received 24 April 2015 Received in revised form 19 July 2015 Accepted August 2015 This paper presents an analytical approach to investigate the nonlinear stability analysis of eccentrically stiffened thin FGM cylindrical panels on elastic foundations subjected to mechanical loads, thermal loads and the combination of these loads The material properties are assumed to be temperature-dependent and graded in the thickness direction according to a simple power law distribution Governing equations are derived basing on the classical shell theory incorporating von Karman–Donnell type nonlinearity, initial geometrical imperfection, the Lekhnitsky smeared stiffeners technique and Pasternak type elastic foundations Explicit relations of load–deflection curves for FGM cylindrical panels are determined by applying stress function and Galerkin method The effects of material and geometrical properties, imperfection, elastic foundations and stiffeners on the buckling and postbuckling of the FGM panels are discussed in detail The obtained results are validated by comparing with those in the literature & 2015 Elsevier Ltd All rights reserved Keywords: Nonlinear mechanical and thermal postbuckling Eccentrically stiffened FGM cylindrical panels Imperfection Elastic foundations Introduction Composite panels are commonly used in aerospace, mechanics, naval and other high-performance engineering applications due to their light weight, high specific strength and stiffness, excellent thermal characteristics At high temperatures, composite panels are found to buckle without the application of mechanical loads Therefore, the buckling and postbuckling response of composite panels have to be well understood Recently, a new class of composite materials known as functionally graded materials (FGMs) attracts special attention of a lot of authors in the world FGM is a new generation of composite material in which its mechanical properties vary smoothly and continuously from one surface to the other Functionally graded structures such as cylindrical panels in recent years, play the important part in the modern industries As a result, static response of FGM cylindrical panels has been the subject of many studies for a long period of time Shen and Wang [1] presented thermal postbuckling analysis for FGM cylindrical panels resting on elastic foundations They [2] also studied the nonlinear bending analysis of simply supported FGM cylindrical panel resting on an elastic foundation in thermal environments n Corresponding author E-mail address: ducnd@vnu.edu.vn (N.D Duc) http://dx.doi.org/10.1016/j.tws.2015.08.005 0263-8231/& 2015 Elsevier Ltd All rights reserved Lee et al [3] investigated the thermomechanical behaviors of FGM panels in hypersonic airflows Alibeigloo and Chen [4] developed the three-dimensional elasticity solution for static analysis of a FGM cylindrical panel with simply supported edges Tung and Duc [5] studied the nonlinear response of thick FGM doubly curved shallow panels resting on elastic foundations and subjected to some conditions of mechanical, thermal, and thermomechanical loads They [6] also investigated the nonlinear response of pressure-loaded FGM cylindrical panels with temperature effects Aghdam et al [7] considered bending of moderately thick clamped FGM conical panels subjected to uniform and non-uniform distributed loadings Du et al [8] studied the nonlinear forced vibration of infinitely long functionally graded cylindrical shells is using the Lagrangian theory and the multiple scale method A semi-analytical solution for static response of fully clamped sheardeformable FGM doubly curved panels is presented by Shahmansouri et al [9] Kiani et al [10] focused on the static, dynamic and free vibration analysis of a FGM doubly curved panel Bich et al [11] researched the linear buckling of FGM truncated conical panels subjected to axial compression, external pressure and the combination of these loads Static and dynamic stabilities of FGM panels which are subjected to combined thermal and aerodynamic loads are investigated in work of Sohn and Kim [12] based on the first order shear deformation theory Yang et al [13] published the 156 N.D Duc et al / Thin-Walled Structures 96 (2015) 155–168 results on thermo-mechanical postbuckling analysis of FGM cylindrical panels with temperature-dependent properties Recently, in 2014, Duc [14] published a valuable book “ Nonlinear static and dynamic stability of functionally graded plates and shells”, in which the results about nonlinear static stability of shear deformable FGM panels are presented Tung [15] introduced an analytical approach to investigate the effects of tangential edge constraints on the buckling and postbuckling behavior of FGM flat and cylindrical panels subjected to thermal, mechanical and thermomechanical loads and resting on elastic foundations However, since this area is relatively new, there are very little researches on nonlinear static problems of FGM cylindrical panels and cylindrical shells reinforced by stiffeners Duc and Quan [16] investigated the nonlinear response of eccentrically stiffened FGM cylindrical panels on elastic foundations subjected to mechanical loads Najafizadeh et al [17] considered the elastic buckling of FGM stiffened cylindrical shells by rings and stringers subjected to axial compression loading Dung et al [18] analyzed the nonlinear buckling and postbuckling of FGM stiffened thin circular cylindrical shells surrounded by elastic foundations in thermal environments and under torsional load Bich et al [19] presented an analytical approach to investigate the nonlinear static and dynamic buckling of imperfect eccentrically stiffened FGM thin circular cylindrical shells subjected to axial compression To the knowledge of the authors, there is limited publication on the stability of FGM structures reinforced by eccentrically stiffeners in thermal environments The most difficult part in this type of problem is to calculate the thermal mechanism of FGM structures as well as stiffeners under thermal loads Duc et al [20,21] investigated the nonlinear postbuckling of an eccentrically stiffened thin FGM plate and circular cylindrical shell resting on elastic foundation in thermal environments Development of the results in these researches, this paper deals with the nonlinear postbuckling of imperfect eccentrically stiffened thin FGM cylindrical panels on elastic foundations under mechanical loads, thermal loads and the combination of these loads The material properties are assumed to be temperature-dependent and graded in the thickness direction according to a simple power law distribution Both of the panels and the stiffeners are assumed to be deformed due to the presence of temperature Using Galerkin method and stress function, the effects of geometrical and material properties, imperfection, elastic foundations and stiffeners on the nonlinear response of the imperfect eccentrically stiffened FGM cylindrical panels are analyzed Problem statement Consider an eccentrically stiffened functionally graded cylindrical panel with the radii of curvature, thickness, axial length and arc length of the panel are R , h, a and b, respectively and is defined in coordinate system (x, y, z ) , as shown in Fig The panel is reinforced by eccentrically longitudinal and transversal stiffeners The width and thickness of longitudinal and transversal stiffeners are denoted by dx , hx and dy , hy respectively; sx , sy are the spacings of the longitudinal and transversal stiffeners The quantities Ax , Ay are the cross-section areas of stiffeners and Ix, Iy, zx , z y are the second moments of cross-section areas and the eccentricities of stiffeners with respect to the middle surface of panel, respectively E0 is Young's modulus of ring and stringer stiffeners In order to provide continuity between the panel and stiffeners, suppose that stiffeners are made of full metal (E0 = Em ) The panel is made from a mixture of ceramic and metal, and the material constitution is varied gradually by a simple power law distribution, in which the volume fractions of the ceramic and metal are expressed as ⎛ 2z + h ⎞ N Vm (z ) = ⎜ ⎟ , ⎝ 2h ⎠ Vc (z ) = − Vm (z ), (1) where N is volume fraction index ( ≤ N < ∞), subscripts m and c stand for the metal and ceramic constituents, respectively Effective properties Preff of FGM panel, such as the elastic modulus E and the thermal expansion coefficient α are determined by linear rule of mixture as Preff (z ) = Prc Vc (z ) + Prm Vm (z ), (2) in which Pr denotes a temperature-dependent material property The effective properties of the FGM panel are obtained by substituting Eq (1) into Eq (2) as [E (z, T ), α (z, T )] = ⎡⎣ Ec (T ), αc (T ) ⎤⎦ ⎛ 2z + h ⎞ N ⎟ , + ⎡⎣ Emc (T ), αmc (T ) ⎤⎦ ⎜ ⎝ 2h ⎠ (3) where Emc (z, T ) = Em (z, T ) − Ec (z, T ), αmc (z, T ) = αm (z, T ) − αc (z, T ), (4) and the Poisson's ratio is assumed to be constant ν (z ) = v = const A material property Pr can be expressed as a nonlinear function of temperature [1,2,13] Fig Configuration and the coordinate system of an eccentrically stiffened cylindrical panel on elastic foundations N.D Duc et al / Thin-Walled Structures 96 (2015) 155–168 ( ) Pr = P0 P−1T−1 + + P1T + P2 T2 + P3 T , (5) in which T = T0 + ΔT, ΔT is the temperature increment in the environment containing the panel and T0 = 300 K (room temperature), P0, P−1, P1, P2 and P3 are coefficients characterizing of the constituent materials The panel–foundation interaction of Pasternak model is given by qe = k1w − k2 ∇2w, Nx = (A11 + 157 E0 A xT ) εx + A12 εy0 + (B11 + CxT ) k x + B12 k y + Φ1, sxT Ny = A12 εx0 + (A22 + E0 A yT ) εy0 + B12 k x + (B22 + CyT ) k y + Φ1, syT + 2B66 k xy, Nxy = A66 γxy Mx = (B11 + CxT ) εx0 + B12 εy0 + (D11 + E0 IxT ) k x + D12 k y + Φ2, sxT (6) where ∇2 = ∂ 2/∂x + ∂ 2/∂y2, w is the deflection of the panel, k1 is Winkler foundation modulus and k2 is the shear layer foundation stiffness of Pasternak model My = B12 εx0 + (B22 + CyT ) εy0 + D12 k x + (D22 + E0 I yT syT ) k y + Φ 2, + 2D66 k xy Mxy = B66 γxy (11) where Theoretical formulation Taking into account the von Karman–Donnell geometrical nonlinearity terms, the strains at the middle surface and curvatures relating to the displacement components u, v, w based on the classical thin shell theory are [22,23] ⎛ ε 0⎞ ⎛ ⎞ u, x + w ,2x/2 ⎜ x⎟ ⎜ ⎟ ⎜ εy0 ⎟ = ⎜ ⎟, ⎜ ⎟ ⎜ v, y − w /R + w , y/2⎟ ⎜ γ ⎟ ⎜⎝ u, y + v, x + w , x w , y ⎟⎠ ⎝ xy ⎠ ⎛ kx ⎞ ⎛ − w ⎞ , xx ⎜ ⎟ ⎜ ⎟ ⎜ k y ⎟ = ⎜ − w , yy ⎟, ⎜ ⎜ ⎟ ⎝ − w , xy ⎟⎠ ⎝ k xy ⎠ E1 E1ν E1 , A12 = , A66 = , (1 + ν ) − ν2 − ν2 E2 E2 ν E2 B11 = B22 = , B12 = , B66 = , (1 + ν ) − ν2 − ν2 E3 E3 ν E3 D11 = D22 = , D12 = , D66 = , (1 + ν ) − ν2 − ν2 h /2 E (z ) α (z )ΔT (1, z ) dz, (Φ1, Φ2 ) = − − ν −h /2 A11 = A22 = ∫ (7) and where εx0 and εy0 are normal strains, γxy is the shear strain at the middle surface of the panel and k ij , ij = x, y, xy are the curvatures The strain components across the panel thickness at the distance z from the mid-plane are given by dyT (hyT )3 dxT (hxT )3 + A xT (z xT )2 , I yT = + A yT (z yT )2 , 12 12 E0 A yT z yT E AT zT CxT = Tx x , CyT = , sx syT ⎛ 0⎞ ⎛ kx ⎞ ⎛ εx ⎞ ⎜ εx ⎟ ⎜ εy ⎟ = ⎜ ε ⎟ + z ⎜⎜ k ⎟⎟ y ⎜γ ⎟ ⎜ y ⎟ ⎜ ⎟ ⎝ xy ⎠ ⎜ γ ⎟ 2k xy ⎠ ⎝ ⎝ xy ⎠ hyT + hT hxT + hT , z yT = , 2 ⎛ ⎞ E h − E1 = Ec h + mc , E2 = Emc h2 ⎜ ⎟, ⎝N+2 N+1 (N + 1) ⎠ sh x , sh σxy ) σysh = IxT = z xT = (8) Hooke's law for cylindrical panel taking into account the temperature-dependent properties is defined as (σ (12) E3 = ⎡ ⎤ Ec h3 1 + Emc h3 ⎢ − + ⎥, ⎣N+3 N+2 12 (N + 1) ⎦ (13) E (z, T ) [(εx, εy ) − ν2 with the geometric shapes of stiffeners after the thermal deformation process in Eq (13) can be determined as the follows: + ν ( εy, εx ) − (1 + ν ) α (z, T ) ΔT (1, 1)], dxT = dx (1 + αm T (z )), dyT = dy (1 + αm T (z )), E (z, T ) γ , = (1 + ν ) xy (9) hyT = hy (1 + αm T (z )), z xT = z x (1 + αm T (z )), and for stiffeners (σxst , σyst ) = E0 (T )(εx, εy ) − hxT = hx (1 + αm T (z )), E0 (T ) α0 (T )ΔT (1, 1), − 2ν z yT = z y (1 + αm T (z )), sxT = sx (1 + αm T (z )), (10) where E0 (T ) , α0 (T ) are the Young's modulus and thermal expansion coefficient of the stiffeners, respectively Unlike other publications, in this paper, material properties of the eccentrically outside stiffeners are assumed to depend on temperature All elastic moduli of FGM panels and stiffeners are assumed to be temperature dependence and they are deformed in the presence of temperature Therefore, the geometric parameters, the panel's shape and stiffeners vary through the deforming process due to the temperature change However, because the thermal stress of stiffeners is subtle which distributes uniformly through the whole panel structure, we can ignore it The contribution of stiffeners can be accounted for using the Lekhnitsky smeared stiffeners technique Then integrating the stress–strain equations and their moments through the thickness of the panel, the expressions for force and moment resultants of an eccentrically stiffened FGM cylindrical panel are obtained as syT = sy (1 + αm T (z )), (14) The nonlinear equilibrium equations of FGM cylindrical panels based on classical shell theory are given as [22,23] Nx, x + Nxy, y = (15a) Nxy, x + Ny, y = (15b) Mx, xx + 2Mxy, xy + My, yy + − k1w + k2 ∇2w = 0, Ny + Nx w , xx + 2Nxy w , xy + Ny w , yy + q R (15c) where q is an external pressure uniformly distributed on the surface of the panel The geometrical compatibility equation for an imperfect FGM cylindrical panel is written as [22,23] 158 N.D Duc et al / Thin-Walled Structures 96 (2015) 155–168 εx0, yy + εy0, xx − γxy w ,2xy − w , xx w , yy + 2w , xy w ,⁎xy − w , xx w ,⁎yy , xy = − w , yy w ,⁎xx − w , xx R ⁎ ⁎ ⁎ ⁎ ⁎ − 2A12 ) f, xxyy + B21 A11 f, xxxx + A22 f, yyyy + (A66 w , xxxx (16) The first two equations of the nonlinear motion Eqs (15a), (15b) are automatically satisfied by choosing the stress function f (x, y ) as Nx = f, yy , Ny = f, xx , Nxy = − f, xy (17) Substituting relation (17) into Eq (11), we obtain ⁎ ⁎ ⁎ ⁎ + B12 + B22 − 2B66 ) w , xxyy w , yyyy + (B11 ⎛ − ⎜ w ,2xy − w , xx w , yy + 2w , xy w ,⁎xy − w , xx w ,⁎yy − w , yy w ,⁎xx ⎝ − w , xx ⎞ ⎟ = R ⎠ (23) Eqs (22) and (23) are nonlinear equations in terms of variables w and f and they are used to investigate the nonlinear stability of FGM eccentrically stiffened cylindrical panels on elastic foundations ⁎ ⁎ ⁎ ⁎ ⁎ ⁎ εx0 = A22 f, yy − A12 f, xx + B11 w , xx + B12 w , yy − (A22 − A12 ) Φ1, ⁎ ⁎ ⁎ ⁎ ⁎ ⁎ εy0 = A11 f, xx − A12 f, yy + B21 w , xx + B22 w , yy − (A11 − A12 ) Φ1, ⁎ ⁎ γxy f, xy + 2B66 w , xy, = − A66 Solution procedures (18) In the present study, the edges of eccentrically stiffened FGM cylindrical panel are assumed to be simply supported Depending on the in-plane restraint at the edges, three cases of boundary conditions, labeled as Cases 1, and will be considered [5] where ⁎ A11 = E0 Ay ⎞ E A ⎞ A 1⎛ 1⎛ ⁎ ⁎ ⎟, = 12 , A22 = ⎜ A22 + ⎜ A11 + x ⎟, A12 sx ⎠ sy ⎠ Δ⎝ Δ Δ⎝ ⁎ A66 = ⎛ E0 Ay ⎞ E A ⎞⎛ ⎟ − A12 , Δ = ⎜ A11 + x ⎟ ⎜ A22 + ⎝ A66 sx ⎠ ⎝ sy ⎠ Case Four edges of the FGM cylindrical panel are simply supported and freely movable (FM) The associated boundary conditions are ⁎ ⁎ B11 = A22 ( B11 + Cx ) − A12⁎ B12, B22⁎ = A11⁎ ( B22 + Cy ) − A12⁎ B12, ⁎ ⁎ ⁎ B12 B12 − A12 = A22 ( B22 + Cy ), B21⁎ = A11⁎ B12 − A12⁎ ( B11 + Cx ), ⁎ B66 = B66 A66 w = Nxy = Mx = 0, Nx = Nx0 at x = 0, a (19) Substituting once again Eq (18) into the expression of Mx, My, Mxy in Eq (11), then Mx, My, Mxy into the Eq (15c) leads to w = Nxy = My = 0, Ny = Ny0 at y = 0, b, (24) Case Four edges of the FGM cylindrical panel are simply supported and immovable (IM) In this case, boundary conditions are ⁎ ⁎ ⁎ ⁎ ⁎ B21 f, xxxx + B12 f, yyyy + (B11 + B22 − 2B66 ) f, xxyy w = u = Mx = 0, Nx = Nx0 at x = 0, a ⁎ ⁎ − D11 w , xxxx − D22 w , yyyy w = v = My = 0, Ny = Ny0 at y = 0, b, ⁎ ⁎ ⁎ − (D12 + D21 + 4D66 ) w , xxyy + Nx w , xx + 2Nxy w , xy + Ny w , yy + Ny + q − k1w + k2 ∇2w = 0, R (20) where ⁎ D11 = D11 + ⁎ D22 = D22 + E0 IxT ⁎ ⁎ − B11 + CxT B11 − B12 B21 , sxT ( E0 I yT syT ( w = Nxy = Mx = 0, Nx = Nx0 at x = 0, a ) ) − (B w = v = My = 0, Ny = Ny0 at y = 0, b, ⁎ ⁎ ⁎ D12 = D12 − B11 + CxT B12 − B12 B22 , ⁎ ⁎ D21 = D12 − B12 B11 ⁎ D66 = D66 − 22 ) ⁎ + CyT B21 , ⁎ B66 B66 (21) For an imperfect cylindrical panel, Eq (20) is modified into form as ⁎ ⁎ ⁎ ⁎ ⁎ − D11 w , xxxx − D22 w , yyyy − (D12 + D21 + 4D66 ) w , xxyy + f, yy w , xx − 2f, xy w , xy + f, xx w , yy + ⁎ f, xx R (26) where Nx0, Ny0 are in-plane compressive loads at movable edges (i.e Case and the first of Case 3) or are fictitious compressive edge loads at immovable edges (i.e Case and the second of Case 3) The mentioned conditions (24)–(26) can be satisfied identically if the panel deflection w is chosen by [5,15,16] w = W sin λm x sin δ n y , (27) where λm = mπ /a , δn = nπ /b , m , n = 1, 2, are natural numbers representing the number of half waves in the x and y directions, respectively; W is the amplitude of deflection Concerning with the initial imperfection w ⁎, we introduce an assumption it has the same form like the panel deflection w , i.e ⁎ ⁎ ⁎ ⁎ ⁎ B21 f, xxxx + B12 f, yyyy + (B11 + B22 − 2B66 ) f, xxyy + k2 ∇2w = 0, Case All edges of the FGM cylindrical panel are simply supported Two edges x = 0, a are freely movable, whereas the remaining two edges y = 0, b are immovable For this case, the boundary conditions are defined as ) ⁎ ⁎ − B12 B12 − B22 + CyT B22 , ( (25) + q − k1w (22) where w (x, y ) is a known function representing initial small imperfection of the panel Setting Eq (18) into Eq (16) gives the compatibility equation of an imperfect eccentrically stiffened FGM cylindrical panel as w ⁎ (x, y) = μh sin λm x sin δ n y , (28) where the coefficient μ varying between and represents imperfection size Introduction of Eqs (27) and (28) into the compatibility Eq (23), we define the stress function as N.D Duc et al / Thin-Walled Structures 96 (2015) 155–168 f = A1 cos 2λm x + A2 cos 2δ n y + A3 sin λm x sin δ n y + 4.1 Mechanical stability analysis Nx0 y2 Ny0 x2 , + (29) with λ m2 δn2 W (W + 2μh), A2 = W (W + 2μh) ⁎ ⁎ δn2 32A11λ m 32A22 A1 = λ m2 A3 = − ( (B ⁎ 21λ m ( ⁎ ⁎ ⁎ ⁎ δn4 + (B11 + B12 + B22 − 2B66 ) λ m2 ⁎ λm A11 + ⁎ δn4 A22 + ⁎ (A66 − ⁎ 2A12 ) λ m2 δn2 Consider a simply supported eccentrically stiffened FGM cylindrical panel with all movable edges and resting on elastic foundations Two cases of mechanical loads will be analyzed 4.1.1 Eccentrically stiffened FGM cylindrical panel under uniform external pressure Consider an eccentrically stiffened FGM cylindrical panel with movable edges and only subjected to uniform external pressure on the upper surface of the panel In this case, Nx0 = Ny0 = 0, and Eq (31) leads to W ) δ ) W ) ⁎ ⁎ ⁎ ⁎ λ m + A22 δn4 + (A66 R A11 − 2A12 ) λ m2 δn2 n (30) Setting Eqs (25)–(27) into Eq (22) and applying the Galerkin procedure for the resulting equation we obtain equation for determining nonlinear static analysis of eccentrically stiffened FGM cylindrical panels on elastic foundations ( ( − + ) ) + ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦ ( 16Bh4 (m B 2 a + n2 ) 2 ⎡ ⁎ 4 ⁎ ⁎ ⁎ ⁎ 4⎤ m3nπ 4Ba2 Rb ⎣ B21m Ba + (B11 + B22 − 2B66 ) m n Ba + B12 n ⎦ ⎡ A ⁎ m4 B + (A ⁎ − 2A ⁎ ) m2n2B + A ⁎ n4 ⎤ 8Bh3 ⎣ 11 ⎦ a a 22 66 12 ⁎ 16Bh2 ⎡⎣ A11 m4 Ba4 m5nπ 2Ba4 Rb2 ⁎ ⁎ ⁎ + (A66 − 2A12 ) m2n2Ba2 + A22 n4 ⎤⎦ 2m4 n2π 2Ba4 Rb ⁎ ⁎ ⁎ ⁎ m4 Ba4 + (A66 n4 ⎤⎦ 3Bh3 ⎡⎣ A11 − 2A12 ) m2n2Ba2 + A22 ⁎ ⁎ ⁎ ⁎ ⁎ 4⎤ m4 Ba4 + (B11 n ⎦ 2m2n2π 4Ba2 ⎡⎣ B21 + B22 − 2B66 ) m2n2Ba2 + B12 ⁎ ⁎ ⁎ ⁎ m4 Ba4 + (A66 n4 ⎤⎦ 3Bh4 ⎡⎣ A11 − 2A12 ) m2n2Ba2 + A22 , ⁎ B⁎ ⎞ m2n2π 4Ba2 ⎛ B21 π 2n2Rb ⎜⎜ ⁎ + 12 ⎟, + ⁎ ⎟ ⁎ A22 24A11 Bh 12Bh ⎝ A11 ⎠ mnπ ⎛ m4 Ba4 n4 ⎞ ⎜ + ⁎ ⎟⎟, ⎜ A⁎ A11 ⎠ 256Bh ⎝ 22 (33) k1a4 k a2 ⁎ ⁎ ⁎ ⁎ ⁎ ⁎ = hA11 = hA22 = hA12 , K2 = ⁎ , A11 , A22 , A12 , ⁎ D11 D11 ⁎ ⁎ ⁎ A66 = hA66 = , B11 ⁎ B11 ⁎ B⁎ B⁎ B22 ⁎ ⁎ = 12 , B21 = 21 , , B12 h h h ⁎ ⁎ D12 D22 ⁎ = , D12 = , h h ⁎ = , B22 h ⁎ D11 ⁎ = = , D22 , h h ⁎ D⁎ D66 ⁎ ⁎ ¯ = W /h, = 21 = D21 , D , Bh = b/h, Ba = b/a, W 66 h3 h3 ⁎ B66 ⁎ B66 ⁎ D11 Rb = b/ R δ4 ⎞ mnπ ⎛ λ m4 ⎜ ⁎ + n⁎ ⎟ W (W + μh)(W + 2μh) A11 ⎠ 64λm δ n ⎝ A22 mnπ Nx0 λ m2 + Ny0 δn2 (W + μh) 4λ m δ n Ny0 4q + + = 0, λm δ n R λm δ n ⁎ mnπ 4Ba2 D11 K2 16Bh4 K1 = W (W + μh) ⁎ ⎡ ⎤ B⁎ B δn +⎢ − ( 21 + 12 ) λm δ n ⎥ W (W + 2μh) ⁎ ⁎ ⁎ A22 A11 ⎣ 6A11Rλm ⎦ + and ) ( (32) ⁎ ⁎ ⁎ ⁎ ⁎ + mn5π 6D22 + m3n3π 6Ba2 (D12 + D21 + 4D66 ) m5nπ 6Ba4 D11 b41 = − 16Bh4 b21 = − + − ⁎ mnπ 2Ba4 D11 K1 b31 = − ( ) 2 2 ⎡ ⁎ 4 ⁎ ⁎ ⁎ ⁎ 4⎤ mnπ ⎣ B21m Ba + (B11 + B22 − 2B66 ) m n Ba + B12 n ⎦ , + ⎡ A ⁎ m4 B + (A ⁎ − 2A ⁎ ) m2n2B + A ⁎ n4 ⎤ 16Bh4 ⎣ 11 ⎦ a a 22 66 12 W ( )( ) where ) 8λ m δ n ⎡ λ m2 ⎢ ⁎ ⁎ ⁎ ⁎ ⎢ A11 λ m + A22 δn4 + (A66 − 2A12 ) λ m2 δn2 R ⎢ ⁎ ⁎ ⁎ ⁎ ⁎ ⎢ B21 λ m + B12 δn4 + (B11 + B22 − 2B66 ) λ m2 δn2 ⎢− ⁎ ⁎ ⁎ ⁎ 2 ⎢⎣ A11λ m + A22 δn + (A66 − 2A12 ) λ m δn ( ¯ W ¯ +μ W ¯ + 2μ , + b41 W + mnπ 4λ m δ n ⎧ ⁎ ⁎ ⁎ ⁎ 2⎤ ⎫ ⎡ ⁎ ⎪ λ m ⎣ B21λ m + B12 δn + (B11 + B22 − 2B66 ) λ m δn ⎦ ⎪ ⎪ R ⎡ A ⁎ λ + A ⁎ δ + (A ⁎ − 2A ⁎ ) λ δ ⎤ ⎪ ⎣ 11 m 22 n 66 12 m n ⎦ ⎪ ⎪ ⎪ ⎪ ⁎ ⁎ ⁎ ⁎ ⁎ λ m + B12 δn4 + (B11 + B22 − 2B66 ) λ m2 δn2 ⎤⎦ ⎪ ⎪ ⎡⎣ B21 ⎪− ⎪ ⎡ A ⁎ λ + A ⁎ δ + (A ⁎ − 2A ⁎ ) λ δ ⎤ ⎪ ⎪ ⎣ 11 m 22 n 66 12 m n ⎦ ⎨ ⎬ ⎪ λ4 ⎪ ⎪− m ⎪ ⎡ ⁎ ⁎ ⁎ ⁎ ⎪ R ⎣ A11λ m + A22 ⎪ δn4 + (A66 − 2A12 ) λ m2 δn2 ⎤⎦ ⎪ ⎪ ⎪ −D ⁎ λ − D ⁎ δ − (D ⁎ + D ⁎ + 4D ⁎ ) λ δ ⎪ m m n 22 n 11 12 21 66 ⎪ ⎪ 2 ⎪ − k2 λ m + δn − k1 ⎪ ⎩ ⎭ ) ¯ + b21W ¯ W ¯ + μ + b31W ¯ W ¯ + 2μ q = b11W b11 = ( 159 (34) Eq (32) may be used to trace postbuckling load–deflection curves of FGM cylindrical panels resting on elastic foundations subjected to uniform external pressure For a perfect panel (μ = 0) , Eq (32) leads to ) (31) where m , n are odd numbers Hereafter, we will consider in detail three problems corresponding to three mentioned loading types ¯ + (b21 + b31) W ¯ + b41 W ¯ q = b11W (35) 160 N.D Duc et al / Thin-Walled Structures 96 (2015) 155–168 4.1.2 Eccentrically stiffened FGM cylindrical panel under axial compressive loads A movable edges eccentrically stiffened cylindrical panel supported by elastic foundations and subjected to axial compressive loads Fx uniformly distributed at two curved edges x = 0, a in the absence of external pressure and thermal loads is considered In this case, the prebuckling force resultants are q = 0, Ny0 = 0, Nx0 = − Fx h (36) ( ) ¯ W ¯ + 2μ ¯ W W ¯ + b32 ¯ W ¯ + 2μ , + b22 W + b42 W ¯ ¯ W+μ W+μ ( ) (37) where b12 = + π ⎧⎡ ⎤ λm ⎪⎢ ⎥ δ n2 ⎪⎢ ⎥ ⁎ ⁎ ⁎ ⁎ 2 R ⎪ ⎢ A11λm + A22 δ n + (A66 − 2A12 ) λm δ n ⎥ ⎪⎢ ⎥ ⎪⎢ + B ⁎ δ + (B ⁎ + B ⁎ − B ⁎ ) λ δ ⎥ B ⁎ λm n m n ⎪⎢ 21 12 11 22 66 ⎨ − Nx0 = Φ1 + δn ⎥ ⎢ ⎥ ⁎ λ + A ⁎ δ + (A ⁎ − 2A ⁎ ) λ δ mnπ ⎪ A ⎥⎦ ⎪ ⎢⎣ 11 m 22 n 66 12 m n ⎪ ⎡ A⁎ ⎤ ⎪ + (B ⁎ A ⁎ + B ⁎ A ⁎ ) δ − 12 ⎥ ⎢ (B ⁎ A ⁎ + B ⁎ A ⁎ ) λm ⎪ 21 12 12 11 22 12 n ⁎ ⎢⎣ 11 11 R ⎥⎦ ⎪ ⁎ ⁎ ⎩ (A11A22 − A12 ) ( ) ( ⁎ m2π 2Ba2 D11 Bh2 + ⁎ n4 π 2D22 + m2Ba2 Bh2 4 ⁎ ⎣ A11 m Ba + Bh2 ⁎ ⁎ n4 ⎤⎦ − 2A12 ) m2n2Ba2 + A22 ⁎ (A66 ) ( ⁎ ⁎ ⁎ n2π (D12 + D21 + 4D66 ) m2Ba2 Rb2 2⎡ (41) Substitution of Eqs (27)–(29) into Eq (41) and then the result into Eq (40) give fictitious edge compressive loads The introduction of Eq (36) into Eq (31) gives Fx = b12 ∂u ⁎ ⁎ ⁎ ⁎ ⁎ ⁎ f, yy − A12 f, xx + B11 w, xx + B12 w, yy − (A22 = A22 − A12 ) Φ1 ∂x − w ,2x − w , x w ,⁎x, ∂v ⁎ ⁎ ⁎ ⁎ ⁎ ⁎ f, xx − A12 f yy + B22 w, yy + B21 w, xx − (A11 = A11 − A12 ) Φ1 ∂y w − w ,2y − w , y w ,⁎y + Ry + ) + A ⁎ δ ) W (W + 2μh) , (A ⁎ λm 11 12 n (A ⁎ A ⁎ − A ⁎ ) 11 22 12 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬W ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (42) + 2 ⎡ ⁎ 4 ⁎ ⁎ ⁎ ⁎ 4⎤ π ⎣ B21m Ba + (B11 + B22 − 2B66 ) m n Ba + B12 n ⎦ ⎡ A ⁎ m4 B + (A ⁎ − 2A ⁎ ) m2n2B2 + A ⁎ n4 ⎤ m2Ba2 Bh2 ⎣ 11 ⎦ a a 22 66 12 ⎧⎡ ⎤ λm ⎪⎢ ⎥ ⎪⎢ ⎥ ⁎ ⁎ ⁎ ⁎ 2 R ⎪ ⎢ A11λm + A22 δ n + (A66 − 2A12 ) λm δ n ⎥ ⎪⎢ ⎥ ⎪⎢ ⁎ ⁎ ⁎ ⁎ ⁎ 4 2 ⎥ B λ m + B δ n + (B + B − B ) λ m δ n ⎪⎢ 21 12 11 22 66 ⎨ − Ny0 = Φ1 + λm⎥ ⎥ ⁎ λ + A ⁎ δ + (A ⁎ − 2A ⁎ ) λ δ mnπ ⎪ ⎢ A ⎥⎦ ⎪ ⎢⎣ 11 m 22 n 66 12 m n ⎪ ⎡ A⁎ ⎤ ⎪ ⁎ ⁎ ⁎ ⁎ ⁎ ⁎ ⎢ (B A + B A ) λm + (B A + B ⁎ A ⁎ ) δ n2 − 22 ⎥ ⎪ 21 22 12 12 22 22 R ⎥⎦ ⎪ (A ⁎ A ⁎ − A ⁎ ) ⎢⎣ 11 12 ⎩ 11 22 12 + A ⁎ δ ) W (W + 2μh) + (A ⁎ λm 12 22 n (A ⁎ A ⁎ − A ⁎ ) 11 22 12 ( + ( ) ⁎ D11 m2Ba2 + n2 K2 m2Bh2 b22 = − + ⁎ Ba2 D11 K1 m2π 2Bh2 , 32mnBa2 Rb b42 = + ⁎ B⁎ ⎞ 8n ⎛ B21 ⎜ ⎟, + 12 ⁎ A⁎ A22 3mBh ⎝ 11 ⎠ ⎛ m4 B π2 n4 ⎞ ⎜ ⁎ a + ⁎ ⎟⎟ 2 2⎜ A11 ⎠ 16m Ba Bh ⎝ A22 (38) Eq (37) is employed to trace postbuckling load–deflection curves of the imperfect eccentrically stiffened FGM panel subjected to axial compressive loads For a perfect cylindrical panel (μ = 0) only subjected to axial compressive load Fx , Eq (37) leads to ¯ + b42 W ¯ 2, Fx = b12 + (b22 + b32 ) W (39) From which upper buckling compressive load may be obtained with W → as Fx = b12 4.2 Thermal stability analysis A simply supported eccentrically stiffened FGM cylindrical panel on elastic foundations with all immovable edges is considered The panel is subjected to uniform external pressure q and simultaneously exposed to temperature environments The in-plane condition on immovability at all edges, i.e u = at x = 0, a and v = at y = 0, b, is fulfilled in an average sense as [5] b ∫0 ∫0 a ∂u dxdy = 0, ∂x a ∫0 ∫0 b ∂v dydx = ∂y ) (43) Introducing Nx0, Ny0 at Eqs (42), (43) into Eq (31) gives ⁎ ⁎ ⁎ ⁎ m4 Ba4 + (A66 n4 ⎤⎦ 3π 2Bh ⎡⎣ A11 − 2A12 ) m2n2Ba2 + A22 2nRb ⁎ 3m3π 2Ba2 Bh A11 ) ( 2 ⎡ ⁎ 4 ⁎ ⁎ ⁎ ⁎ 4⎤ 32n ⎣ B21m Ba + (B11 + B22 − 2B66 ) m n Ba + B12 n ⎦ , + ⁎ ⁎ ⁎ ⁎ 3mBh2 ⎡⎣ A11 m4 Ba4 + (A66 n4 ⎤⎦ − 2A12 ) m2n2Ba2 + A22 b32 = − ) ( 2 ⎡ ⁎ 4 ⁎ ⁎ ⁎ ⁎ 4⎤ 2Rb ⎣ B21m Ba + (B11 + B22 − 2B66 ) m n Ba + B12 n ⎦ − ⎡ A ⁎ m4 B + (A ⁎ − 2A ⁎ ) m2n2B2 + A ⁎ n4 ⎤ Bh ⎣ 11 ⎦ a a 22 66 12 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬W ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (40) From Eqs (7) and (18) one can obtain the following expressions in which initial imperfection has been included ⎧ ⎡ ⁎ ⁎ ⁎ ⁎ ⁎ 2⎤ ⎪ λ ⎣⎢ B21λm + B12 δ n + (B11 + B22 − 2B66 ) λm δ n ⎦⎥ ⎪2 m ⎡ ⁎ ⁎ δ + (A ⁎ − A ⁎ ) λ δ ⎤ ⎪ R + A A λ ⎣⎢ 11 m 22 n 66 12 m n ⎦⎥ ⎪ ⎪ ⎪ ⎡ ⁎ ⁎ δ + (B ⁎ + B ⁎ − B ⁎ ) λ δ ⎤ B B λ + ⎪ ⎣⎢ 21 m 12 n 11 22 66 m n ⎦⎥ ⎪− ⎡ ⎤ ⎨ ⁎ ⁎ ⁎ ⁎ Φ1 = 2 2⎪ ⎣⎢ A11λm + A22 δ n + (A66 − 2A12 ) λm δ n ⎦⎥ λm ⎪ ⎪ λm ⎪− + A ⁎ δ + (A ⁎ − 2A ⁎ ) λ δ ⎤ ⎪ R ⎡⎢ A ⁎ λm ⎣ 11 22 n 66 12 m n ⎦⎥ ⎪ ⎪ ⁎ 2δ2 − k λ2 + δ2 − k ⎪ − D λm − D ⁎ δ n4 − (D ⁎ + D ⁎ + 4D ⁎ ) λm m n n 11 22 12 21 66 ⎩ ( ⎡ ⎡ λm ⎢ ⎢ ⎢ ⎢ + A ⁎ δ + (A ⁎ − 2A ⁎ ) λ δ R A ⁎ λm n m n ⎢ ⎢ 22 66 12 32δ n2 ⎢ 11 +⎢ + B ⁎ δ + (B ⁎ + B ⁎ − 2B ⁎ ) λ δ ⎢ 3mnπ ⎢ B ⁎ λm 66 m n 21 12 n 11 22 ⎢ ⎢− ⎢ ⎢ ⁎ λ + A ⁎ δ + (A ⁎ − 2A ⁎ ) λ δ A m n ⎢⎣ ⎢⎣ 11 22 66 12 m n ( ) ( ( ) ) ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦ ⎧⎡ ⎤ λm ⎪⎢ ⎥ δ n2 ⎪⎢ ⎥ ⁎ ⁎ ⁎ ⁎ 2 R ⎪ ⎢ A11λm + A22 δ n + (A66 − 2A12 ) λm δ n ⎥ ⎪⎢ ⎥ ⎪⎢ ⁎ ⁎ ⁎ ⁎ ⁎ 4 2 ⎥ B λm + B δ n + (B + B − 2B ) λ m δ n ⎪⎢ 21 12 11 22 66 ⎨ − − δn ⎥ ⎥ ⁎ λ + A ⁎ δ + (A ⁎ − 2A ⁎ ) λ δ mnπ ⎪ ⎢ A ⎥⎦ ⎪ ⎢⎣ 11 m 22 n 66 12 m n ⎪ ⎡ ⎪ A⁎ ⎤ ⁎ ⁎ ⁎ ⁎ ⁎ ⁎ ⎢ ⎪+ (B A + B A ) λm + (B A + B ⁎ A ⁎ ) δ n2 − 12 ⎥ 21 12 12 11 22 12 R ⎥⎦ ⎪ (A ⁎ A ⁎ − A ⁎ ) ⎢⎣ 11 11 ⎩ 11 22 12 ⎡ ⎤ ⁎ B⁎ δ n2 ⎥ W (W + 2μh) ⎢ B + − ( 21 + 12 ) δ n2 ⎥ ⎢ ⁎ A⁎ (W + μh) A⁎ mnπ ⎣⎢ 6A11Rλm 11 22 ⎦⎥ ⎡ ⎤ ⎛ ⁎ ⎞ ⁎ ⁎ ⁎ (A λm + A δ n2 ) ⎥ ⎢ ⎜ A11λm + A22 δ n ⎟ 11 12 −⎢ ⁎ A⁎ ⎥ W (W + 2μh) ⎟⎟ + ⁎ ⁎ ⁎ ⎜⎜ A λ 16 ⎢⎣ m⎝ 11 22 ⎠ (A11A22 − A12 ) ⎥⎦ ( ) ( ) ( ) ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ W ⎬ ⎪ (W + μh) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬W ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (44) In this paper, the eccentrically stiffened FGM cylindrical panel is exposed to temperature environments uniformly raised from stress free initial state Ti to final value Tf and temperature increment ΔT = Tf − Ti is considered to be independent from thickness variable The thermal parameter is obtained from Eq (12) as Φ1 = − PhΔT, where (45) N.D Duc et al / Thin-Walled Structures 96 (2015) 155–168 P= ⎡ Emc αc + Ec αmc E α ⎤ + mc mc ⎥ ⎢ Ec αc + −v⎣ N+1 2N + ⎦ 161 (46) Setting Eq (45) into Eq (44) gives ΔT = b13 ( ) ¯ W ¯ + 2μ ¯ W W ¯ + b33 ¯ W ¯ + 2μ , + b23 W + b43 W ¯ ¯ W+μ W+μ ( ) (47) bi3 in which specific expressions of coefficients (i = 1, 4) are given in Appendix A Eq (47) shows the relationship of thermal load–deflection of the eccentrically stiffened FGM panel in postbuckling state and used to trace postbuckling curves of the FGM panel under thermal load The two sides of Eq (47) are temperature dependence which makes it very complex The iterative algorithm is used to determine the deflection–load relations in the buckling period of the FGM panel To be more specific, given the volume fraction index N , the geometrical parameters (b /a, b /h, b /R ) and the value of W /h, we can use these values to determine ΔT in Eq (47) as the follows: we choose an initial step for ΔT1 on the right side in Eq (47) with ΔT = ( T = T0 = 300 K ) In the next iterative step, we replace the known value of ΔT found in the previous step to determine the right side of Eq (47), ΔT2 This iterative procedure will stop at the kth -steps if ΔTk satisfies the condition |ΔT − ΔTk | ≤ ε Here, ΔT is a desired solution for the temperature and ε is a tolerance used in the iterative steps If the imperfection μ = and W → 0, from above expression (47) gives ΔT = b13 Fig Comparisons of nonlinear load–deflection curves with results of Tung [15] for the unstiffened FGM cylindrical panel under axial compressive loads 4.3 Thermo-mechanical stability analysis The simply supported FGM cylindrical panel with tangentially restrained edges is assumed to be subjected to external pressure q uniformly distributed on the outer surface of the panel and exposed to uniformly raised temperature field Subsequently, setting Eq (45) into Eqs (42) and (43) then the result into Eq (31) give ( ) ( ¯ + b24 W ¯ W ¯ + μ + b34 W ¯ W ¯ + 2μ q = b14 W + ¯ b44 W ( W¯ + μ)( W¯ + 2μ) − b54 P ΔT , ) (48) in which specific expressions of coefficients bi4 (i = 5) are given in Appendix B Eq (48) expresses explicit relation of pressure-deflection curves for eccentrically stiffened FGM cylindrical panels rested on elastic foundations and under combined action of uniformly raised temperature field and uniform external pressure Numerical results and discussion 5.1 Validation of the present approach To validate the present study, firstly, Fig compares the results of this paper for an unstiffened FGM cylindrical panel under axial compressive loads with the results given in work of Tung [15] with different values of elastic foundation stiffness K1 and K2 in the case of temperature independent properties Fx is found from Eq (37) and the data base in this case is taken: b /a = 1, b /h = 50, b /R = 0.1, N = 1, μ = 0.1, K1 = K2 = Secondly, Fig compares the present results with those of Duc et al [16] for stiffened and unstiffened FGM cylindrical panel under uniform external pressure based on classical shell theory in the case of temperature independent properties The input Fig Comparisons of nonlinear load–deflection curves with results of Duc et al [16] for the unstiffened and stiffened FGM cylindrical panel under uniform external pressure 162 N.D Duc et al / Thin-Walled Structures 96 (2015) 155–168 Fig Comparisons of thermal postbuckling behavior with results of Shen and Wang [1] for the unstiffened FGM cylindrical panels under uniform temperature rise parameters are: b/a ¼1, b/h¼ 50, b/R ¼0.5, K1 ¼10, K2 ¼30 where q is found from Eq (32) Finally, the thermal postbuckling behavior of Si3N4/SUS304 unstiffened FGM cylindrical panels under uniform temperature rise with two values of Winkler elastic foundation stiffness are compared in Fig with the theoretical results of Shen and Wang [1] based on the higher order shear deformation shell theory The temperature dependent properties are taken into account The input parameters are chosen as: N = 0.5, m = n = 1, h = mm , b /h = 40, a/b = 1.2, a/R = 0.5, Fig Effects of temperature increment on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under uniform external pressure (movable edges) 5.2 Effects of temperature Figs 5–7 show effects of temperature increment ΔT on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under uniform external pressure (movable edges), axial compressive loads (movable edges) and uniform external pressure (immovable edges), respectively Obviously, the load-carrying capacity of the panel with temperature independent properties is higher than the one of the panel with temperature dependent properties Moreover, the increase of temperature increment leads to the reduction of load-carrying capacity of the panel μ = 0.05 The elastic foundation stiffness in this comparison are identified as K1⁎ = k1b4 Em h3 , K2⁎ = k b2 Em h3 with Em is determined at room temperature As can be seen that good agreements are obtained in these three comparisons Next, we will investigate the effects of the volume fraction index, the geometrical dimensions, elastic foundations, imperfections and stiffeners on the nonlinear response of the eccentrically stiffened FGM cylindrical panel The effective material properties with dependent temperature in Eq (5) are listed in Table [1,2,13] The Poisson's ratio is v = 0.3 The parameters for the stiffeners are [16] 5.3 Effects of elastic foundations and initial imperfection Figs 8–11 indicate effects of elastic foundations on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under uniform external pressure (movable edges), axial compressive loads (movable edges), uniform temperature rise (immovable edges) and uniform external pressure (immovable edges), respectively Obviously, the load-carrying capacity of the panel becomes considerably higher due to the support of elastic foundations In addition, the beneficial effect s1 = s2 = 0.4 m, z1 = z2 = 0.0225 m, h1 = h2 = 0.003 m, d1 = d2 = 0.004 m Table Material properties of the constituent materials of the considered FGM panel Material Property P0 PÀ1 P1 P2 P3 Si3N4 (Ceramic) E (Pa) ρ (kg/m3) 348.43e9 2370 5.8723eÀ 0 À 3.70eÀ 9.095e À 2.160e À 0 À 8.946e À 11 0 13.723 0 0 201.04e9 8166 12.330eÀ 0 3.079eÀ 8.086e À À 6.534e À 0 0 15.379 0 0 α (K−1) k (W/mK) SUS304 (Metal) E (Pa) ρ (kg/m3) α (K−1) k (W/mK) N.D Duc et al / Thin-Walled Structures 96 (2015) 155–168 Fig Effects of temperature increment on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under axial compressive loads (movable edges) 163 Fig Effects of elastic foundations on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under uniform external pressure (movable edges) Fig Effects of temperature increment on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under uniform external pressure (immovable edges) of the Pasternak foundation on the postbuckling response of the eccentrically stiffened FGM cylindrical panels is better than the Winkler one The effects of initial imperfection with the coefficient μ on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under different type of loads are also shown in Figs 8–11 It can be seen that the perfect cylindrical panel has a better mechanical and thermal loading capacity than those of the imperfect panel 5.4 Effects of volume fraction index Figs 12–15 show effect of volume fraction index N on the Fig Effects of elastic foundations on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under axial compressive loads (movable edges) nonlinear response of the imperfect and perfect eccentrically stiffened FGM cylindrical panels under uniform external pressure (movable edges), axial compressive loads (movable edges), uniform temperature rise (immovable edges) and uniform external pressure (immovable edges), respectively As expected, the loadcarrying capacity of the FGM panel gets better if the volume N increases This is reasonable because when N is increased, the ceramic volume fraction is increased; however, elastic module of ceramic is higher than metal ( Ec > Em ) The results from these 164 N.D Duc et al / Thin-Walled Structures 96 (2015) 155–168 Fig 12 Effect of volume fraction index N on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under uniform external pressure (movable edges) Fig 10 Effects of elastic foundations on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under uniform temperature rise (immovable edges) Fig 13 Effect of volume fraction index N on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under axial compressive loads (movable edges) figures also show that the buckling and postbuckling load is very sensitive to the change of initial imperfection 5.5 Effects of stiffeners Fig 11 Effects of elastic foundations on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under uniform external pressure (immovable edges) The influences of stiffeners as well as initial imperfection on the nonlinear postbuckling response of FGM cylindrical panels under uniform external pressure (movable edges), axial compressive loads (movable edges), uniform temperature rise (immovable edges) and uniform external pressure (immovable edges) N.D Duc et al / Thin-Walled Structures 96 (2015) 155–168 165 Fig 14 Effect of volume fraction index N on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under uniform temperature rise (immovable edges) Fig 16 Effects of stiffeners on the nonlinear response of FGM cylindrical panels under uniform external pressure (movable edges) Fig 15 Effect of volume fraction index N on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under uniform external pressure (immovable edges) are shown in Figs 16–19, respectively We can see that postbuckling load-carrying capability of the stiffened panel is higher than that of unstiffened panel In other words, the stiffeners can enhance the loading capacity for the cylindrical FGM panel Moreover, from these figures, the initial imperfection considerably impact on the nonlinear response of stiffened and unstiffened FGM cylindrical panel Fig 17 Effects of stiffeners on the nonlinear response of FGM cylindrical panels under axial compressive loads (movable edges) 166 N.D Duc et al / Thin-Walled Structures 96 (2015) 155–168 Fig 20 Effects of ratio b/a on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under uniform external pressure (movable edges) Fig 18 Effects of stiffeners on the nonlinear response of FGM cylindrical panels under uniform temperature rise (immovable edges) Fig 21 Effects of ratio b/h on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under axial compressive loads (movable edges) Fig 19 Effects of stiffeners on the nonlinear response of FGM cylindrical panels uniform external pressure (immovable edges) 5.6 Effects of geometrical parameters Figs 20–22 analyze the effects of geometrical parameters on the nonlinear response of eccentrically stiffened FGM cylindrical panels Specifically, Figs 20 and 21 indicate the influences of ratios b /a, b /h on the nonlinear response of eccentrically stiffened FGM panels under uniform external pressure (movable edges) and axial compressive loads (movable edges), respectively The thermal postbuckling behavior of the eccentrically stiffened FGM panels with various values of ratio b /R is illustrated in Fig 22 As can be observed, the load-carrying capacity of the eccentrically stiffened FGM panel increases when increasing the ratios b /a, b /h, and b /R The influences of initial imperfection on the nonlinear postbuckling of the eccentrically stiffened FGM cylindrical panels also N.D Duc et al / Thin-Walled Structures 96 (2015) 155–168 167 Appendix A ⎡ ⁎ 4 ⎤ ⎧ ⎫ ⁎ ⁎ ⁎ B m B + B ⁎ n4 + (B11 + B22 − 2B66 ) m2n2Ba2 ⎦ ⎪ Rb ⎣ 21 a 12 ⎪ ⎡ ⎤ ⁎ ⁎ ⁎ ⁎ 4 2 B ⎪ h ⎣ A11 m Ba + A22 n + (A66 − 2A12 ) m n Ba ⎦ ⎪ ⎪ ⎪ ⎡ ⁎ 4 ⁎ ⁎ ⁎ ⁎ ⎪ 2 2⎤ ⎪ ⎣ B21m Ba + B12 n + (B11 + B22 − 2B66 ) m n Ba ⎦ π ⎪− ⎪ 2 ⎡ ⁎ 4 ⁎ ⁎ ⁎ 2 2⎤ m B B ⎪ ⎪ a h ⎣ A11 m Ba + A22 n + (A66 − 2A12 ) m n Ba ⎦ ⎪ ⎪ ⎪ ⎪ m2B R ⎨ ⎬ a b b1 = − P ⎪ − π ⎡ A⁎ m4B + A⁎ n4 + (A ⁎ − 2A⁎ ) m2n2B ⎤ ⎪ a a ⎣ 11 ⎦ 22 66 12 ⎪ ⎪ 2⎪ ⎪ π2 4 ⁎ ⁎ ⁎ ⁎ ⁎ n π ⎪ − m2B B D11m Ba + D22 n − (D12 + D21 + 4D66 ) B ⎪ a h h ⎪ ⎪ ⎪ D ⁎ K2 m2Ba2 + n2 ⎪ ⁎ D11 K1Ba2 ) 11 ( ⎪ ⎪ − 2 ⎪ ⎪ B m Bh ⎩ ⎭ h ( ) , b23 Fig 22 Effects of ratio b/R on the nonlinear response of the eccentrically stiffened FGM cylindrical panels under uniform temperature rise (immovable edges) shown in Figs 20–22 Concluding remarks The paper presents an analytical investigation on the buckling and postbuckling of imperfect eccentrically stiffened thin FGM cylindrical panels resting on elastic foundations subjected to mechanical loads, thermal loads and the combination of these loads The formulations are based on the classical shell theory taking into account geometrical nonlinearity, initial imperfection, the Lekhnitsky smeared stiffeners technique and Pasternalk elastic foundations The Galerkin method is used to obtain explicit expressions of load–deflection curves The results show that elastic foundations and stiffeners have a beneficial influence on the buckling loads and postbuckling load carrying capacity of the FGM cylindrical panels The study also shows the effects of volume fraction index, imperfection and geometrical parameters on the nonlinear response of FGM cylindrical panels ⎧ ⎡ 32mnB R ⎫ ⎤ a b ⎪⎢ ⎪ ⎥ ⁎ ⁎ ⁎ ⁎ 4 2 ⎪ ⎢ 3π Bh ( A11 m Ba + A22 n + (A66 − 2A12 ) m n Ba ) ⎥ ⎪ ⎪⎢ ⎪ ⁎ ⁎ ⁎ ⁎ ⁎ 4 2 ⎥ ⎪ ⎢ 32n ( B21m Ba + B12 n + (B11 + B22 − 2B66 ) m n Ba ) ⎥ ⎪ − ⎪ ⎢ 3mBh2 A⁎ m4B + A⁎ n4 + (A ⁎ − 2A⁎ ) m2n2B ⎥ ⎪ a a ( ) 11 22 66 12 ⎦ ⎪⎣ ⎪ ⎪⎧⎡ ⎪ ⎫ ⎤ 4m n Ba Rb ⎪⎪ ⎪ ⎪ ⎥ ⎢ 4 2 ⁎ ⁎ ⁎ ⁎ ⎪ ⎪ π Bh ( A11 m Ba4 + A22 n + (A66 − 2A12 ) m n Ba2 ) ⎪⎪ ⎥ ⎢ 1⎪ ⎢ ⎪⎪ ⎥ ⁎ ⁎ ⁎ ⎬ = − ⎨⎪ 4n B ⁎ m4B + B ⁎ n4 + (B11 + B22 − 2B66 ) m2n2Ba2 ) ⎪⎪ , b2 ⎥ P ⎪ ⎪ ⎢ − ( 21 a 12 ⎪ ⎪ 4 2 ⁎ ⁎ ⁎ ⁎ ⎪ ⎢⎣ mBh2 ( A11 m Ba + A22 n + (A66 − 2A12 ) m n Ba ) ⎥⎦ ⎪ ⎪⎪ ⎪⎪ ⁎ ⎬ ⎨ 4A12 Rb ⎪ ⎪ ⎪⎪ ⎪ ⎪ − mnπ 2B (A⁎ A⁎ − A⁎ 2) h 11 22 ⎪ ⎪ 12 ⎪ ⎪ ⎪⎪ ⎪⎪+ ⎪ ⎪ ⁎ ⁎ ⁎ ⎪ ⎪ mnBh2 (A11 A22 − A12 ) ⎪⎪ ⎪⎪ ⎪ ⎪ 2 ⁎ ⁎ ⁎ ⁎ ⁎ ⁎ ⁎ ⁎ ⎡ ⎤ ⎪ ⎪ ⎣ (B A + B A ) m Ba + (B A + B22 A ) n ⎦⎪⎪ 12 11 12 11 11 21 12 ⎭⎭ ⎩⎩ ⎫ ⎧ ⎡ 32mnB R ⎤ a b ⎪ ⎪⎢ ⎥ 4 2 ⁎ ⁎ ⁎ ⁎ B π A m B A n A A m n B + + ( − ) h a) ⎥ ( 11 a 22 ⎪ ⎪⎢ 66 12 ⎪ ⎪⎢ 4 2 ⎥ ⁎ ⁎ ⁎ ⁎ ⁎ B m B B n B B B m n B + + ( + − ) a) ⎪ ⎪ ⎢ 32n ( 21 a 12 11 22 66 ⎥ − ⎪ ⎪ ⎢ 3mBh2 A⁎ m4B + A⁎ n4 + (A ⁎ − 2A⁎ ) m2n2B ⎥ a a) ( 11 22 66 12 ⎦ ⎣ ⎪ ⎪ ⎪⎧⎡ ⎫⎪ ⎤ 4m n Ba2 Rb ⎪⎪ ⎪⎪ ⎥ ⁎ ⁎ ⁎ ⁎ ⎪ ⎪ ⎢ π 2Bh ( A11 m4Ba4 + A22 n + (A66 − 2A12 ) m2n2Ba2 ) ⎪⎪ ⎥ ⎢ 1⎪⎪⎢ ⎪⎪ ⎥ ⎬ ⁎ ⁎ ⁎ = − ⎨⎪ 4n B ⁎ m4B + B ⁎ n4 + (B11 + B22 − 2B66 ) m2n2Ba2 ) ⎪⎪ , b3 ⎥ P ⎪ ⎢ − ( 21 a 12 ⎪⎪ ⁎ ⁎ ⁎ ⁎ m4Ba4 + A22 n + (A66 − 2A12 ) m2n2Ba2 ) ⎥ ⎪ ⎪ ⎢⎣ mBh2 ( A11 ⎦ ⎪⎪ ⎪⎪ ⁎ ⎬⎪ 4A12 Rb ⎪⎨ − ⎪⎪ ⎪⎪ ⁎ ⁎ ⁎ mnπ 2Bh (A11 A22 − A12 ) ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ + ⎪⎪ ⎪ ⎪ mnB (A⁎ A⁎ − A⁎ 2) 12 h 11 22 ⎪⎪ ⎪⎪ ⎡ (B ⁎ A ⁎ + B ⁎ A ⁎ ) m2B2 + (B ⁎ A ⁎ + B ⁎ A ⁎ ) n2⎤⎪ ⎪ ⎪⎪ ⎪ ⎦⎪ a 22 12 21 12 12 11 ⎭⎭ ⎩ ⎩ ⎣ 11 11 = − = ⁎ ⁎ ⁎ ⁎ ⎤ ⎡ 1⎢ 2n R b 8n B21A22 + B12 A11 ⎥ − ( ) , b4 ⁎ ⁎ ⁎ P ⎢⎣ 3m3π 2A11 ⎥⎦ A11 A22 3mBh2 Ba Bh ⎡ ⁎ ⁎ ⎤ ⎛ A ⁎ m4 B + A ⁎ n4 ⎞ π (A11 Ba + A12 n) ⎥ π2 1⎢ a 22 ⎟⎟ + ⎜⎜ 11 ⁎ ⁎ 2 ⁎ ⁎ ⁎ ⎥ P ⎢⎣ 16m Ba Bh ⎝ A11 A22 ⎠ 8Bh2 (A11 − A12 )⎦ A22 168 N.D Duc et al / Thin-Walled Structures 96 (2015) 155–168 Appendix B b14 ⎤ ⎡ ⎡ B ⁎ m4B + B ⁎ n4 + (B ⁎ + B ⁎ − 2B ⁎ ) m2n2B ⎤ a⎦ 11 22 66 ⎥ ⎢ m nπ Ba Rb ⎣ 21 a 12 ⎡ ⁎ 4 ⁎ ⁎ ⁎ 2 2⎤ B ⎥ ⎢ h ⎣ A11 m Ba + A22 n + (A66 − 2A12 ) m n Ba ⎦ ⎥ ⎢ ⎡ ⁎ 4 ⁎ ⁎ ⁎ ⁎ 2 2⎤ ⎥ ⎢ B m Ba + B12 n + (B11 + B22 − 2B66 ) m n Ba ⎦ ⎥ ⎢ − mnπ4 ⎣ 21 ⎡ ⎤ ⁎ ⁎ ⁎ ⁎ m4Ba4 + A22 n + (A66 − 2A12 ) m2n2Ba2 ⎦ ⎥ ⎢ 16Bh ⎣ A11 ⎥ ⎢ 2 = − ⎢ m nπ Ba Rb ⎥ − ⎡ ⁎ 4 ⁎ ⁎ ⁎ 2 2⎤ ⎥ ⎢ 16Bh2 ⎣ A11 m Ba + A22 n + (A66 − 2A12 ) m n Ba ⎦ ⎥ ⎢ ⎢ mnπ mnπ 4 2 2⎥ ⁎ ⁎ ⁎ ⁎ ⁎ ⎢ − 16B D11m Ba + D22 n − 16B (D12 + D21 + 4D66 ) m n Ba ⎥ h h ⎥ ⎢ ⁎ ⁎ Ba mnπ 2K1D11 Ba ⎥ ⎢ mnπ 4K2 D11 2 + − m B n a ⎥⎦ ⎢⎣ 16Bh4 16Bh4 ( ) ( ) ⎡ ⎡ ⎤⎤ m4n2π 2Ba4 Rb ⎢ ⎢ ⎥⎥ A⁎ m4B + A⁎ n4 + (A ⁎ − 2A⁎ ) m2n2B B a) ⎢ ⎢ h ( 11 a 22 66 12 ⎥⎥ ⎢ 12 ⎢ ⎥⎥ 2 ( B ⁎ m4Ba4 + B ⁎ n4 + (B ⁎ + B ⁎ − 2B ⁎ ) m2n2Ba2 ) 12 11 22 66 ⎢ ⎢ m n π Ba 21 ⎥⎥ − ⁎ ⁎ ⁎ ⁎ ⎥ ⎢ ⎢ Bh A11 m4Ba4 + A22 n + (A66 − 2A12 ) m2n2Ba2 ) ⎥ ( ⎦⎥ ⎣ , b24 = − ⎢⎢ ⎥, b3 ⎡ (B ⁎ A⁎ + B ⁎ A⁎ ) m4B + (B ⁎ A⁎ + B ⁎ A⁎ ) m2n2B π a a 11 11 21 12 12 11 22 12 ⎥ ⎢ ⎢ ⁎ ⁎ ⁎ ⎥ ⎢ ⎣ Bh4 A22 − A12 (A11 ) ⎥ ⎢ ⎤ 2 ⁎ ⎥ ⎢ A12 m π Ba Rb ⎥ ⎥ ⎢ − ⁎ ⁎ ⁎ Bh (A11 A22 − A12 ) ⎦ ⎦ ⎣ ⁎ ⎡ n2π 2R B ⁎ m2n2π 4Ba2 ⎤ B b ⎥, b4 = −⎢ − ( 21 + 12 ) ⁎ ⁎ ⁎ ⎢⎣ 24A11 Bh ⎥⎦ A11 A22 Bh4 ⎡ 4 2 ⎤ ⁎ ⁎ mnπ ⎛ m4 Ba4 n4 ⎞ mnπ (A11 m B + A12 m n Ba ) ⎥ ⎜⎜ ⁎ + ⁎ ⎟⎟ + =⎢ ,b ⎢⎣ 256Bh4 ⎝ A22 ⎥⎦ ⁎ ⁎ ⁎ A11 ⎠ 128Bh4 A22 (A11 − A12 ) = m3nπ 4Ba2 (W + μh) 16Bh2 References [1] H.S Shen, H Wang, Thermal postbuckling of FGM cylindrical panels resting on elastic foundations, Aero 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(22) and (23) are nonlinear equations in terms of variables w and f and they are used to investigate the nonlinear stability of FGM eccentrically stiffened cylindrical panels on elastic foundations. .. initial imperfection considerably impact on the nonlinear response of stiffened and unstiffened FGM cylindrical panel Fig 17 Effects of stiffeners on the nonlinear response of FGM cylindrical panels