Composite Structures 94 (2012) 1379–1384 Contents lists available at SciVerse ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct Buckling of functionally graded conical panels under mechanical loads Dao Huy Bich a, Nguyen Thi Phuong b, Hoang Van Tung c,⇑ a Vietnam National University, Ha Noi, Viet Nam University of Transport Technology, Ha Noi, Viet Nam c Faculty of Civil Engineering, Hanoi Architectural University, Ha Noi, Viet Nam b a r t i c l e i n f o Article history: Available online 27 November 2011 Keywords: Buckling Functionally graded materials Truncated conical panels a b s t r a c t This paper presents an analytical approach to investigate the linear buckling of truncated conical panels made of functionally graded materials and subjected to axial compression, external pressure and the combination of these loads 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 linear stability equations in terms of displacement components for conical panels are derived by using the classical thin shell theory Approximate analytical solutions are assumed to satisfy simply supported boundary conditions and Galerkin method is applied to obtain closed-form relations of bifurcation type buckling loads An analysis is carried out to show the effects of material and geometrical properties and combination of loads on the linear stability of conical panels Ó 2011 Elsevier Ltd All rights reserved Introduction Due to special geometrical shapes, conical shells and panels have been widely used in a variety of engineering fields as major load carrying parts These special shells can find their important applications in aerospace and marine industries, such as aircraft fuselages, ship shell, and various structural components As a result, static and dynamic behavior including problems relating to stability and vibration of conical shells and panels have received considerable attention of researchers Lu and Chang [1,2] used an analytical approach to study the buckling and postbuckling behavior of simply supported isotropic truncated conical shells under thermal loadings Their analysis was also compared with experimental results Tani investigated dynamic instability of isotropic truncated conical shells under periodic axial load neglecting bending deformations before instability [3] and under periodic pressure including these deformations [4] by using Donnell shell theory and a finite difference procedure In the same approach, he analyzed the effects of axisymmetric initial deflections and large prebuckling deflections on the static buckling of isotropic truncated conical shells subjected to uniform heating [5] and combined pressure and heating [6] The dynamic and static behavior of laminated crossply truncated conical shells have been analyzed in works [7,8] by using shear deformation shell theories Xu et al [7] used Galerkin procedure and the method of harmonic balance to study free vibration of laminated thick conical shells, whereas Patel et al [8] uti- ⇑ Corresponding author E-mail address: hoangtung0105@gmail.com (H.V Tung) 0263-8223/$ - see front matter Ó 2011 Elsevier Ltd All rights reserved doi:10.1016/j.compstruct.2011.11.029 lized semi-analytical finite element approach to investigate static buckling and postbuckling of laminated cross-ply truncated conical shells under uniform temperature rise The free vibration analysis of isotropic thin truncated conical panels has been carried out by Lam et al [9] using generalized differential quadrature method and by Zhao et al [10] using the mesh-free kp-Ritz method Also, Paczos and Zielnica [11] applied Ritz method to study the stability of bilayered elastic–plastic conical shell panels under simply supported conditions and the action of compressive and pressure loads Functionally graded materials (FGMs) are special composites commonly composed of metal and ceramic materials Due to smooth and continuous variation in the volume fractions of constituents in a specific direction, FGMs possess potential advantages in comparison with conventional composites and satisfy many rigorous requisitions of modern engineering By high performance temperature resistance capacity, FGMs are widely used as temperature shielding components in structures of aircraft, missile and nuclear plants Conical shells and panels have a wide application in many aerospace vehicles which increasingly made of functionally graded materials This necessitates more investigations on both static and dynamic behavior of FGM shell structures in general, and FGM conical shells and panels in particular Bhangale et al [12] used a finite element formulation based on first order shear deformation theory to study the linear thermal buckling and free vibration behavior of FGM truncated conical shells assumed to be clamped at two ends Sofiyev [13–15] adopted an analytical approach to investigate the linear mechanical and thermal buckling of simply supported FGM truncated conical shells with and without the effects of elastic foundations Recently, he 1380 D.H Bich et al / Composite Structures 94 (2012) 1379–1384 also analyzed nonlinear buckling behavior of FGM truncated conical shells subjected to axial load by using modified Donnell theory taking geometrical nonlinearity into consideration [16] An analytical investigation on the linear stability of simply supported FGM truncated conical shells under mechanical and thermal loads has been carried out by Naj et al [17] using the classical thin shell theory Also by using the classical theory and generalized Galerkin method, Mahmoudkhani et al [18] determined the aeroelastic instability boundaries of simply supported FGM conical shells at high supersonic Mach numbers and thermal environments with the temperature dependence of material properties taken into consideration In spite of practical importance, investigations on static and dynamic responses of FGM conical panels are scarce Zhao and Liew extended their previous works on isotropic conical panels to analyze the free vibration of functionally graded conical panels by using a meshless method [19] Also, Aghdam et al [20] employed extended Kantorovich method to predict bending behavior of clamped FGM conical panels under uniform and nonuniform distributed loadings First order shear deformation shell theory has been adopted for formulation in both these studies To the best of authors’ knowledge, there is no investigations on the buckling of FGM truncated conical panels The present paper aims to investigate the linear buckling behavior of simply supported truncated conical panels made of functionally graded materials and subjected to mechanical loads The classical shell theory is used to derive equilibrium and linear stability equations in terms of displacement components Approximate analytical solutions are assumed to satisfy boundary conditions and Galerkin procedure is applied to obtain closed-form relations of bifurcation type buckling loads An analysis is carried out to show the effects of material and geometrical properties and combination of loads on the linear stability of FGM truncated conical panels Functionally graded conical shell panels Consider a truncated conical panel of thickness h, length L, semi-vertex angle a, subtended angle b and the radii at the two ends a and b as shown in Fig The panel is made from a mixture of ceramics and metals, and is defined in coordinate system (x, h, z) whose origin is located in the middle surface of the shell panel, x is in the generatrix direction measured from the vertex of conical panel, h is in the circumferential direction and z is perpendicular to the middle surface and points outwards (Àh/2 z h/2) Suppose that the material composition of the shell varies smoothly along the thickness in such a way that the inner surface is metal-rich and the outer surface is ceramic-rich by following a simple power law in terms of the volume fractions of the constituents as V c zị ẳ k 2z ỵ h ; 2h V m zị ¼ À V c ðzÞ ð1Þ where Vc and Vm are the volume fractions of ceramic and metal constituents, respectively, and volume fraction index k is a nonnegative number that defines the material distribution It is assumed that the effective modulus of elasticity E of FGM conical panels changes in the thickness direction z and can be determined by the linear rule of mixture as Ezị ẳ Em ỵ Ecm k 2z ỵ h 2h 2ị where Ec, Em are moduli of elasticity of ceramic and metal constituents, respectively, Ecm = Ec À Em, and Poisson’s ratio m is assumed to be constant Theoretical formulation In the present study, the shell panels are assumed to be thin and the classical shell theory based on Love’s hypotheses is used to investigate the buckling of FGM conical panels under mechanical loads According to this theory, the strains across the shell thickness at a distance z from the midplane are ex ¼ exm À zvx eh ¼ ehm À zvh cxh ẳ cxhm zvxh 3ị where exm, ehm are the normal strains and cxhm is the shear strain at the middle surface of the shell panel, whereas vx, vh and vxh are the change of curvatures and twist that are related to the displacement components u, v, w in the x, h, z coordinate directions, respectively, as [21] exm ẳ u;x ỵ w2;x u w v ;h ỵ ỵ cot a ỵ 2 w2;h x sin a x x 2x sin a v ẳ u;h ỵ v ;x ỵ w;x w;h x sin a x sin a x ehm ¼ cxhm 1 w;hh ỵ w;x x x2 sin a 1 ¼ w;xh À w;h x sin a x sin a 4ị vx ẳ w;xx ; vh ẳ vxh ð5Þ where subscript (,) indicates the partial derivative Hooke law for a conical shell panel is defined as ðrx ; rh ị ẳ Ezị ẵex ; eh ị ỵ meh ; ex ị; m2 rxh ẳ Ezị c 21 ỵ mị xh 6ị The force and moment resultants of an FGM conical panel are expressed in terms of the stress components through the thickness as ðNx ; Nh ; Nxh ị ẳ Z h=2 rx ; rh ; rxh Þdz; ðMx ; M h ; Mxh Þ h=2 ẳ Z h=2 rx ; rh ; rxh ịzdz ð7Þ Àh=2 Fig Configuration and the coordinate system of a truncated conical panel Introduction of Eqs (2), (3) and (6) into Eq (7) gives the constitutive relations as 1381 D.H Bich et al / Composite Structures 94 (2012) 1379–1384 E1 ; E2 ị E2 ; E3 ị exm ỵ mehm ị v ỵ mvh ị m2 À m2 x ðE1 ; E2 Þ ðE2 ; E3 ị Nh ; M h ị ẳ ehm ỵ mexm ị v ỵ mvx ị m2 À m2 h ðE1 ; E2 Þ ðE ; E Þ c À vxh ðNxh ; M xh ị ẳ 21 ỵ mị xhm 1ỵm 8ị E2 ẳ Ecm h ẵ1=k ỵ 2ị 1=2k ỵ 2Þ; ð9Þ The nonlinear equilibrium equations of a geometrically perfect conical panel based on the classical shell theory are given by Brush and Almroth [21] Nxh;h ỵ N x Nh ẳ sin a Nh;h ỵ xNxh;x þ 2Nxh ¼ sin a 1 M xh;xh ỵ Mxh;h ỵ Mh;hh xMx;xx ỵ 2Mx;x þ sin a x x sin a À M h;x ỵ cot aNh ỵ xNx w;x ỵ Nxh w;h ị;x sin a 1 ỵ Nxh w;x ỵ N h w;h ị;h ẳ sin a x sin a 10ị v ẳ v0 ỵ v 1; w ẳ w0 ỵ w1 15ị 1 w1;hh ỵ w1;x x x2 sin a 1 ¼ w1;xh À w1;h : x sin a x sin a vxh1 ð16Þ In the present study, the FGM conical panel is assumed to be free simply supported at all edges and subjected to combination of external pressure q (Pascal) uniformly distributed on the outer surface and axial compressive load p (Pascal) uniformly distributed on the two end edges of the panel By solving the membrane form of equilibrium equations, prebuckling force resultants are determined as [11] q x2 x0 Nx0 ¼ À x tan a À 02 À ph ; x x Nh0 ¼ Àqx tan a; ¼0 Nxh0 ð17Þ Substitution of Eqs (14)–(17) into Eq (13) gives stability equations in terms of displacement components as Stability equations of FGM conical panels may be established by the adjacent equilibrium criterion [21] It is assumed that equilibrium state of the FGM conical panel under applied load is represented by displacement components u0, v0 and w0 The state of adjacent equilibrium differs that of stable equilibrium by u1, v1 and w1, and the total displacement components of a neighboring configuration are u ẳ u0 ỵ u1 ; u w v 1;h ỵ ỵ cot a x sin a x x v1 ¼ v 1;x À u1;h ; ỵ x x sin a vx1 ẳ w1;xx ; vh1 ẳ E3 ẳ Em h =12 ỵ Ecm h ẵ1=k ỵ 3ị 1=k ỵ 2ị ỵ 1=4k ỵ 4ị: xNx;x ỵ 14ị exm1 ẳ u1;x ; ehm1 ¼ where etc: in which cxhm1 E1 ¼ Em h ỵ Ecm h=k ỵ 1ị; E1 ; E2 ị E2 ; E3 ị exm1 ỵ mehm1 ị v þ mvh1 Þ; À m2 À m2 x1 N x1 ; M x1 ị ẳ Nx ; M x ị ẳ l11 u1 ị ỵ l12 v ị ỵ l13 w1 ị ẳ 18ị l21 u1 ị ỵ l22 v ị ỵ l23 w1 ị ẳ 19ị l31 u1 ị ỵ l32 v ị ỵ l33 w1 ị ỵ ql34 w1 ị ỵ pl35 w1 ị ẳ 20ị where the detail of operators lij are displayed in Appendix A The edges of truncated conical panel are free simply supported and associated boundary conditions are 11ị v ẳ 0; w1 ẳ 0; w1;xx ¼ at x ¼ x0 ; Similarly, the force and moment resultants of a neighboring state are represented by u1 ¼ 0; w1 ¼ 0; w1;hh ¼ at h ẳ 0; Nx ẳ Nx0 ỵ Nx1 ; From boundary conditions (21) approximate solutions for Eqs (18)– (20) are assumed as Mx ẳ Mx0 ỵ M x1 ; Nh ẳ Nh0 ỵ Nh1 ; Mh ẳ M h0 þ Mh1 ; Nxh ¼ N xh0 þ Nxh1 M xh ẳ M xh0 ỵ Mxh1 12ị where terms with subscripts correspond to the u0, v0, w0 displacements and those with subscripts represents the portions of the increments of force and moment resultants that are linear in u1, v1, w1 Subsequently, introduction of Eqs (11) and (12) into Eq (10) and subtracting from the resulting equations terms relating to stable equilibrium state, neglecting nonlinear terms in u1, v1, w1 or their counterparts in the form of Nx1, Nh1, etc and prebuckling rotations yield stability equations xNx1;x ỵ N xh1;h ỵ Nx1 Nh1 ẳ sin a Nh1;h ỵ xNxh1;x ỵ 2Nxh1 ẳ sin a 1 xMx1;xx ỵ 2M x1;x ỵ M xh1;xh ỵ M xh1;h ỵ M h1;hh sin a x x sin a Nxh0 À M h1;x ỵ cot aNh1 ỵ xNx0 w1;x ỵ w1;h sin a ;x 1 ỵ Nxh0 w1;x ỵ N h0 w1;h ¼0 sin a x sin a ;h mpðx À x0 Þ nph sin L b mpðx À x0 ị nph cos v ẳ V sin L b mpx x0 ị nph w1 ẳ W sin sin L b where, as mentioned above, force and moment resultants relating to stability state are 21ị u1 ẳ U cos ð22Þ where m, n are numbers of half waves in axial direction and waves in circumferential direction, respectively Due to x0 x x0 + L and for sake of convenience in integration, Eqs (18), (19) are multiplied by x and Eq (20) by x2 Subsequently, introduction of solutions (22) into obtained equations and applying Galerkin method for the resulting, that are Z 13ị x0 ỵ L b x0 ỵL x0 Z x0 ỵL x0 Z x0 ỵL x0 Z b R1 cos mpðx À x0 Þ nph sin x sin a dh dx ¼ L b R2 sin mpðx À x0 Þ nph cos x sin a dh dx ¼ L b R3 sin mpðx À x0 Þ nph sin x sin a dh dx ¼ L b Z b Z b ð23Þ where R1, R2, R3 are the left hand sides of Eqs (18)–(20) after these equations are multiplied by x, x and x2, respectively, and substituted into by solutions (22), we obtain the following equations 1382 D.H Bich et al / Composite Structures 94 (2012) 13791384 a11 U ỵ a12 V ỵ a13 W ẳ a21 U ỵ a22 V ỵ a23 W ẳ 24ị Table Effects of volume fraction index k and semi-vertex angle a on the critical buckling compressive loads pcr (GPa) of FGM conical panels (L/a = 2, a/h = 100, b = 300) a31 U ỵ a32 V ỵ a33 ỵ a34 q ỵ a35 pịW ẳ a () where the detail of coefficients aij may be found in Appendix B Because the solutions (22) are nontrivial, the determinant of coefficient matrix of Eq (24) must be zero a11 a21 a31 a12 a22 a32 ¼0 a23 a33 ỵ a34 q ỵ a35 p 15 30 40 50 a13 ð25Þ Solving Eq (25) for p and q yields a31 a12 a23 a13 a22 ịỵa32 a13 a21 a11 a23 ị ỵ a33 a11 a22 a12 a21 ị a34 q ỵ a35 p ẳ a12 a21 À a11 a22 ð26Þ Eq (26) is used for determining the buckling loads of FGM conical panels under uniform axial compression, external pressure and combined loads For given values of the material and geometrical properties of the FGM panel, critical buckling loads are determined by minimizing loads with respect to values of m, n a k 0.5 0.4242(11, 1)a 0.7656(11, 1) 0.5413(9, 1) 0.8912(9, 1) 0.4368(7, 1) 0.1987(11, 1) 0.4319(11, 1) 0.2876(9, 1) 0.5316(9, 1) 0.2362(7, 1) 0.1219(11, 1) 0.3060(11, 1) 0.1954(9, 1) 0.3899(9, 1) 0.1624(7, 1) 0.1177(11, 1) 0.2581(11, 1) 0.1832(9, 1) 0.2961(9, 1) 0.1467(7, 1) The numbers in brackets indicate the buckling mode (m, n) Table Effects of subtended angle b and ratio a/h on the critical buckling compressive loads pcr (GPa) of FGM conical panels (L/a = 2, k = 1, a = 300) b (°) a/h 30 50 70 90 120 150 80 100 150 0.5629(9, 1) 0.2756(5, 3) 0.1563(7, 3) 0.1023(5, 5) 0.1319(7, 5) 0.2377(7, 7) 0.1954(9, 1) 0.0982(3, 3) 0.0740(9, 1) 0.0629(9, 1) 0.0141(3, 7) 0.0520(9, 1) 0.0948(11, 1) 0.0348(9, 3) 0.0401(11, 1) 0.0351(11, 1) 0.0039(5, 9) 0.0227(7, 11) Results and discussion To verify the present study, the buckling of an FGM truncated conical shell made of silicon nitride and nickel and subjected to uniform axial compressive loads is considered The elasticity moduli of silicon nitride (Si3N4) and nickel at room temperature may be calculated be 348.43 GPa and 223.95 GPa, respectively Table shows the results of the present study, at which subtended angle b ? 2p and m = 0.31, in comparison with those reported by Sofiyev [14] As can be seen, a good agreement is obtained in this comparison To illustrate the proposed approach, we consider a ceramic– metal functionally graded conical panel that consists of aluminum and alumina with the following elasticity moduli Table Effects of volume fraction index k and semi-vertex angle a on the critical buckling external pressures qcr  104 (GPa) of FGM conical panels (L/a = 2, a/h = 100, b = 300) Em ¼ 70 GPa; Ec ¼ 380 GPa The critical buckling loads for FGM conical panels under simultaneous action of axial compression and external pressure are displayed in Table for different combinations of geometry parameters It is evident that the combination of mechanical loads and/or increase in length to small radius ratio L/a make FGM conical panels buckle sooner Also, the variation of buckling compressive and pressure loads versus L/a ratio is separately illustrated in Figs and As can be observed, there is a considerable difference between buckling loads as L/a is small In contrast, this difference becomes small when L/a ratio to be larger Finally, the variation trend of the critical compressive loads versus a/h ratio is analyzed in Fig This example shows that for various values of angles a and b, decreasing tendency of buckling loads versus a/h ratio is quite similar ð27Þ and Poisson’s ratio is chosen to be 0.3 Table shows the effects of volume fraction index k and semivertex angle a on the critical buckling axial compressive loads of FGM conical panels As expected, the critical compressive loads are reduced when the k index is increased However, there is no a definite trend of variation of critical loads versus various values of semi-vertex angle a Furthermore, for given geometrical properties, the panels buckle in the same a mode (m, n) for different material distributions The effects of subtended angle b and small radius to thickness ratio a/h on the critical buckling compressive loads are analyzed in Table As can be seen, there exist no a steady variation tendency in the critical loads for different values of angle b The critical buckling forces are increased or decreased in various ranges of subtended angle In addition, the critical forces are reduced and buckling modes are changed as a/h is enhanced Table considers the effects of k index and semi-vertex angle a on the critical loads for FGM conical panels under uniform external pressure It is similar to case of axially compressed panels, variation trend of the critical pressures versus increase in angle a is not stable a (°) 15 30 40 50 k 0.5 5.673(1, 1) 54.779(1, 3) 28.340(1, 3) 18.798(1, 3) 12.541(1, 3) 3.254(1, 1) 35.583(1, 3) 18.478(1, 3) 12.280(1, 3) 8.199(1, 3) 2.333(1, 1) 27.395(1, 3) 14.258(1, 3) 9.487(1, 3) 6.337(1, 3) 1.933(1, 1) 18.084(1, 3) 9.419(1, 3) 6.269(1, 3) 4.188(1, 3) Concluding remarks Due to practical importance of truncated conical shell panels and lack of investigations on stability of these structures, the present paper aims to propose an analytical approach to study the linear Table Comparison of dimensionless critical buckling compressive loads for an FGM truncated conical shell (a/h = 100, L/a = 2, a = 300) a pcr  103/Ec Si3N4 k = 0.5 k=1 k=2 Nickel Sofiyev [14] Present 3.244 3.132(11, 3)a 2.653 2.658(11, 3) 2.510 2.510(11, 3) 2.365 2.333(11, 1) 2.128 2.175(11, 5) The numbers in brackets indicate the buckling mode (m, n) 1383 D.H Bich et al / Composite Structures 94 (2012) 1379–1384 Table Critical buckling loads (qcr  104, pcr) (GPa) of FGM conical panels under combined action of axial compression and external pressure (k = 1, a/h = 100) (a, b) (°) L/a (10, 45) (10, 90) (30, 45) (30, 90) (21.395, 0) (14.318, 0.286) (7.952, 0.398) (0, 0.537) (3.482, 0) (1.335, 0.027) (0.694, 0.035) (0, 0.043) (12.174, 0) (4.283, 0.086) (1.939, 0.097) (0, 0.106) (2.815, 0) (1.297, 0.026) (0.717, 0.036) (0, 0.043) 1.5 (18.022, 0) (11.417, 0.228) (4.838, 0.242) (0, 0.252) (2.677, 0) (0.942, 0.019) (0.478, 0.024) (0, 0.029) (7.973, 0) (2.873, 0.057) (1.382, 0.069) (0, 0.080) (3.091, 0) (0.816, 0.016) (0.388, 0.019) (0, 0.022) p (GPa) p (GPa) 1.5 k = 1.0, a/h = 100 1: α = 40, β = 30, (m,n) = (9,1) 2: α = 30, β = 40, (m,n) = (9,1) 3: α = 20, β = 60, (m,n) = (7,3) 2: α = 30, β = 40, (m,n) = (9,1) 0.5 4: α = 10, β = 90, (m,n) = (5,5) 3: α = 20, β = 60, (m,n) = (7,3) 2 3 4 −0.5 −1 0.5 L/a 1.5 Fig Variation of buckling compressive loads of FGM conical panels versus L/a ratio q x 10−3 (GPa) 1: α = 40, β = 30 2: α = 30, β = 40 2.5 3: α = 60, β = 50 2 −1 100 150 200 250 300 350 400 a/h Fig Variation of buckling compressive loads of FGM conical panels versus a/h ratio ical panels is complex and very sensitive to variation of material and geometrical parameters The study shows that there is no a definite variation trend in critical buckling loads for various values of semivertex and subtended angles The results also indicate that for given geometrical parameters FGM conical panels buckle in the same mode with different material distributions k = 1.0, a/h = 100, m = 1, n = 3.5 Acknowledgements 4: α = 50, β = 60 This paper was supported by the National Foundation for Science and Technology Development of Vietnam–NAFOSTED, project code 107.02-2010.08 The authors are grateful for this financial support 1.5 0.5 4: α = 10, β = 90, (m,n) = (5,5) 0.5 k = 1.0, L/a = 1.5 1: α = 40, β = 30, (m,n) = (9,1) Appendix A 1.5 2.5 The detail of operators lij in Eqs (18)–(20) L/a Fig Variation of buckling external pressures of FGM conical panels versus L/a ratio l12 buckling of simply supported FGM conical panels subjected to mechanical loads Formulation is based on the classical thin shell theory and adjacent equilibrium criterion Approximate solutions are assumed to satisfy simply supported boundary conditions and Galerkin method is applied to derive closed-form relations of buckling loads from which linear buckling behavior of FGM conical panels may be predicted In general, buckling behavior of FGM con- ! @2 @ 1 À m @2 ỵ ỵ @x2 @x x 2x sin2 a @h2 " # E1 @2 @ ¼ l21 ¼ ỵ mị ỵ m 3ị x@h sin a @x@h " @3 @2 E1 @ @3 ẳ E2 x ỵ m cot a ỵ ị ỵ x @x x sin a @x@h2 @x @x E2 # @2 E1 À þ cot a 2 x E2 x2 sin a @h l11 ¼ E1 x l13 1384 D.H Bich et al / Composite Structures 94 (2012) 1379–1384 " # E1 ð1 À mÞ @2 @ @2 l23 x 2ỵ ỵ l22 ẳ @x x mÞx sin2 a @h2 @x " # @3 @2 @3 E1 cot a @ ¼ ÀE2 À þ þ sin a @x2 @h x sin a @x@h x2 sin3 a @h3 E2 x sin a @h " @3 @2 E1 @ @3 l31 ẳ E2 x ỵ 2 m cot a ỵ x E2 @x x sin a @x@h2 @x @x # @2 E1 cot a ỵ ỵ ỵ 2 x2 E2 x x2 sin a @h " @3 @2 l32 ¼ E2 À sin a @x @h x sin a @x@h # E1 cot a 1 @ @3 ỵ ỵ ỵ x sin a @h x2 sin3 a @h3 E2 x " @4 @3 @2 @4 l33 ẳ E3 x ỵ ỵ x @x2 x sin2 a @x2 @h2 @x @x 2E2 cot a @ @3 @4 ỵ ỵ ỵ 2 4 x @x x2 sin a @x@h E3 x x3 sin a @h # @2 E2 cot a E1 cot2 a ỵ ỵ ị 2 E3 x2 E3 x x3 sin a @h ! @ x2 x20 @ @2 ỵ l34 ẳ m ị tan a x ỵ @x @x2 sin2 a @h2 l35 ¼ Àð1 À m2 Þhx0 @2 @x2 Appendix B The detail of coefficients aij in Eq (24) " # p2 n2 ð1 À mÞ k ỵ ị ỵ 8ð1 À kÞ2 b2 sin a " # 3 p mn1 ỵ mị k kị mị1 kị2 n ỵ ẳ a21 ẳ ỵ 4b sin a1 kị 8mb sin a 2p2 m2 ! " 3 E2 p m dð1 À k Þ E2 pmdð1 À k ị p2 n2 ỵ ẳ ỵ 2 b sin a 4E1 ð1 À kÞ 8E1 ð1 À kÞ # mpmð1 À k Þ ð1 kị ỵ mị cot a ỵ cot a þ 6ð1 À kÞ 4pm ! p2 m2 ð1 À mÞð1 À k4 Þ ð1 À k2 Þ À m p n2 ẳ ỵ ỵ 2 4 16ð1 À kÞ b sin a " E2 p3 m2 ndð1 À k3 Þ E2 p3 n3 dð1 kị pn1 k2 ị ẳ ỵ ỵ 4b sin a 6E1 ð1 À kÞ2 b sin a 2E1 b3 sin a a11 ¼ a12 a13 a22 a23 p2 m2 ð1 À k4 Þ cot a " a31 a32 a33 E2 ¼ d E1 p3 m3 ð1 À k5 Þ pmð1 À k3 Þ p2 n2 ! kị2 2pm # # B1 ị 1ỵ ỵ 61 kị 101 kị3 b sin a " mpmð1 À k Þ ð2 3mị1 kị ỵ k2 ị cot a 1Àk pm " !# E2 p3 m2 nð1 À k4 Þ pnð1 À k2 Þ p2 n2 ẳ d ỵ 4b sin a E1 8ð1 À kÞ2 b sin a b2 sin a " # 3 pn cot a À k ð1 À kÞ À À 2b sin a 2p2 m2 " 3E2 E3 p4 m4 ð1 À k5 Þ p2 m2 k3 ị 2p2 n2 ẳ cot a1 k2 ịd d2 ỵ ỵ Þ 2 4E1 E1 10ð1 À kÞ 6ð1 À kÞ b sin a !# " # ð1 À kÞ 6p2 n2 p n4 1 À k3 kị3 ỵ cot2 a 1ỵ À À 2 2p2 m2 b sin a b sin a ! 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(18)–(20) L/a Fig Variation of buckling external pressures of FGM conical panels versus L/a ratio l12 buckling of simply supported FGM conical panels subjected to mechanical loads Formulation is based... Variation of buckling compressive loads of FGM conical panels versus a/h ratio ical panels is complex and very sensitive to variation of material and geometrical parameters The study shows that... present paper aims to investigate the linear buckling behavior of simply supported truncated conical panels made of functionally graded materials and subjected to mechanical loads The classical shell