Computational Materials Science 49 (2010) S313–S316 Contents lists available at ScienceDirect Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci Nonlinear analysis of stability for functionally graded cylindrical panels under axial compression 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: Received October 2009 Received in revised form December 2009 Accepted 18 December 2009 Available online 25 January 2010 Keywords: Nonlinear analysis Functionally Graded Materials Postbuckling Cylindrical panel a b s t r a c t This report presents an analytical approach to investigate the stability of functionally graded cylindrical panels under axial compression Equilibrium and compatibility equations for functionally graded panels are derived by using the classical shell theory taking into account both geometrical nonlinearity in von Karman–Donnell sense and initial geometrical imperfection The resulting equations are solved by Galerkin procedure to obtain explicit expressions of buckling loads and postbuckling load–deflection curves Stability analysis for a simply supported functionally graded cylindrical panel shows the effects of material and geometric parameters as well as imperfection on buckling and postbuckling behaviors of the panel Ó 2009 Elsevier B.V All rights reserved Introduction Functionally graded cylindrical panels Flat and curved panel elements constitute a major portion of the structure of aerospace vehicles They are found in the aircraft components as primary load carrying structures such as wing and 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 Some investigations on buckling and postbuckling of laminated composite cylindrical panels are reported in works [2–5] Recently, a new composite is known as Functionally Graded Materials (FGMs) with high performance heat resistance capacity has been developed Some works have published relating to the stability of FGM structures such as [6–9] In this report, the buckling and postbuckling of FGM cylindrical panels subjected to axial compressive loads are investigated by an analytical approach The 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 the buckling loads and postbuckling load–deflection curves Stability analysis for a simply supported panel shows the effects of material and geometric parameters and imperfection on the buckling and postbuckling behaviors of the panel Consider a functionally graded cylindrical panel with radius of curvature R, thickness h, axial length a and arc length b as is shown in Fig The panel is made from a mixture of ceramics and metals, and is defined in a coordinate system (x, h, z), 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 (Àh/ z h/2) The effective modulus of elasticity is assumed to vary only in the thickness direction according to simple power law distribution as [8] * Corresponding author E-mail address: ducnd@vnu.edu.vn (N.D Duc) 0927-0256/$ - see front matter Ó 2009 Elsevier B.V All rights reserved doi:10.1016/j.commatsci.2009.12.030 k 2z ỵ h Ezị ẳ Em ỵ Ec Em ị ; 2h mzị ẳ m 1ị where Ec and Em to be elastic moduli of ceramic and metal, respectively, and m Poisson ratio assumed to be a constant It is evident that, the inner surface (z = h/2) of the panel is ceramic-rich and the outer surface (z = Àh/2) is metal-rich Governing equations In the framework of the classical shell theory [1], the equilibrium and compatibility equations are derived as follows h Dr4 w À f;xx =R f;yy w;xx ỵ w;xx 2f ;xy w;xy ỵ w;xy i ỵ f;xx w;yy ỵ w;yy q ẳ 2ị S314 N.D Duc, H.V Tung / Computational Materials Science 49 (2010) S313–S316 x wà ¼ lh sin b a " # 4 E ðmbB Þ4 16E1 m3 nb B4a a 2 W Dp m Ba ỵ n ỵ 3W þ 4lhÞW 3R R2 2 m2 B2a ỵ bn2 W ỵ lhị ỵ N x0 p2 b m2 B2a ỵ n2 y R Fig Configuration and the coordinate system of the FGM cylindrical panel r4 f À E1 w2;xy À w;xx w;yy À w;xx =R ỵ 2w;xy w;xy w;xx w;yy w;yy w;xx ẳ0 3ị where y = Rh, r4 ẳ @ =@x4 ỵ 2@ =@x2 @y2 ỵ @ =@y4 , and q is lateral pressure positive inwards Also, w to be deflection of panel, wà initial imperfection representing a small initial deviation of the panel surface from a cylindrical shape, f Airy stress function, and ðE1 ; E2 ; E3 ị ẳ Z h=2 Ezị1; z; z2 Þdz ð4Þ Àh=2 Eqs (2) and (3) are the basic equations used to investigate the stability of functionally graded panels subjected to axial compressive loading They are nonlinear equations in terms of two dependent unknowns w and f 512E1 m2 n2 B4a W ỵ lhịW ỵ 2lhịW 2 16b 2 ỵ m B ỵ n bNx0 =R qị ẳ a mnp2 ỵ Ba ¼ b=a; b ¼ Ny0 =Nx0 2 2 2 2 D p m B ỵ n a E B R m W pẳ4 ỵ a a 2 W ỵl m2 B2a B2h p2 m2 B ỵ n2 a 16E1 mnB3a Ra 3p Stability analysis w ¼ M x ¼ Nxy ¼ 0; Nx ¼ Nx0 w ¼ M y ¼ Nxy ¼ 0; Ny ¼ N y0 on x ¼ 0; a on y ¼ 0; b ð5Þ where N x0 ; Ny0 are prebuckling force resultants in directions x and y, respectively To solve two Eqs (2) and (3) for two unknowns w and f, we assume the following approximate solutions satisfying simply supported boundary conditions on all edges [5] mpx npy sin w ¼ W sin a b mpx npy sin À hxị kyị f ẳ F sin a b 6ị where m, n = 1, 2, are number of half-waves, W and F are constant coefficients depending on m and n Also, h(x) and k(y) to be preselected functions such that solutions (6) satisfy force boundary conditions, thus 2 dx ¼À Ny0 ; F d kyị dy ẳ N x0 F 7ị Considering the boundary conditions (5), the imperfections of the panel are assumed as follows which are in the shape of the buckling mode ð10Þ Eq (9) is used to investigate the buckling and postbuckling behaviors of FGM cylindrical panels subjected to various conditions of loading However, in the present report, simply supported FGM cylindrical panel is assumed to be under only in-plane axial compressive load p (in Pascals), uniformly distributed along curved edges x = 0, a In this case Nx0 = Àph, Ny0 = q = and Eq (9) yields 2B h In this section, an analytical approach is used to investigate the stability of FGM cylindrical panels under mechanical loads The functionally graded cylindrical panel is assumed to be simply supported on all edges and, in general case, subjected to in-plane compressive loads, uniformly distributed along the edges, and lateral pressure uniformly distributed on the outer surface of the panel The displacement and force boundary conditions for a simply supported panel are defined as ð9Þ where m, n are odd numbers, and À d hðxÞ ð8Þ z E1 E3 À E22 ; E1 ð1 À m2 Þ m; n ¼ 1; 2; where À1 l to be imperfection size Introduction of Eqs (6)–(8) into Eqs (2) and (3), then applying Galerkin method for the resulting equations, we obtain nonlinear algebraic equations for W and F Subsequently, eliminating variable F from these equations gives h Dẳ mpx npy sin ; a b ỵ m2 B2a ỵ n2 2 3W ỵ 4lịW W þl 512E1 B2a n2 2 ðW þ 2lÞW 9p2 B2h m2 B2a ỵ n2 11ị where D ẳ D=h ; E1 ¼ E1 =h; Bh ¼ b=h; Ra ¼ a=R; W ¼ W=h ð12Þ For a perfect panel (l = 0), Eq (11) leads to p¼ 2 Dp2 m2 B2a ỵ n2 m2 B2a B2h ỵ E1 B4a R2a m2 2 p2 m2 B2a ỵ n2 16E1 mnB3a Ra 512E1 B2a n2 2 W ỵ 2 W 2 2 2 2 p Bh m Ba ỵ n 9p Bh m Ba ỵ n 13ị from which buckling compressive load may be obtained at W ¼ as pu ẳ 2 Dp2 m2 B2a ỵ n2 m2 B2a B2h ỵ E1 B4a R2a m2 p2 m2 B2a þ n2 Þ2 ð14Þ Above equation represents upper buckling compressive load of FGM panel The critical upper buckling load pucr is obtained for the values of m and n that make the preceding expression a minimum Eq (13) points out that the pWị curve has an extremum at dp=dW ẳ 0, i.e at W ẳ 9mBa Ra Bh =64nị By examining the sign d p=dW one obtains the conclusion that the pðWÞ curve reaches minimum at W and pl ẳ pW ị ẳ 2 Dp2 m2 B2a ỵ n2 m2 B2a B2h E1 B4a R2a m2 2 8p2 m2 B2a ỵ n2 ð15Þ N.D Duc, H.V Tung / Computational Materials Science 49 (2010) S313–S316 represents lower buckling compressive load of FGM panel This analysis shows that the panel subjected to axial compression can exhibit a snap-through to a new equilibrium position at W The intensity of the snap-through is given by the difference between the upper and lower buckling loads, i.e 9E1 m2 B4a R2a = 2 8p2 m2 B2a ỵ n2 Eq (13) determines postbuckling equilibrium paths pðWÞ of the perfect panel with values of m and n that make buckling loads a minimum In case of imperfect panel, postbuckling pðWÞ curves represented by Eq (11) originate from coordinate origin This indicates imperfection sensitivity of axially loaded panel and no bifurcation buckling point exists for imperfect panels Results and discussion To illustrate the proposed approach, we consider a ceramic–metal functionally graded panel that consist of aluminum and alumina with the following properties Ec ¼ 380 GPa; Em ¼ 70 GPa; m ¼ 0:3 As shown in Ref [2], the most pronounced buckling and postbuckling responses for deformation modes with half-wave numbers m = n = Thus, the results presented in this section also correspond to values of m = n = Effects of some material and geometric parameters on the postbuckling behavior of the perfect and imperfect FGM cylindrical panels are shown in Figs 2–5 It is noted that in all figures W/h denotes the dimensionless maximum deflection of the panel Fig shows the postbuckling load–deflection curves of FGM cylindrical panels under axial compressive loads with different values of volume fraction index k (=0, and 5) As can be seen, both well-known snap-through behavior and imperfection sensitivity of the panels are exhibited in this figure Both bifurcation-type buckling loads and postbuckling equilibrium paths become lower for higher values of k representing panels with the greater percentage of metal, as expected Furthermore, the severity of snapthrough response, which is measured by difference between upper (bifurcation point) and lower buckling loads, is decreased when k increases Fig shows the effect of width-to-thickness ratio b/h (=20, 30 and 40) on the postbuckling behavior of the FGM panels under axial compression with k = Fig shows the effect of length-towidth ratio a/b (=0.75, 1.0 and 1.5) on the postbuckling behavior Fig Postbuckling paths of the panel vs k Fig Postbuckling paths of the panel vs b/h ð16Þ Fig Postbuckling paths of the panel vs a/b Fig Postbuckling paths of panel vs a/R S315 S316 N.D Duc, H.V Tung / Computational Materials Science 49 (2010) S313–S316 of the panels under similar conditions It is evident from these figures that postbuckling load carrying capacity of the panels is considerably reduced when b/h and a/b ratios increase It is also seen that panels with small a/b ratio experience a severe snap-through response, although their buckling loads are comparatively high Furthermore, the postbuckling equilibrium paths become more stable, i.e exhibit a more benign snap-through behavior, for small values of b/h or large values of a/b standing for shallower panels The effect of panel curvature on the postbuckling response of axially-loaded FGM cylindrical panels is illustrated in Fig with three various values of length-to-radius ratio a/R (=0.2, 0.5 and 0.75) As can be observed, the buckling loads and postbuckling load bearing capacity of the panels are increased when a/R increases and the deflection is small and a converse trend occurs when the deflection is sufficiently large In addition, it is shown that the panels with small a/R ratio (i.e a/R = 0.2) have stable postbuckling equilibrium paths due to its flatted configuration Concluding remarks The report presents a simple analytical approach to investigate the buckling and postbuckling behaviors of functionally graded cylindrical panels under axial compressive loads By using Galerkin method, closed-form relations of buckling loads and postbuckling load–deflection curves for a simply supported FGM cylindrical panel under axial compression, with and without imperfection, are determined The results show the snap-through behavior, imperfection sensitivity and complex postbuckling behavior of axially loaded panels The study also confirms that the postbuckling behaviors of FGM cylindrical panels are greatly influenced by material and geometric parameters, initial geometric imperfection as well Acknowledgements This report is supported by the science researching project of Vietnam National University – Hanoi, coded QGTD.09.01 The authors are grateful for this financial support References [1] D.O Brush, B.O Almroth, Buckling of Bars, Plates and Shells, McGraw-Hill, New York, 1975 [2] M.Y Chang, L Librescu, Int J Mech Sci 37 (2) (1995) 121–143 [3] N Jaunky, N.F Knight, Int J Solids Struct 36 (1999) 3483–3496 [4] H.-S Shen, Compos Struct 79 (2007) 390–403 [5] D.H Bich, Nonlinear analysis on stability of reinforced composite shallow shells, in: N.V Khang, D Sanh (Eds.), Proceedings of National Conference on Engineering Mechanics and Automation, Bach Khoa Publishing House, Hanoi, Vietnam, 2006, pp 9–22 [6] H.-S Shen, Int J Solids Struct 39 (2002) 5991–6010 [7] H.-S Shen, J Eng Mech ASCE 129 (4) (2003) 414–425 [8] J Yang, K.M Liew, Y.F Wu, S Kitipornchai, Int J Solids Struct 43 (2006) 307– 324 [9] X Zhao, K.M Liew, Int J Mech Sci 51 (2009) 131–144 ... then applying Galerkin method for the resulting equations, we obtain nonlinear algebraic equations for W and F Subsequently, eliminating variable F from these equations gives h D¼ mpx npy sin... compressive loads, uniformly distributed along the edges, and lateral pressure uniformly distributed on the outer surface of the panel The displacement and force boundary conditions for a simply supported... buckling load pucr is obtained for the values of m and n that make the preceding expression a minimum Eq (13) points out that the pðWÞ curve has an extremum at dp=dW ¼ 0, i.e at W ¼ 9mBa Ra Bh =ð64nÞ