DSpace at VNU: Nonlinear response of pressure-loaded functionally graded cylindrical panels with temperature effects

Composite Structures 92 (2010) 1664–1672 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct Nonlinear response of pressure-loaded functionally graded cylindrical panels with temperature effects Nguyen Dinh Duc a, Hoang Van Tung b,* a b University of Engineering and Technology, Vietnam National University, Ha Noi, Viet Nam Faculty of Civil Engineering, Hanoi Architectural University, Ha Noi, Viet Nam a r t i c l e i n f o Article history: Available online December 2009 Keywords: Nonlinear analysis Functionally graded materials Imperfection Temperature effects Cylindrical panel a b s t r a c t This paper presents an analytical approach to investigate nonlinear response of functionally graded cylindrical panels under uniform lateral pressure with temperature effects are incorporated Material properties are assumed to be temperature-independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents Equilibrium and compatibility equations for cylindrical panels are derived by using the classical shell theory with both geometrical nonlinearity in von Karman–Donnell sense and initial geometrical imperfection are taken into consideration The resulting equations are solved by Galerkin method to determine explicit expressions of nonlinear load-deflection curves Stability analysis for a simply supported panel shows the effects of material and geometric parameters, in-plane restraint and temperature conditions, and imperfection on the nonlinear response of the panel Ó 2009 Elsevier Ltd All rights reserved Introduction Curved panel elements such as cylindrical panels constitute a major portion of the structure of aerospace vehicles They are found in the aircraft components as primary load carrying structures such as fuselage sections as well as in spacecraft and missile structural applications Moreover, these elements can also be found in various industries such as shipbuilding, transportation, and building constructions Therefore, problems relating to stability of such structural elements have practical importance and attracted many attentions of science community Mahayni [2] established the large-deflection equations for a thin shallow shell and treated buckling and postbuckling behaviors of a simply supported isotropic cylindrical panel subjected to thermal loads Palazotto and his co-workers reported a series of investigations on stability of cylindrical panels made of laminated composite materials [3–8] They considered many different effects such as moisture and temperature [3], higher order shear deformation, and nonlinearity [5–7] on the stability of cylindrical panels which are assumed to be subjected to different loads such as transverse concentrated and distributed loads, shear and dynamic loads as well Yamada and Croll [9] studied buckling of isotropic cylindrical panels under uniform external pressure by using a fully nonlinear Ritz solution procedure Birman and Bert [10] investigated effects of temperature on buckling and postbuckling of reinforced and unstiffened plates and shells subjected to the simultaneous action * Corresponding author E-mail address: htung0105@gmail.com (H Van Tung) 0263-8223/$ - see front matter Ó 2009 Elsevier Ltd All rights reserved doi:10.1016/j.compstruct.2009.11.033 of a thermal field and an axial loading Geier and Singh [11] presented a simple analytical solution for computing bifurcation buckling loads of thin and moderately thick orthotropic cylindrical shells and panels subjected to compression and normal pressure Jaunky and Knight [12] have assessed the accuracy of different shell theories, i.e Sanders-Koiter, Love and Donnell shell theories for buckling of anisotropic and isotropic curved panels Bukling loads using these shell theories were obtained by using a Rayleigh–Ritz method and compared with finite element results The dynamic stability of simply supported, isotropic cylindrical panels under combined static and periodic axial forces has been investigated by Ng et al [13] using the generalized Donnell’s shell theory Librescu and Chang [14,15] and Librescu and Lin [16] used analytical approach to investigate postbuckling behavior of laminated composite flat and curved panels under various loading conditions such as axial, transverse, and combined loads with effect of shear deformation, imperfection, and elastic foundation are included This approach is extended by Hause et al [17] for anisotropic flat and curved sandwich panels Librescu et al [18] presented an excellent analytical investigation on nonlinear response of flat and curved panels subjected to thermomechanical loads Shen [20] reported a postbuckling analysis of a shear deformable laminated cylindrical panel with piezoelectric actuators subjected to the combined action of mechanical, electric and thermal loads Functionally Graded Materials (FGMs) are microscopically composites usually made from a mixture of metals and ceramics By gradually varying the volume fraction of constituent materials, their material properties exhibit a smooth and continuous change from one surface to another, thus eliminating interface problems 1665 N Dinh Duc, H Van Tung / Composite Structures 92 (2010) 1664–1672 and mitigating thermal stress concentrations Due to high performance heat resistance capacity, FGMs are now developed as structural components in ultrahigh temperature environments and extremely large thermal gradients such as aircraft, space vehicles, nuclear plants, and other engineering applications Despite the evident importance in practical applications, investigations in the buckling and postbuckling of FGM cylindrical panels are still limited in number A postbuckling analysis of axially loaded FGM cylindrical panels in thermal environments was reported by Shen [21] He used higher order shear deformation shell theory in conjunction with a boundary layer theory of shell buckling and a asymptotic perturbation technique to determine buckling loads and postbuckling paths of FGM cylindrical panels Both initial imperfection and temperature-dependent properties are accounted for in his work Following this direction, Shen and Leung [22] analyzed postbuckling behavior of FGM cylindrical panels subjected to uniform lateral pressure on point of view of the existence of bifurcation-type buckling Yang et al [23] studied postbuckling of FGM cylindrical panels subjected to combination of axial load and a uniform temperature change They employed the classical shell theory and a differential quadrature iteration algorithm to predict critical bifurcation-type buckling load (which only occur for completely clamped panels) and to trace postbuckling paths of perfect and imperfect panels with temperature-dependent properties Recently, the nonlinear response of functionally graded cylindrical shell panels under mechanical and thermal loads has been investigated by Zhao and Liew [24] using the element-free kp-Ritz method and the formulation is based on a modified version of Sanders nonlinear shell theory In this paper, the nonlinear response of FGM cylindrical panels under uniform lateral pressure with and without temperature effects is investigated by an analytical approach Formulation is based on the classical shell theory with both von Karman–Donnell type of kinematic nonlinearity and initial geometrical imperfection are taken into consideration The resulting equations are solved by Galerkin procedure to obtain closed-form expressions of nonlinear load-deflection curves Stability analysis for a simply supported panel shows the effects of material and geometric parameters, inplane restraint and temperature conditions, and imperfection on the nonlinear response of the panels Functionally graded cylindrical panels  k 2z ỵ h ; 2h V m zị ẳ V c ðzÞ ð2Þ where P denotes a temperature-independent material property, and subscripts m and c stand for the metal and ceramic constituents, respectively From Eqs (1) and (2) the effective properties of FGM cylindrical panel can be written as follows in which Poissons ratio m is assumed to be constant ẵEzị; azị; Kzị ẳ ẵEm ; am ;K m ỵ ½Ecm ; acm ; K cm Š  k 2z ỵ h ; 2h mzị ẳ m; 3ị where Ecm ¼ Ec À Em ; acm ¼ ac À am ; K cm ¼ K c À K m : ð1Þ where V c and V m are the volume fractions of ceramic and metal constituents, respectively, and volume fraction index k is a nonnegative number that defines the material distribution We assume that the effective properties Peff of functionally graded panel, such as the modulus of elasticity E, the coefficient of thermal expansion a, and the coefficient of thermal conduction K, change in the thickness direction z and can be determined by the linear rule of mixture as ð4Þ It is evident that E ¼ Ec ; a ¼ ac ; K ¼ K c at z ¼ h=2 and E ¼ Em ; a ¼ am ; K ¼ K m at z ¼ Àh=2 Governing equations In the present study, the classical shell theory is used to obtain the equilibrium and compatibility equations as well as expressions of buckling loads and nonlinear load-deflection curves of FGM cylindrical panels The strains across the panel thickness at a distance z from the mid-plane are ex ẳ exm ỵ zkx ; ey ẳ eym ỵ zky ; cxy ẳ cxym ỵ 2zkxy ; 5ị where exm and eym are the normal strains, cxym is the shear strain at the middle surface of the panel, and kij are the curvatures According to the classical shell theory, the strains at the middle surface and the curvatures are related to the displacement components u; v ; w in the x; y; z coordinate directions as [1] exm ẳ u;x ỵ w2;x =2; eym ẳ v ;y w=R ỵ w2;y =2; cxym ẳ u;y ỵ v ;x ỵ w;x w;y ; kx ¼ Àw;xx ; ky ¼ Àw;yy ; kxy ¼ Àw;xy ; 6ị where y ẳ Rh, and geometrical nonlinearity in von Karman–Donnell sense is accounted for, also, subscript (,) indicates the partial derivative Hooke law for a panel is defined as rx ; ry ị ẳ ẵE=1 m2 ịẵex ; ey ị ỵ mey ; ex ị þ mÞaDTð1; 1ފ; Consider a functionally graded cylindrical panel with radius of curvature R, thickness h, axial length a and arc length b The panel is made from a mixture of ceramics and metals, and is defined in coordinate system ðx; h; zÞ, where x and h are in the axial and circumferential directions of the panel and z is perpendicular to the middle surface and points inwards ðÀh=2 z h=2Þ Suppose that the material composition of the panel varies smoothly along the thickness in such a way that the inner surface is ceramic-rich and the outer surface is metal-rich by following a simple power law in terms of the volume fractions of the constituents as [23] V c zị ẳ Peff zị ẳ P c V c zị ỵ Pm V m zị; sxy ẳ ẵE=21 ỵ mފcxy : ð7Þ The force and moment resultants of a panel are expressed in terms of the stress components through the thickness as Nij ; M ij ị ẳ Z h=2 rij 1; zịdz; ij ẳ x; y; xy: 8ị Àh=2 Substituting Eqs (3), (5) and (7) into Eq (8) gives the constitutive relations E1 E2 Um exm ỵ meym ị ỵ kx ỵ mky ị ; m2 m2 1m E1 E2 Um eym ỵ mexm ị ỵ ky ỵ mkx ị ; Ny ¼ À m2 À m2 1Àm E1 E c ỵ kxy ; N xy ẳ 21 ỵ mị xym ỵ m E2 E3 Ub exm ỵ meym ị ỵ kx ỵ mky ị ; Mx ¼ À m2 À m2 1Àm E2 E3 Ub eym ỵ mexm ị ỵ ky ỵ mkx ị À ; My ¼ À m2 À m2 1m E2 E c ỵ kxy ; Mxy ẳ 21 ỵ mị xym ỵ m Nx ẳ 9ị ð10Þ 1666 N Dinh Duc, H Van Tung / Composite Structures 92 (2010) 16641672 where E1 ẳ Em h ỵ Ecm h=k ỵ 1ị; Substituting the above equations into Eq (19), with the aid of Eqs (6) and (16), leads to the compatibility equation of an imperfect FGM cylindrical panel as E2 ẳ Ecm h ẵ1=k ỵ 2ị 1=2k ỵ 2ị; E3 ẳ Em h =12 ỵ Ecm h ẵ1=k ỵ 3ị 1=k ỵ 2ị ỵ 1=4k ỵ 4ị; Um ; Ub ị ẳ Z h=2 r4 f À E1 ðw2;xy À w;xx w;yy À w;xx =R ỵ 2w;xy w;xy w;xx w;yy w;yy w;xx ị ẳ 0: h=2  k #"  k # 2z ỵ h 2z ỵ h Em ỵ Ecm am ỵ acm DT1; zịdz: 2h 2h 21ị " 11ị The nonlinear equilibrium equations of a perfect cylindrical panel based on the classical shell theory are given by [1] N x;x ỵ N xy;y ẳ 0; N xy;x ỵ Ny;y ẳ 0; M x;xx ỵ 2M xy;xy ỵ M y;yy þ Ny =R þ N x w;xx þ 2Nxy w;xy ỵ Ny w;yy ỵ px; yị ẳ 0; 12ị where pðx; yÞ is lateral pressure positive inwards When Eqs (9), (10) are substituted into Eqs (12), the equilibrium equations can be written in terms of deflection variable w and force resultants as N x;x ỵ N xy;y ẳ 0; N xy;x ỵ Ny;y ẳ 0; 13ị Dr w Ny =R Nx w;xx ỵ 2Nxy w;xy ỵ Ny w;yy ị px; yị ẳ 0; where r2 ẳ @ =@x2 ỵ @ =@y2 , and E22 : 2ị E1 E3 Dẳ E1 m ð14Þ For an imperfect panel, let wà ðx; yÞ denotes a known small imperfection This parameter represents a small initial deviation of the panel surface from a cylindrical shape When imperfection is considered, the equilibrium Eqs (13) is modified into form as N x;x ỵ N xy;y ẳ 0; N xy;x ỵ Ny;y ẳ 0; Dr4 w Ny =R Nx w;xx ỵ w;xx ị 2N xy w;xy þ wÃ;xy Þ; ð15Þ À Ny ðw;yy þ wÃ;yy Þ px; yị ẳ 0: Eqs (17) and (21) are the basic equations used to investigate the stability of functionally graded cylindrical panels They are nonlinear equations in terms of two dependent unknowns w and f Stability analysis In this section, an analytical approach is used to investigate the nonlinear stability of FGM cylindrical panels under mechanical transverse and combined thermomechanical loads In general case, the FGM cylindrical panel is assumed to be simply supported on all edges and subjected to in-plane compressive loads, uniformly distributed along the edges, and lateral pressure uniformly distributed on the outer surface of the panel Depending on the in-plane behavior at the edges, three cases of boundary conditions, labelled Cases (1), (2) and (3) will be considered [14,17,25] Case (1) The edges are simply supported and freely movable (FM) The associated boundary conditions are w ¼ M xx ¼ Nxy ¼ 0; Nx ¼ Nx0 on x ¼ 0; a; w ¼ M yy ¼ Nxy ¼ 0; Ny ¼ Ny0 on y ¼ 0; b: Case (2) The edges are simply supported and immovable (IM) The associated boundary conditions are w ¼ u ¼ M xx ¼ 0; N x ¼ Nx0 on x ¼ 0; a; w ¼ v ¼ Myy ¼ 0; Ny ¼ Ny0 on y ¼ 0; b: w ¼ M xx ¼ Nxy ¼ 0; Nx ¼ f;yy ; w ¼ v ¼ Myy ¼ 0; Nxy ¼ Àf;xy : ð16Þ Substituting Eq (16) into the third of Eqs (15) leads to Dr w À f;xx =R f;yy w;xx ỵ w;xx ị ỵ 2f ;xy w;xy ỵ w;xy ị f;xx w;yy ỵ w;yy ị px; yị ẳ 0: 17ị The Eq (17) includes two dependent unknowns, w and f To obtain a second equation relating these two unknowns, the compatibility equation may be used The geometrical compatibility equation of a cylindrical panel is written as [1] exm;yy ỵ eym;xx cxym;xy ẳ w2;xy À w;xx w;yy À w;xx =R: ð18Þ For a imperfect panel, the above equation may be modified into form as exm;yy ỵ eym;xx cxym;xy ẳ w2;xy w;xx w;yy w;xx =R ỵ 2w;xy w;xy w;xx w;yy w;yy wÃ;xx : ð19Þ From the constitutive relations (9), one can write exm ; eym ị ẳ cxym ẳ ½ðNx ; Ny Þ À mðNy ; Nx Þ À E2 kx ; ky ị ỵ Um 1; 1ị; E1 ẵ1 ỵ mịNxy E2 kxy : E1 20ị ð23Þ Case (3) The edges are simply supported Axial edge loads are applied in the two curved edges The curved edges x ¼ 0; a are considered freely movable, the remaining two straight edges being unloaded and immovable For this case, the boundary conditions are Considering the first two of Eqs (15), a stress function f may be defined as Ny ẳ f;xx ; 22ị Nx ẳ Nx0 Ny ẳ Ny0 on x ¼ 0; a; on y ¼ 0; b; ð24Þ where Nx0 ; Ny0 are prebuckling force resultants in directions x and y, respectively, for Case (1) and the first of Case (3), and are fictitious compressive edge loads rendering the edges immovable for Case (2) and the second of Case (3) To solve two Eqs (17) and (21) for unknowns w and f, and with the consideration of the boundary conditions (22)–(24), we assume the following approximate solutions [14,15] w ¼ W sin km x sin ln y; f ẳ A1 cos 2km x ỵ A2 cos 2ln y ỵ A3 sin km x sin ln y 1 ỵ A4 cos 2km x cos 2ln y þ N x0 y2 þ N y0 x2 ; 2 25ị where km ẳ mp=a; ln ẳ np=b; m; n ¼ 1; 2; are number of halfwaves in x and y directions, respectively, and W is amplitude of deection Also, Ai i ẳ 4ị are coefficients to be determined When only the third term and the last two ones in expression of f is retained, this solution form is consistent with that in [19] Considering the boundary conditions (22)–(24), the imperfections of the panel are assumed as [14,15,25] 1667 N Dinh Duc, H Van Tung / Composite Structures 92 (2010) 1664–1672 wà ¼ lh sin km x sin ln y; m; n ¼ 1; 2; ð26Þ where the coefficient l varying between values À1 and represents imperfection size By substituting Eqs (25), (26) into Eq (21), the coefficients Ai are determined as A1 ẳ A3 ẳ E1 l2n WW ỵ 2lhÞ; 32k2m E1 k2m Þ2 n Rðk2m E1 k2m A2 ẳ WW ỵ 2lhị; 32l2n ỵl W; lateral pressure plays on the postbuckling response of cylindrical panels the role of an initial geometric imperfection in the sense that the structure undergoes bending at the onset of loading For perfect panels, the limit points of qðWÞ curves in Eq (32) are determined by condition dq ẳ A 2BW ỵ CW ẳ 0; dW 27ị A4 ẳ 0: 33ị which yields W 1;2 ¼ Introduction of Eqs (25) and (26) into Eq (17) and applying Galerkin method for the resulting equation yield where Dk2m ỵ l2n ị2 W ỵ ẵ2k2m l2n A1 ỵ A2 ị ỵ k2m Nx0 ỵ l2n Ny0 W ỵ lhị ! 32k2m l2n 64k2m A1 16Ny0 16p A3 W ỵ lhị k m A3 ỵ ẳ 0: R 3mnp2 mnp2 3mnp2 mnp2 Aẳ Bẳ 28ị Bầ mnp2 16B4h p B2 À AC ; C ð34Þ " Dp4 ðm2 B2a þ n2 Þ2 þ E1 m4 B6a B2h R2a ðm2 B2a ỵ n2 ị2 2E1 p2 m4 n2 B5a Ra E1 p2 n2 Ba Ra ;  2 ỵ 24B3h 3B3h m2 B2a ỵ n2 Cẳ # ; 3E1 p6 mnm4 B4a ỵ n4 ị 256B4h : 35ị Eq (28) , derived for odd values of m; n, is used to determine buckling loads and nonlinear equilibrium paths of FGM cylindrical panels under uniform lateral pressure with and without effect of temperature By examining the sign d q=dW one obtains the conclusion that the curve represented by Eq (32) reaches maximum at W with corresponding value 4.1 Mechanical stability analysis qu ẳ qW ị ẳ The simply supported FGM cylindrical panel with freely movable edges (that is, Case (1)) is assumed to be subjected to lateral pressure q (in Pascals), uniformly distributed on the outer surface of the panel in the absence of edge loads and temperature In this case, we have p ¼ q; Nx0 ¼ Ny0 ¼ 0: ð29Þ Introduction of Eqs (27) and (29) into Eq (28) gives qẳ " mnp2 Dp 16B4h ỵ m2 B2a 2 ỵn ị ỵ 2E1 p2 m4 n2 B5a Ra 3B3h m2 B2a ỵ n2 ị2 E1 m4 B6a B2h R2a 256B4h W m2 B2a ỵ n2 ị2 WW ỵ lị E1 p6 mnm4 B4a þ n4 Þ # E1 p2 n2 Ba Ra 24B3h WW ỵ lịW ỵ 2lị; WW ỵ 2lị 30ị where Ba ¼ b=a; Bh ¼ b=h; E1 ¼ E1 =h; Ra ¼ a=R; W ¼ W=h: For a perfect panel, qẳ mnp 31ị l ẳ 0, Eq (30) leads to " Dp 16B4h D ¼ D=h ; m2 B2a 2 ỵn ị ỵ E1 m4 B6a B2h R2a m2 B2a ỵ n2 ị2 E1 p ỵ mnm4 B4a 256B4h ỵn ị W 3: 3C 2 ẵB3AC 2B2 ị ỵ 2B2 À ACÞ3=2 Š; ð36Þ and reaches minimum value at W with ql ẳ qW ị ẳ 3C ẵB3AC 2B2 ị 2B2 ACị3=2 ; ð37Þ provided B2 À AC > 0: ð38Þ If the material and geometric parameters of the panel are such that condition (38) is satisfied, snap-through behavior of the panel may be predicted Specifically, Eqs (36) and (37) represent upper and lower limit buckling loads of perfect FGM panel under lateral pressure, respectively, and the intensity of the snap-through response is given by the difference between two these values, i.e 4ðB2 ACị3=2 =3C In case of B2 ẳ AC, the qðWÞ curve has only one stationary point at W ¼ B=C which be an inflexion point as can be shown For the panels such that B2 < AC, there exists only one equilibrium shape and load-deflection curve is stable For imperfect panels, a similar discussion can be given 4.2 Thermomechanical stability analysis A simply supported FGM cylindrical panel with immovable edges (that is, Case (2)) under simultaneous action of uniform lateral pressure q (in Pascals) and thermal loads is considered The condition expressing the immovability on the edges, u ¼ (on x ¼ 0; a) and v ¼ (on y ¼ 0; b), is fulfilled on the average sense as [14,15] # W E1 p2 n2 Ba Ra 2E1 p m n Ba Ra À4 5W  2 þ 24B3h 3B3h m2 B2a þ n2 Z ð32Þ Eqs (30) and (32) show that there exists no bifurcation-type buckling for both perfect and imperfect FGM cylindrical panels under uniform lateral pressure, and the cylindrical panels only exhibit extremum-type buckling when the material and geometric parameters satisfy specific conditions In other words, loss of stability occurs at a limit point rather than at a bifurcation point for panels subjected to lateral pressure This behavior of FGM panels is consistent with Chang and Librescu’s discussions [15] that the b Z a @u dxdy ¼ 0; @x Z a Z b @v dydx ¼ 0: @y ð39Þ From Eqs (6) and (9) one can obtain the following relations in which Eq (16) and imperfection have been included @u E2 Um ; ¼ ðf;yy À mf;xx ị ỵ w;xx w2;x w;x w;x ỵ @x E1 E1 E1 @v E2 w Um : ẳ f;xx mf;yy ị ỵ w;yy w2;y w;y w;y ỵ ỵ R @y E1 E1 E1 ð40Þ 1668 N Dinh Duc, H Van Tung / Composite Structures 92 (2010) 1664–1672 Substituting Eqs (25) and (26) into Eqs (40) and then into Eqs (39) yield Nx0 ¼ À Um for FGM cylindrical panels with all immovable edges, and q¼ 1Àm " E1 ð1 À m2 ịk2m l2n # ỵ p2 mnR1 m2 ị þ E1 ðk2 þ ml2n ÞWðW þ 2lhÞ; 8ð1 À m2 ị m k2m ỵ l2n ị2 mE1 ỵ E2 Rk2m ỵ ml2n ị W p2 mn 16B4h ỵ " Dp4 m2 B2a ỵ n2 ị2 ỵ " E1 m2 ịk4m p2 4B4h 3m2 B2a ỵ n2 ị2 ỵ p2 mnR1 m2 ị ỵ E1 l2 ỵ mk2m ịWW ỵ 2lhị; 81 m2 ị n k2m ỵ l2n ị2 Nx0 ẳ 0; N y0 ẳ Um ỵ " E1 k2m k2m ml2n ị p2 mnR k2m ỵ l2n ị2 # E1 ỵ E2 Rl2n " mn E1 m4 B6a B2h R2a 64E1 m2 B6a B2h R2a À ¼ D m2 B2a ỵ n2 ị2 ỵ 2 2 16Bh m Ba ỵ n ị n2 m2 B2a ỵ n2 ị2 # 2 2 64E1 B B R 64E2 Ba Bh Ra ð m2 Ba þ n2 Þ þ 2 a h a2 À W m2 n2 ð1 À Þ m n ð1 À Þ p p m À p m " p 4B4h ð1 À m2 Þ m 2E1 ịm4 n2 B5a Bh Ra 3m2 B2a ỵ n2 ị2 m ỵ E1 Ba Bh Ra mm2 B2a ỵ n2 ị # ỵ ỵ E1 Ba Ra ẵ3 m2 B2a ỵ ịn2 WW ỵ ị 24B3h ị E1 mn ẵ3 ịm4 B4a ỵ n4 Þ 256B4h ð1 À Þ m2 n2 B2a WW ỵ ịW ỵ ị   mnP 16Ba Bh Ra 2 m B ỵ n ịW ỵ ị DT; a mn 16B2h À Þ p m m l m p m m l m p m l l 256B4h n2 W ỵ lị m2 B2a ỵ 3n2 ị  16Ba Bh Ra DT; p4 mn E2 ¼ E2 =h : ð45Þ 4.2.2 Through the thickness temperature gradient In this case, the temperature through thickness is governed by the one-dimensional Fourier equation of steady-state heat conduction   d dT KðzÞ ¼ 0; dz dz Tðz ¼ h=2Þ ¼ T c ; Tz ẳ h=2ị ẳ T m ; 46ị where T c and T m are temperatures at ceramic-rich and metal-rich surfaces, respectively The solution of Eq (46) can be obtained by means of polynomial series Taking the first seven terms of the series, the solution for temperature distribution across the panel thickness becomes [25] Tzị ẳ T m ỵ DT P n¼0 P n¼0 ðÀr k K cm =K m ịn nkỵ1 47ị ; K cm =K m ịn nkỵ1 where r ẳ 2z ỵ hị=2h and DT ¼ T c À T m is defined as the temperature difference between ceramic-rich and metal-rich surfaces of the panel Assuming the metal surface temperature as reference temperature and substituting Eq (47) into Eq (11) give P Um ¼ HDTh; H ¼ n¼0 ðÀK cm =K m Þn nkỵ1 h Em am nkỵ2 P nẳ0 acm ỵEcm am Ecm acm ỵ Emnỵ1ịkỵ2 ỵ nỵ2ịkỵ2 i : 48ị K cm =K m ịn nkỵ1 By following the same procedure as the preceding loading case we obtain explicit expressions of qðWÞ curves for two cases of inplane restraints as Eqs (43) and (44), provided P is replaced by H Such detail expressions are omitted here for sake of brevity Results and discussion E2 p2 m4 B4a ỵ n4 ỵ 2mm2 n2 B2a WW ỵ lị Em acm ỵ Ecm am Ecm acm ỵ ; kỵ1 2k þ r 4.2.1 Uniform temperature rise Under mentioned boundary conditions, environment temperature can be uniformly raised from initial value T i to final one T f and temperature difference DT ¼ T f À T i is a constant The thermal parameter Um can be expressed in terms of the DT from Eqs (11) Subsequently, employing this expression of Um in Eqs (41) and (42) and then introduction of the results into Eq (28) yield p P ¼ Em am ỵ 42ị In what follows, specic expressions of thermal parameter and the nonlinear response of FGM cylindrical panels under uniform external pressure and two types of thermal loads will be analyzed p 16B2h E1 p6 mn for panels with two edges y ¼ 0; b are immovable, where E1 l2n WW ỵ 2lhị:  ỵ E1 n Ba Bh Ra À E2 p n WðW þ 2lÞ þ p4 mnP # ð44Þ which represent the compressive stresses making the edges immovable and depending on thermal parameter and prebuckling deflection Obviously, when prebuckling deflection is ignored, i.e a membrane state analysis, only the first terms in Eqs (41) are retained Subsequently, a simply supported panel with movable edges x ¼ 0; a and immovable edges y ¼ 0; b (that is, Case (3)) and subjected to combined action of uniform lateral pressure q (in Pascals) and thermal loads in the absence of edge loads is considered Employing Eqs (25) and (26) in the second of Eqs (40) and then introduction of the result into the second of Eqs (39) yield q 6B3h E1 ỵ E2 Rl2n ỵ mk2m ị W 41ị Wỵ E1 p2 n2 Ba Ra WW ỵ lịW ỵ 2lị # # 64E2 Ba Bh Ra À W p2 m2 p4 m2 n2 m2 B2a ỵ n2 ị2 " E1 m2 n2 B3a Bh Ra 5m2 B2a ỵ 3mn2 ị Um 1m m2 B2a ỵ n2 ị2 64E1 B2a B2h R2a ẵn4 ỵ ỵ mịm2 n2 B2a WW ỵ lị N y0 ẳ E1 m4 B6a B2h R2a p ð43Þ In this section, the nonlinear response of the cylindrical panels made of functionally graded materials is analyzed The panels are assumed to be simply supported on all edges and, unless otherwise stated, edges are freely movable In characterizing the behavior of the panels, deformations in which the central region of a panel moves towards the plane that contains the four corners of the panel are referred to as inward deflections Deformations in the opposite direction are referred to as outward deflection [18] As shown in references [15,17,18], the most pronounced buckling and postbuckling responses for deformation modes with half-wave num- 1669 N Dinh Duc, H Van Tung / Composite Structures 92 (2010) 1664–1672 bers m ¼ n ¼ Thus, the results presented in this section also correspond to values of m ¼ n ¼ To validate the present formulation in the nonlinear analysis of pressure-loaded FGM cylindrical panels, the nonlinear response of a simply supported FGM cylindrical panel under uniform external pressure is analyzed, which was considered by Zhao and Liew [24] using the element-free kp-Ritz method and modified version of Sander’s nonlinear shell theory The nonlinear load-deflection curves of perfect panels made of zirconia ðZrO2 Þ and Aluminum ðAlÞ with different values of volume fraction index k are compared in Fig with Zhao and Liew’s results As can be observed, a very good agreement is obtained in this comparison study To illustrate the proposed approach, we consider a ceramic-metal functionally graded panel that consist of zirconia and aluminum with the following properties [24] Em ¼ 70 GPa; am ¼ 23:10À6  CÀ1 ; K m ¼ 204 W=mK; Ec ¼ 151 GPa; ac ¼ 10:10À6  CÀ1 ; K c ¼ 2:09 W=mK; q (GPa) 0.05 Ref [24], k = 0.2 Ref [24], k = 1.0 Ref [24], Al Present, k = 0.2 Present, k = 1.0 Present, Al 0.04 0.03 a/b = 1.0, b/h = 20, a/R = 0.2 0.02 0.01 0 0.5 1.5 W/h ð49Þ and Poisson’s ratio is chosen to be 0.3 Effects of material and geometric parameters, temperature and in-plane boundary conditions as well as imperfection on the nonlinear response of the perfect and imperfect FGM cylindrical panels are graphically shown in Figs 2–12 in which Figs 2–6, 11 are plotted for case of mechanical stability analysis, i.e in the absence of temperature It is noted that in all figures W=h denotes the dimensionless maximum deflection of the panel Fig gives the nonlinear load-deflection curves for FGM cylindrical panels under uniform lateral pressure and with three different values of volume fraction index k ð¼ 0; and 5Þ A benign snap-through behavior of the panels is shown in this figure It is also seen that nonlinear curves become higher for smaller values of k representing panels with the greater volume percentage of zirconia, as expected Fig shows the effect of width-to-thickness ratio b/h (= 20, 30 and 40) on the nonlinear behavior of the FGM panels with k ¼ 1:0 Fig shows the effect of length-to-width ratio a/b (= 0.75, 1.0 and 1.5) on the nonlinear behavior of the panels under similar conditions It is evident from two these figures that loading carrying capacity of the panels is considerably reduced when b=h and a=b ratios increase Furthermore, the nonlinear equilibrium paths become more stable for larger values of a=b standing for shallower panels The effect of panel curvature on the nonlinear response of pressure-loaded FGM panels is illustrated in Fig with three various values of length-to-radius ratio a/R (= 0.5, 0.75 and 1.0) As can be seen, the load bearing capability of the panels is increased when a=R ratio increases and the deflection is small and a converse trend occurs when the deflection is sufficiently large In addition, the results indicate that the panels with large a=R exhibit a benign snapthrough response and the panels with small one have stable equilibrium paths due to its flatted configuration The effect of in-plane boundary conditions on the nonlinear response of FGM panels under uniform lateral pressure is depicted in Fig In this figure, the nonlinear equilibrium paths of the panels with all freely movable edges (Case (1)) are plotted in comparison with their counterparts when all edges are immovable (Case (2)) and two edges y ¼ 0; b are immovable (Case (3)) As can be observed, the load bearing capability of the panel in the Case (2) is higher than that of panels in two the remaining cases Furthermore, the panel in Case (3) experiences a snap-through response in contrast to comparatively stable behavior of the panels in Cases (1) and (2) Fig shows the effect of temperature field on the nonlinear response of FGM cylindrical panels under uniform lateral pressure with all immovable edges The counterpart of the case Fig Comparisons of nonlinear load-deflection curves for FGM cylindrical panels in this figure for panels with two immovable edges is illustrated in Fig As can be seen, in the presence of temperature field the pressure-loaded panels exhibit a bifurcation behavior with no deflection occuring until a bifurcation point is reached This is contrast to their isothermal counterparts in which curves originate from coordinate origin This behavior may be explained that the temperature field produce the deformations that cause the central region of a panel to deflect outward (negative deflection) prior to applying the mechanical loads With the application of lateral pressure, the outward deflection is reduced and when lateral pressure exceeds bifurcation point loads, which are represented by the last terms with W ¼ in Eqs (43) and (44) respectively, a inward deflection occurs It is also seen that the increase in buckling loads and postbuckling load carrying capacity in small range of deflection due to the presence of temperature as compared with their isothermal counterparts is paid by an unstable postbuckling behavior, that is, a snap-through response The intensity of the snap-through response is more severe when environment temperature is increased Furthermore, the panels with two immovable edges have weaker capacity of load carrying accompanied by more intense snap-through response in comparison with the panels with all immovable edges The effect of through the thickness temperature gradient on the nonlinear response of pressure-loaded panels is plotted in Fig q (GPa) −3 x 10 imperfect (μ = 0.1) perfect 2.5 a/b = 1.1, b/h = 50, a/R = 0.5 1.5 k=0 k=1 k=5 0.5 0 W/h Fig Effect of volume fraction index on the nonlinear response of FGM cylindrical panels 1670 N Dinh Duc, H Van Tung / Composite Structures 92 (2010) 1664–1672 q (GPa) q (GPa) 0.04 0.03 imperfect (μ = 0.1) perfect 0.035 imperfect (μ = 0.1) perfect 0.025 a/b = 1.0, a/R = 0.5, k = 1.0 0.03 a/b = 1.0, b/h = 50, 1: b/h = 20 0.025 0.02 a/R = 0.5, k = 1.0 2: b/h = 30 0.02 1: IM (all edges) 0.015 3: b/h = 40 2: IM (y = 0, b) 0.015 3: FM 0.01 0.01 0.005 0.005 3 0 0.5 1.5 2.5 3 W/h W/h Fig Effect of width-to-thickness ratio on the nonlinear response of FGM cylindrical panels Fig Effect of in-plane boundary conditions on the nonlinear response of FGM cylindrical panels q (GPa) q (GPa) 0.03 0.05 imperfect (μ = 0.1) perfect imperfect (μ = 0.1) perfect 0.025 0.04 b/h = 30, a/R = 0.5, k = 1.0 a/b = 1.0, b/h = 50, a/R = 0.5, k = 1.0 0.02 1: a/b = 0.75 0.03 1: ΔT = o 2: ΔT = 200 C 2: a/b = 1.0 0.015 3: a/b = 1.5 o 3: ΔT = 400 C 0.02 0.01 2 0.01 0.005 0 0.5 1.5 2.5 W/h W/h Fig Effect of length-to-width ratio on the nonlinear response of FGM cylindrical panels q (GPa) Fig Effect of temperature field on the nonlinear response of FGM cylindrical panels (all immovable edges) q (GPa) 0.07 0.014 imperfect (μ = 0.1) perfect 0.06 a/b = 1.0, b/h = 50, a/R = 0.5, k = 1.0 a/b = 1.0, b/h = 30, k = 1.0 0.05 imperfect (μ = 0.1) perfect 0.012 0.01 1: ΔT = 1: a/R = 0.5 0.04 2: a/R = 0.75 o 2: ΔT = 200 C 0.008 o 3: ΔT = 400 C 3: a/R = 1.0 0.03 0.006 0.02 0.004 0.01 0.002 0 W/h Fig Effect of length-to-radius ratio on the nonlinear response of FGM cylindrical panels 0 W/h Fig Effect of temperature field on the nonlinear response of FGM cylindrical panels (immovable y ¼ 0; b) 1671 N Dinh Duc, H Van Tung / Composite Structures 92 (2010) 1664–1672 q (GPa) q (GPa) 0.1 0.03 imperfect (μ = 0.1) perfect 0.025 1: μ = −0.5 2: μ = −0.2 3: μ = 0.0 4: μ = 0.2 5: μ = 0.5 0.08 a/b = 1.0, b/h = 50, a/R = 0.5, k = 1.0 o 1: T = 27 C 0.02 c 0.06 o 2: T = 400 C a/b = 1.0, b/h = 40, a/R = 0.5 c 0.015 o k = 1.0, ΔT = 300 C o 3: T = 800 C c 0.01 3 0.02 0.005 0.04 1 0 Fig Effect of temperature gradient on the nonlinear response of FGM cylindrical panels q (GPa) 0.025 ceramic−rich inner suface ceramic−rich outer suface 0.02 a/b = 1.0, b/h = 50, a/R = 0.5, k = 1.0, μ = 0.015 o 1: T = 400 C c o 2: T = 800 C c 0.01 0.005 0 W/h Fig 10 Effect of type of temperature gradient on the nonlinear response of FGM cylindrical panels q (GPa) 0.025 1: μ = −0.5 2: μ = −0.2 3: μ = 0.0 4: μ = 0.2 5: μ = 0.5 0.02 0.015 a/b = 1.0, b/h = 40, a/R = 0.5, k = 1.0 0.005 1 Fig 12 Effect of imperfection on the nonlinear response of FGM cylindrical panels with all immovable edges with different values of temperature T c at ceramic-rich surface when metal-rich surface temperature is retained at T m ¼ 27  C (room temperature) It seems that through the thickness temperature gradient produce smaller outward deflection and the panels have more stable postbuckling behavior, that is, more benign snap-through response under this temperature condition Fig 10 shows the effect of type of through the thickness temperature gradient on the nonlinear response of pressure-loaded FGM perfect cylindrical panels Specifically, together the present FGM cylindrical panel having inner surface (i.e concave side) is ceramic-rich, another panel with interchanged properties, which possesses ceramic-rich outer surface (i.e convex side), is considered The temperature is conducted from ceramic pure surface to metal pure one through the thickness As can be observed, although lower bifurcation point and postbuckling curves in small range of deflection, temperature gradient from the outer surface of panel makes the postbuckling behavior of the panel to be more stable Finally, the effect of initial imperfection on the nonlinear response of panels is considered Fig 11 depicts the nonlinear loaddeflection curves of FGM panels under uniform lateral pressure A similar consideration is shown in Fig 12 for panels with all immovable edges and temperature field is included In two these figures, the nonlinear equilibrium paths are plotted for various values of imperfection size l in which, as mentioned in [18], negative or positive imperfections produce perturbations in the panel geometry that move the central region of a panel outward or inward, respectively As can be seen, the nonlinear load-deflection curves become lower when l increases from À0.5 to 0.5 and the deflection is small However, a inverse trend occurs when the deflection exceeds a specific value As above, the nonlinear equilibrium paths start from bifurcation point rather than coordinate origin due to the presence of temperature field Concluding remarks 0.01 W/h W/h W/h Fig 11 Effect of imperfection on the nonlinear response of FGM cylindrical panels An analytical study of the nonlinear response of functionally graded cylindrical panels subjected to uniform lateral pressure with and without temperature effects has been presented The formulation is based on the classical shell theory with both von Karman– Donnell nonlinear terms and initial geometrical imperfection are included By using Galerkin method, explicit expressions of nonlinear load-deflection curves for a simply supported panel under mentioned loads are determined From these explicit expressions, the 1672 N Dinh Duc, H Van Tung / Composite Structures 92 (2010) 1664–1672 nonlinear response of the panels is analyzed and the results are given in graphic form The results show that the nonlinear response of the FGM cylindrical panels is greatly influenced by in-plane restraint and temperature conditions Furthermore, the study also confirm that the nonlinear response of pressure-loaded panels are complex and significantly influenced by the material and geometric parameters, initial imperfection as well Acknowledgements This work is supported by the research project of Vietnam National University-Ha Noi, coded QGTD.09.01 The authors are 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nonlinear response of functionally graded cylindrical panels subjected to uniform lateral pressure with and without temperature. .. W/h Fig Effect of length-to-radius ratio on the nonlinear response of FGM cylindrical panels 0 W/h Fig Effect of temperature field on the nonlinear response of FGM cylindrical panels (immovable... W/h Fig Effect of length-to-width ratio on the nonlinear response of FGM cylindrical panels q (GPa) Fig Effect of temperature field on the nonlinear response of FGM cylindrical panels (all immovable