Composite Structures 106 (2013) 104–113 Contents lists available at SciVerse ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct Instability of eccentrically stiffened functionally graded truncated conical shells under mechanical loads Dao Van Dung, Le Kha Hoa ⇑, Nguyen Thi Nga, Le Thi Ngoc Anh Vietnam National University, Hanoi, Viet Nam a r t i c l e i n f o Article history: Available online 13 June 2013 Keywords: Stiffened truncated conical shell Stiffener Functionally graded material Critical buckling load Analytical a b s t r a c t This paper is concerned with the mechanical buckling load of an eccentrically stiffened truncated conical shells made of functionally graded materials and subjected to axial compressive load and external uniform pressure by analytical method Shells are reinforced by stringers and rings The change of spacing between stringers in the meridional direction is taken into account Material properties of shell are graded in the thickness direction according to a volume fraction power-law distribution The equilibrium and linearized stability equations for stiffened shells are derived based on the classical shell theory and smeared stiffeners technique The resulting equations which they are the couple set of three variable coefficient partial differential equations in terms of displacement components are investigated by Galerkin method and the closed-form expression for determining the buckling load is obtained The effects of stiffeners, material and dimensional parameters are analyzed in detail Ó 2013 Elsevier Ltd All rights reserved Introduction A conical shell is one of the common structural elements used in modern airplane, missile, booster and other aerospace vehicles As a result, problems relating to stability and vibration of these structures have received considerable attention of researchers Seide [1,2] investigated the buckling of conical shells under the axial loading Singer [3] presented the buckling of conical shells subjected to the axisymmetrical external pressure Lu and Chang [4] and Chang and Lu [5] examined the themoelastic buckling of conical shells based on linear [4] and nonlinear [5] analyses They used Galerkin method for integrating the equilibrium equation Tani and Yamaki [6] obtained the results of truncated conical shells under axial compression Using the Donnell-type shell theory, the linear buckling analysis of laminated conical shells, with orthotropic stretching-bending coupling, under axial compressive load and external pressure, are studied by Tong and Wang [7] The same authors [8] investigated linear buckling analysis of laminated composite conical shells Eight first-order differential equations were obtained and solved by the numerical integration technique and the multisegment method in their work Wu and Chiu [9] presented a three-dimensional solution for the thermal buckling of the laminated composite conical shells Xu et al [10] presented a solution for nonlinear free vibration of symmetrically laminated cross-ply conical shell with its two ends ⇑ Corresponding author Tel.: +84 989358315 E-mail address: lekhahoa@gmail.com (L.K Hoa) 0263-8223/$ - see front matter Ó 2013 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.compstruct.2013.05.050 both clamped and both simply supported Galerkin procedure and the method of harmonic balance used to analyze dynamic responses of shell Lam et al [11] reported an improved generalized differential quadrature method for the investigation of the effects of boundary conditions on the free vibration characteristics of truncated conical panels The effects of the vertex angles on the frequency parameters were examined in their study Based on the classical thin shell theory, Liew et al [12] considered the free vibration analysis of thin conical shells under different boundary conditions using the element-free kp-Ritz method The kernel particle (kp) functions are employed in hybridized form hamonic functions to approximate the two-dimensional displacement field Civalek [13] proposed a discrete singular convolution method for analyzing the free vibration of rotating conical shells A regularized Shannon’s delta kernel is selected as the singular convolution to illustrate his algorithm In recent years, because functionally graded material (FGM) structures are widely used in modern engineering, the stability and vibration behaviors of FGM plates and shells have attracted increasing research effort Among those available, Sofiyev [14– 16] investigated the linear stability and vibration of unstiffened FGM truncated conical shells with different boundary conditions The same author [17] presented the nonlinear buckling behavior and nonlinear vibration [18] of FGM truncated conical shells, and considered [19] the buckling of FGM truncated conical shells subjected to axial compressive load and resting on Winkler–Pasternak foundations For linear analysis, the general characteristics in his works is that the modified Donnell-type equations are used and Galerkin method is applied to obtain closed-form relations of 105 D Van Dung et al / Composite Structures 106 (2013) 104–113 bifurcation type buckling load or to find expressions of fundamental frequencies, whereas for nonlinear analysis, the large deflection theory with von Karman–Donnell-type of kinetic nonlinearity is used Based on the first-order shell theory by Love–Kirchhoff and the Sanders nonlinear kinetic equations, the thermal and mechanical instability of FGM truncated conical shells also is investigated by Naj et al [20] Bich et al [21] presented results on the buckling of un-stiffened FGM conical panels under mechanical loads The linearized stability equations in terms of displacement components are derived by using the classical shell theory Galerkin method is applied to obtained explicit expression of buckling load Malekzadeh and Heydarpour [22] studied the influences of centrifugal and Coriolis forces in combination with the other geometrical and material parameters on the free vibration behavior of rotating FGM truncated conical shells subjected to different boundary conditions based on the first-order shear deformation theory As can be seen that the above introduced works only relate to unstiffened structures However, in practice, plates and shells including conical shells, usually reinforced by stiffeners system to provide the benefit of added load carrying capability with a relatively small additional weight Thus, the study on static and dynamic behavior of theses structures are significant practical problem Weingarten [23] conducted a free vibration analysis for a ring-stiffened simply supported conical shell by considering an equivalent orthotropic shell and using Galerkin method He also carried out experimental investigations Crenwelge and Muster [24] applied an energy approach to find the resonant frequencies of simply supported ringstiffened, and ring and stringer-stiffened conical shells Mustaffa and Ali [25] studied the free vibration characteristics of stiffened cylindrical and conical shells by applying structural symmetry techniques Some significant results on vibration of FGM conical shells, cylindrical shells and annular plate structures with a fourparameter power-law distribution based on the first-order shear deformation theory are analyzed by Tornabene [26] and Tornabene et al [27] Srinivasan and Krisnan [28] obtained the results on the dynamic response analysis of stiffened conical shell panels in which the effect of eccentricity is taken into account The integral equation for the space domain and mode superposition for the time domain are used in their work Based on the Donnell–Mushtari thin shell theory and the stiffeners smeared technique, Mecitoglu [29] studied the vibration characteristics of a stiffened truncated conical shell by the collocation method The minimum weight design of axially loaded simply supported stiffened conical shells with natural frequency constraints is considered by Rao and Reddy [30] The influence of placing the stiffeners inside as well as outside the conical shell on the optimum design is studied The expressions for the critical axial (buckling) load and natural frequency of vibration of conical shell also are derived For stiffened FGM structures, recently there are some investigations which have been focused on the analysis of the static buckling and postbuckling, vibration and dynamic buckling of plates and shells Najafizadeh et al [31] with the linearized stability equations in terms of displacements studied buckling of FGM cylindrical shell reinforced by rings and stringers under axial compression The stiffeners and skin, in their work, are assumed to be made of functionally graded materials and its properties vary continuously through the thickness direction Bich et al [32] presented an analytical approach to investigated the nonlinear postbuckling of eccentrically stiffened FGM plates and shallow shells based on the classical shell theory in which the stiffeners are assumed to be homogeneous Dung and Hoa [33] obtained the results on the static nonlinear buckling and post-buckling analysis of eccentrically stiffened FGM circular cylindrical shells under external pressure The material properties of shell and stiffeners are as- sumed to be continuously graded in the thickness direction Galerkin method was used to obtain closed-form expressions to determine critical buckling loads Bich et al [34] obtained the results on the nonlinear dynamic analysis of eccentrically stiffened FGM cylindrical panels The governing equations of motion were derived by using the smeared stiffeners technique and the classical shell theory with von Karman geometrical nonlinearity The same authors [35] investigated the nonlinear vibration dynamic buckling of eccentrically stiffened imperfect FGM doubly curved thin shallow shells based on the classical shell theory The nonlinear critical dynamic buckling load is found according to the Budiansky–Roth criterion The review of the literature signifies that few analytical studies have been carried out to investigate on the stability of FGM conical shells and there is no work on the analytical solution for combined loaded stiffened FGM conical shells This may be attributed to the inherent complexity of governing equations of conical shell Those are variable coefficient partial differential equations In addition, for stiffened conical shell, the spacing between stringers in the meridional direction also varies These difficulties have to be got over as investigating the stability of truncated conical shells In this paper, a mechanical buckling of eccentrically stiffened functionally graded (ES-FGM) thin truncated conical shells subjected to axial compressive load and uniform external pressure load is investigated The present novelty is that the shells under combined load are reinforced by rings and stringers attached to their inside or outside The change of spacing between stringers in the meridional direction is taken into account The material properties of shell are graded continuously in the thickness direction The theoretical formulations based on the smeared stiffeners technique and the classical shell theory, are derived The resulting equations which they are the couple set of three variable coefficient partial differential equations in terms of displacement components are solved by Galerkin method The closed-form expressions to determine critical buckling loads are obtained The influences of various parameters such as stiffener, dimensional parameters and volume fraction index of materials on the stability of shell are clarified in detail FGM truncated conical shell and theoretical formulation 2.1 Functionally graded truncated conical shell Consider a thin truncated conical shell of thickness h and semivertex angle a The geometry of shell is shown in Fig 1, where L is the length and R is its small base radius The truncated cone is referred to a curvilinear coordinate system (x, h, z) whose the origin is located in the middle surface of the shell, x is in the generatrix direction measured from the vertex of conical shell, h is in the circumferential direction and the axes z being perpendicular to the plane (x, h), lies in the outwards normal direction of the cone Also, x0 indicates the distance from the vertex to small base, and u, v, and w denote the displacement components of a point in the middle surface in the direction x, h and z, respectively Assume that the truncated conical shell is made from a mixture of a ceramic and a metal (denoted by c and m, respectively) and the material compositions only vary smoothly along its thickness direction with the power law distribution as V m ỵ V c ẳ 1; V c ẳ V c zị ẳ k z ỵ h 1ị where h/2 z h/2 and k P is the volume fraction and takes only non-negative values According to the mentioned law, the Young’s modulus can be expressed by E ẳ Ezị ẳ Em V m ỵ Ec V c ẳ Em ỵ Ecm 2z ỵ h 2h k ð2Þ 106 D Van Dung et al / Composite Structures 106 (2013) 104–113 z O α Ring b2 h2 x0 z2 x R h d2 L b1 d2 d1 Stringer z1 h1 d1 θ h x (a) (b) Fig Geometry of stiffened truncated conical shell where Ecm ¼ Ec À Em ð3Þ The Poisson’s ratio is assumed to be a constant 2.2 Theoretical formulation The present study uses the classical shell theory with the geometrical nonlinearity in von Karman sense and smeared stiffeners technique to establish the governing equations Thus, the normal and shear strains at distance z from the middle surface of shell are [36] ex ¼ exm þ zkx ; eh ¼ ehm þ zkh ; cxh ẳ cxhm ỵ 2zkxh ; 4ị where exm and ehm are the normal strains and cxhm is the shear strain at the middle surface of the shell, and kx, kh and kxh are the change of curvatures and twist, respectively They are related to the displacement components as [36] 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 cxhm ẳ u;h ỵ v ;x þ w;x w;h x sin a x sin a x 1 w;hh À w;x ; kx ¼ Àw;xx ; kh ¼ À x x2 sin a 1 kxh ẳ w;xh ỵ w;h : x sin a x sin a ehm ẳ 5ị rsh x r sh h rsh xh 6ị Nxh ẳ A66 cxhm ỵ 2B66 kxh ; 9ị Mxh ẳ B66 cxhm ỵ 2D66 kxh ; 10ị Es I kx ỵ D12 kh ; Mx ẳ ẵB11 ỵ C xịexm þ B12 ehm þ D11 þ d1 ðxÞ Er I kh ; Mh ẳ B12 exm ỵ ẵB22 ỵ C ehm ỵ D12 kx ỵ D22 ỵ d2 A1 ẳ b1 h1 ; C2 ẳ ặ ð7Þ where the subscripts sh and s denote shell and stiffeners, respectively Es and Er are Young moduli of stiffener in the x-direction C 01 ẳ ặ b1 h1 ỵ A1 z21 ; 12 E1 A11 ẳ A22 ¼ ; À m2 E2 B11 ¼ B22 ¼ ; À m2 E3 ; D11 ¼ D22 ¼ m2 I1 ẳ 8ị C 01 ; x h ỵ h1 h ỵ h2 ; z2 ẳ ; 2 L 2p sin a d1 xị ẳ k0 x; d2 ¼ ; k0 ¼ ; nr ns A2 ¼ b2 h2 ; E r A2 z ; d2 C xị ẳ and for stiffeners rsx ẳ Es ex ; rsh ¼ Er eh ; Es A1 Nx ẳ A11 ỵ exm ỵ A12 ehm ỵ ẵB11 ỵ C xịkx ỵ B12 kh ; d1 xị Er A2 ehm ỵ B12 kx ỵ ẵB22 ỵ C kh ; Nh ẳ A12 exm ỵ A22 ỵ d2 where The stressstrain relations for the FGM conical shell are Ezị ẳ ex ỵ meh ị; m2 Ezị ẳ eh ỵ mex ị; m2 Ezị ẳ c ; 21 ỵ mị xh and h-direction, respectively To guarantee the continuity between the stiffener and shell, the stiffener is taken to be pure-metal if it is located at metal-rich side and is pure-ceramic if it is located at ceramic-rich side Taking into account the contribution of stiffeners by the smeared stiffener technique and omitting the twist of stiffeners because these torsion constants are smaller more than the moments of inertia [36] In addition, the change of spacing between stringers in the meridional direction also is taken into account Integrating the above stress–strain equations and their moments through the thickness of the shell, we obtain the expressions for force and moment resultants of an eccentrically stiffened FGM conical shells and z1 ¼ Es A1 z1 ; k0 I2 ẳ b2 h2 ỵ A2 z22 ; 12 mE1 E1 A12 ¼ ; A66 ẳ ; m2 21 ỵ mị m E2 E2 B12 ¼ ; B66 ¼ ; À m2 21 ỵ mị mE3 E3 D12 ẳ ; D66 ẳ : m2 21 ỵ mị 11ị 107 D Van Dung et al / Composite Structures 106 (2013) 104–113 1 À E2 ¼ Ecm h ðk þ 2Þ ð2k þ 2Þ 1 1 3 ; Em h ỵ Ecm h þ E3 ¼ 12 k þ k þ 4k ỵ E1 ẳ Em h ỵ Ecm h ; kỵ1 ! ; 12ị in which ns, nr are the number of stringer and ring respectively; h1 and b1 are the thickness and width of stringer (x-direction); h2 and b2 are the thickness and width of ring (h-direction) Also, d1 = d1(x) and d2 are the distance between two stringers and two rings, respectively The quantities A1, A2 are the cross-section areas of stiffeners and I1, I2 are the second moments of inertia of the stiffener cross sections relative to the shell middle surface; and z1, z2 represent the eccentricities of stiffeners with respect to the middle surface of shell Using the classical shell theory, the nonlinear equilibrium equations of truncated conical shells are given as follows [36] xNx;x ỵ Nxh;h ỵ Nx Nh ẳ 0; sin a Nh;h ỵ xNxh;x ỵ 2Nxh ẳ 0; sin a xM x;xx ỵ 2M x;x ỵ M xh;xh ỵ M xh;h sin a x 1 M À M À N cot a ỵ xN w ỵ N w ỵ h;hh h;x h x ;x xh ;h sin a x sin a ;x 1 ỵ Nxh w;x ỵ N h w;h qx ẳ 0: 13ị sin a x sin a ;h To establish the stability equations, the adjacent equilibrium criterion is used [36] Assume that the equilibrium state of ESFGM conical shell under mechanical loads is defined in terms of the displacement components u0, v0 and w0 We give an arbitrarily small increments u1, v1 and w1 to the displacement variables, so the total displacement components of a neighboring state are u ẳ u0 ỵ u1 ; v ẳ v0 ỵ v 1; w ẳ w0 ỵ w1 : ð14Þ Similarly, the force and moment resultants of a neighboring state may be related to the state of equilibrium as Nx ẳ Nx0 ỵ Nx1 ; Mx ẳ Mx0 ỵ M x1 ; Nh ẳ Nh0 ỵ Nh1 ; Mh ẳ M h0 ỵ Mh1 ; Nxh ẳ N xh0 ỵ Nxh1 ; M xh ẳ M xh0 þ Mxh1 ; ð15Þ where terms with subscripts correspond to the u0, v0, w0 displacements and those with subscripts represents the portions of increments of force and moment resultants that are linear in u1, v1 and w1 The stability equations may be obtained by substituting Eqs (14) and (15) into Eq (13) and note that the terms in the resulting equations with subscript satisfy the equilibrium equations and therefore drop out of the equations In addition, the nonlinear terms with subscript are ignored because they are small compared to the linear terms The remaining terms form the stability equations as follows xNx1;x ỵ Nxh1;h ỵ N x1 À N h1 ¼ 0; sin a Nh1;h ỵ xNxh1;x ỵ 2Nxh1 ẳ 0; sin a 1 xM x1;xx ỵ 2Mx1;x ỵ M xh1;xh ỵ M xh1;h ỵ M h1;hh sin a x x sin a À M h1;x Nh1 cot a ỵ xNx0 w1;x ỵ Nxh0 w1;h sin a ;x 1 ỵ Nxh0 w1;x þ Nh0 w1;h ¼0 sin a x sin a ;h Es A1 C0 Nx1 ẳ A11 ỵ exm1 þ A12 ehm1 þ B11 þ kx1 þ B12 kh1 ; k0 x x ! C Es I1 kx1 ỵ D12 kh1 ; etc M x1 ẳ B11 ỵ exm1 ỵ B12 ehm1 ỵ D11 þ x k0 x ð17Þ and the linear form of the strains and curvatures in terms of the displacement components are u1 w1 v1;h ỵ ỵ cot a; x sin a x x v cxhm1 ¼ v 1;x ỵ u1;h ; x sin a x 1 kx1 ¼ Àw1;xx ; kh1 ¼ À w1;hh À w1;x ; x x2 sin a 1 kxh1 ẳ w1;xh ỵ w1;h : x sin a x sin a exm1 ẳ u1;x ; ehm1 ẳ 18ị Also assume that a shell only subjected to the axial compressive load P (N) and external uniform pressure q (Pa) Thus, the prebuckling force resultants of shell are found by solving the membrane form of the equilibrium Eq (13), as [36] q P Nx0 ¼ À x tan a À ; px sin 2a Nh0 ¼ Àqx tan a; Nxh0 ẳ 0: 19ị Substituting Eqs (17)(19) into Eq (16), the stability equations in terms of the displacement components are of the form T 11 u1 ị ỵ T 12 v ị ỵ T 13 w1 ị ẳ 0; 20ị T 21 u1 ị ỵ T 22 v ị ỵ T 23 w1 ị ẳ 0; 21ị T 31 u1 ị ỵ T 32 v ị ỵ T 33 w1 ị ỵ qT 34 w1 ị ỵ PT 35 w1 ị ẳ 0; 22ị where Tij are differential operators and are defined as Es A1 @ @2 @ Er A2 T 11 ẳ A11 x ỵ ỵ A66 ỵ A11 A22 ỵ ; 2 @x k0 @x d2 x @h x sin a @ Er A2 @ A12 ỵ A66 ị A22 ỵ A66 ỵ ; T 12 ẳ sin a @x@h x sin a d2 @h @3 @3 T 13 ẳ B11 x ỵ C 01 B12 ỵ 2B66 ị @x3 x sin2 a @x@h2 Er A2 @2 ỵ cot a A22 ỵ B12 ỵ 2B66 ỵ B22 ỵ C Þ 2 x d2 @h x2 sin a @ @ B11 ỵ B22 ỵ C ị ỵ A12 cot a ; x @x @x @2 Er A2 @ A12 ỵ A66 ị ỵ A22 ỵ A66 ỵ ; sin a @x@h x sin a d2 @h E r A2 @ @2 @ ẳ A ỵ ỵ xA66 ỵ A66 A66 ; 22 2 @x x d2 @h @x x sin a T 21 ¼ T 22 @3 @3 B12 ỵ 2B66 ị B22 ỵ C Þ 3 sin a @x @h x2 sin a @h @ Er A2 @ B22 ỵ C ị ỵ cot a A22 ỵ ; x sin a @x@h x sin a d2 @h T 23 ẳ 16ị where the force and moment resultants for the state of stability are given by @3 @3 @2 T 31 ¼ B11 x ỵ C 01 ỵ B12 ỵ 2B66 ị þ 2B11 2 @x @x @x@h x sin a @2 @ ỵ B22 ỵ C ị A12 cot a þ ðB22 þ C Þ þ 2 x @x x @h x sin a E r A2 ; B22 ỵ C ị cot a A22 ỵ x d2 108 D Van Dung et al / Composite Structures 106 (2013) 104–113 T 32 ẳ @3 @3 B12 ỵ 2B66 ị ỵ B22 ỵ C ị sin a x sin a @x @h x2 sin a @h @2 B22 ỵ C ị @x@h 1 Er A2 @ ỵ B22 ỵ C ị cot a A22 ỵ ; x sin a x sin a @h d2 ð27Þ Substituting expressions (25) and (27) into Eq (26), after integrating longer and some rearrangements, may be written in the following form @4 @3 D12 ỵ 2D66 ị ỵ D12 þ 2D66 Þ 2 2 @x @h @x@h2 x sin a x sin a @ Er I @ ỵ 2B12 cot a 2D11 ỵ D22 ỵ x @x d2 @x2 ỵ cot aB22 ỵ C Þ x2 sin a 2 Er I @ D ỵ 2D ỵ D ỵ 12 66 22 d2 @h2 x3 sin a Er I @ À D22 þ þ cot aðB22 þ C Þ x d2 @x x2 E r A2 ; À cot2 a A22 ỵ x d2 ! x2 @ @ @2 ; ¼ À tan a ỵx ỵ @x sin2 a @h2 @x2 T 34 T 35 ¼ À @2 : p sin 2a @x2 ð23Þ The system of Eqs (20)–(22) is used to analyze the stability and find the critical buckling load of ES-FGM truncated conical shells It is difficult that these equations are a couple set of three variable coefficient partial differential equations This problem will be got over below Buckling analysis of ES-FGM truncated conical shell In this section, an analytical approach is given to investigate the stability of ES-FGM truncated conical shells Assume that a shell is simply supported at both ends The boundary conditions in this case, are expressed by v ¼ w1 ¼ 0; M x1 ẳ at x ẳ x0 ; x0 ỵ L: ð24Þ The approximate solution [21] satisfying the boundary conditions (24) may be described as mpðx À x0 Þ nh sin ; L mpðx À x0 Þ nh v ¼ B sin cos ; L mpðx À x0 ị nh sin ; w1 ẳ C sin L u1 ẳ A cos x0 ỵL x0 Z x0 þL x0 Z x0 þL x0 Z 2p X1 cos mpx x0 ị nh sin x sin adhdx ẳ 0; L X2 sin mpðx À x0 Þ nh cos x sin adhdx ¼ 0; L Z 2p Z 2p X3 sin mpðx À x0 Þ nh sin x sin adhdx ¼ 0; L L11 A ỵ L12 B ỵ L13 C ẳ 0; L21 A ỵ L22 B ỵ L23 C ẳ 0; 28ị L31 A ỵ L32 B ỵ L33 ỵ qL34 ỵ PL35 ịC ẳ 0; where the coefcients Lij are defined in Appendix Because the solutions of Eq (28) are nontrivial, the determinant of coefficient matrix of this system must be zero Developing that determinant and solving resulting equation for combination of P and q, leads to L34 q ỵ L35 P ẳ L31 L12 L23 L13 L22 ị þ L32 ðL13 L21 À L11 L23 Þ À L33 L21 L12 À L11 L22 ð29Þ Eq (29) is used to determine the critical buckling load of ES-FGM conical shells subjected to axial compressive load and uniform pressure load The buckling loads P and q still depend on values of m and n, therefore must minimize these expressions with respect to m and n, we obtain the critical values of P and q respectively Numerical results and discussion 4.1 Comparison results To validate the present study, firstly, Table compares the results of this paper for an un-stiffened isotropic truncated conical shell under external pressure with the results given in the monograph of Brush and Almroth [36, pp 217] The data base in this case is taken as: k = 0, h = 0.01 m, R = 100  h, m = 0.3, P = 0, q⁄ = 104Rqcr/ (Eh), where qcr is found from Eq (29) Next, Table compares the present results with those of Naj et al [20] and Baruch et al [37] for a pure isotropic truncated conical shell The input parameters are: k = 0, h = 0.01 m, R = 100  h, pEh cos2 a ffiffiffiffiffiffiffiffiffiffiffiffi m = 0.3, q = 0, P⁄ = Pcr/Pcl with Pcl ¼ 2p [1,20], where Pcr is 3ð1Àm2 Þ found from Eq (29) As can be seen that good agreements are obtained in these two comparisons 4.2 ES-FGM truncated conical shell ð25Þ where m is the number of half-waves along a generatrix and n is the number of full-waves along a parallel circle, and A, B and C are constant coefficients Due to x0 x x0 + L; h 2p and for sake of convenience in integration, Eqs (20) and (21) are multiplied by x and Eq (22) by x2, and applying Galerkin method for the resulting equations, that are Z X1 ẳ xẵT 11 u1 ị ỵ T 12 v ị ỵ T 13 w1 ị; X2 ẳ xẵT 21 u1 ị ỵ T 22 v ị ỵ T 23 w1 ị; X3 ẳ x2 ẵT 31 u1 ị ỵ T 32 v ị ỵ T 33 w1 ị ỵ qT 34 w1 ị ỵ PT 35 ðw1 Þ: Es I @ Er I @ ¼ À D11 x þ À ðD þ Þ 22 k0 @x4 x3 sin4 a d2 @h4 T 33 where In the following subsections, the materials used are Alumina with Ec = 380 GPa and Aluminum with Em = 70 GPa and m = 0.3 Table Comparisons with results of [36] for un-stiffened isotropic conical shells qà ¼ 10 Ehqcr R 10 300 500 700 ð26Þ 850 a [36] Present [36] Present [36] Present [36] Present [36] Present Buckling mode (m, n) L/R = 1/2 L/R = 19.40 19.3867 (1,22)a 14.55 14.4517 (1,23) 8.81 8.6783 (1,21) 3.50 3.4120 (1,17) 0.591 0.5756 (1,9) 8.57 8.5293 5.84 5.7808 3.28 3.2094 1.201 1.1510 0.1710 0.1589 L/R = (1,17) (1,19) (1,19) (1,16) (1,10) 3.74 3.7572 2.24 2.2594 1.164 1.1403 0.399 0.3747 0.0498 0.0445 (1,14) (1,16) (1,17) (1,16) (1,10) 109 D Van Dung et al / Composite Structures 106 (2013) 104–113 Table Comparisons with results of [20] and [37] for un-stiffened isotropic truncated conical shells a 10 50 100 300 600 800 a L/R = 0.2 L/R = 0.5 Naj et al [20] Baruch et al [37] Present Naj et al [20] Baruch et al [37] Present 1.005 (7) 1.006 (7) 1.007 (7) 1.0171 (5) 1.148 (0) 2.492 (0) 1.005 1.006 1.007 1.017 1.144 2.477 1.0002 (1,12)a 1.0001 (1,12) 1.0002 (1,12) 1.0017 (1,7) 1.1299 (1,1) 2.5091 (1,1) 1.0017 (8) 1.001 (8) 1.000 (8) 0.987 (7) 1.045 (7) 1.004 (5) 1.002 1.002 1.002 1.001 1.044 1.015 1.0001 1.0002 1.0005 1.0023 1.0150 1.0266 (7) (7) (7) (5) (0) (0) (2,17) (2,17) (2,17) (2,15) (1,14) (1,4) Buckling mode (m, n) Table Effect of stiffener arrangement on critical axial compressive load Pcr (MN), a (8) (8) (8) (7) (7) (5) Pcr (MN) (q = 0) Outside Inside Un-stiffened Stringer (nst = 30) Ring (nr = 30) Orthogonal (nst = nr = 15) 95.2671 (6,21) a 123.2511 (2,19) 105.4866 (9,1) 132.7063 (2,17) 95.2671 (6,21) 100.1151 (2,19) 95.5330 (7,18) 102.8853 (4,21) Table Effect of stiffener number on critical axial compressive load Pcr (MN), a = 300 Stiffener number (nst = nr) Pcr (MN); q = 10 (nst = nr = 5) 20 30 40 50 110.0880 122.0932 132.7063 142.3474 151.7364 Outside Buckling mode (m, n) The geometrical parameters are taken as h = 0.0127 m, R = 1.27 m, L = 2R, h1 = 0.01375 m, b1 = 0.0127 m, h2 = 0.01 m, b1 = 0.0127 m The stiffener is taken to be pure-metal if it is located at inside surface of shell and is pure-ceramic if it is located at outward surface of shell Inside (3,20) (2,18) (2,17) (2,17) (2,17) 98.1136 (4,22) 100.6099 (4,22) 102.8853 (4,21) 105.1471 (4,21) 107.2141 (3,20) the critical load increases when the number of stiffeners increases and inversely This increase is considerable For example, comparing Pcr = 110.0880 (MN) (ns = 10) with Pcr = 151.7364 (MN) (ns = 50) in Table 5, the value of critical compressive load increases about 27.4% The prime reason is that the presence of stiffeners makes the shells to become stiffer 4.2.1 Effect of stiffener arrangement Using Eq (29) with q = 0, a = 300, the critical axial compressive load of FGM truncated conical shell may be found Table shows that critical compressive load Pcr of shells are reinforced by stiffeners attached to their inside is smaller than outside ones With the same stiffener number (ns = 30), the critical axial compressive load of the orthogonally stiffened shell is the biggest, the second is stringer stiffened shell, the third is ring stiffened shell and the critical load of un-stiffened shell is the smallest In addition, the critical axial compressive loads of externally stiffened shells are bigger than those of internally stiffened shells This is reasonable because the external stiffeners are ceramic (stiffer), while the internal stiffeners are metal having smaller stiffness The effect of stiffener arrangement and pre-loaded axial pression P on critical pressure load qcr is presented in Table using Eq 29 It is observed that the critical pressure qcr for stiffened shells by ring is the biggest, the second is orthogonally stiffened, the third is stiffened by stringer and the smallest is the un-stiffened shell The effect of pre-loaded axial compression P on critical pressure is illustrated in this table As can be seen that the value of qcr decreases in the increase of axial load P For example, as qcr = 512.3773 kPa (P = 0) bigger than qcr = 404.2515 kPa (P = 30 MN) about 21.1% (Table 4, orthogonal stiffener) 4.2.4 Effect of radius-to-thickness ratio R/h The effects of the radius-to-thickness ratios R/h on critical buckling load Pcr and qcr of stiffened FGM truncated conical shells, using Eq (29) are considered in Figs and It is observed that the both critical loads decrease markedly with the increase of R/h ratio This result agrees with the actual property of structure, i.e the shell is thinner the value of critical load is smaller 4.2.2 Effect of stiffener number The effect of stiffener number on critical axial compressive load Pcr and critical pressures qcr are presented in Tables and 6, respectively As expected, the obtained results in both tables show that 4.2.5 Effect of length-to-radius ratio L/R Fig illustrates the effects of L/R ratios on critical pressures load qcr when P = It can be observed, the critical pressures qcr decrease with the increase of length-to-radius ratio L/R 4.2.3 Effect of semi-vertex angle a Tables and illustrate the effect of semi-vertex angle a on critical axial compressive load Pcr and critical pressures qcr, respectively As can be seen that the critical buckling load of truncated conical shell strongly decreases when semi-vertex angle increases For example for an orthogonally stiffened shell in Table 8, when the semi-vertex angle varies the values from 50 to 750, the critical pressure qcr decreases from 974.4742 (kPa) to 53.6600 (kPa) This remark also has been pointed out in Ref [36] Graphically, the effects of semi-vertex angle a on critical axial compressive load Pcr and critical pressures qcr are plotted in Figs and They also show that critical axial compressive load Pcr and critical pressures qcr decrease when a increases and the critical load-angle a curve for an orthogonally stiffened shell is the highest Table Effect of stiffener arrangement on critical pressures qcr (kPa), (inside stiffener) qcr (kPa) P=0 Un-stiffened Stringer (nst = 30) Ring (nr = 30) Orthogonal (nst = nr = 15) 437.4323 437.7963 579.5405 512.3773 (1,17) (1,17) (1,16) (1,16) P = 10 MN P = 20 MN P = 30 MN 405.3130 405.6770 543.4985 476.3353 370.5404 370.8965 507.4566 440.2934 334.4985 334.8545 467.0337 404.2515 (1,17) (1,17) (1,16) (1,16) (1,16) (1,16) (1,16) (1,16) (1,16) (1,16) (1,15) (1,16) 110 D Van Dung et al / Composite Structures 106 (2013) 104–113 Table Effect of stiffener number on critical pressures qcr (kPa), (inside stiffener), P = 0, a = 300 Stiffener number (ns) Stringer (nst = ns) Ring (nr = ns) Orthogonal (nst = nr = ns/2) 10 20 30 40 50 437.5537 437.6750 437.7963 437.9175 438.0386 489.2583 534.8855 579.5405 623.2560 665.5502 463.8526 489.3786 512.3773 535.1296 557.6398 (1,17) (1,17) (1,17) (1,17) (1,17) (1,16) (1,16) (1,16) (1,16) (1,15) (1,17) (1,16) (1,16) (1,16) (1,16) Table Effect of angle a on critical axial compressive load Pcr (MN), L = 2.54 m, q = a Stringer (inside) (nst = 30) Ring (inside) (nr = 30) Orthogonal (inside) (nst = nr = 15) 50 100 200 300 450 600 750 126.8508 (2,15) 125.0687 (2,16) 116.1394 (2,17) 100.1151 (2,19) 67.4521 (2,19) 34.0728 (2,18) 9.4455 (2,14) 125.8018 (9,16) 123.0509 (9,16) 112.2802 (8,17) 95.5330 (7,18) 63.8662 (6,17) 32.0713 (5,14) 8.6198 (3,12) 135.5210 (3,17) 132.5515 (4,19) 121.0889 (4,21) 102.8853 (4,21) 68.7137 (3,20) 34.3557 (3,18) 9.3008 (2,14) Table Effect of angle a on critical pressures qcr (kPa), (inside stiffener), P = 0, L = 2.54 m a Stringer (inside) (nst = 30) Ring (inside) (nr = 30) Orthogonal (inside) (nst = nr = 15) 50 100 200 300 450 600 750 828.0539 (1,13) 735.2565 (1,14) 577.5780 (1,16) 437.7963 (1,17) 268.1861 (1,18) 139.3501 (1,17) 47.3542 (1,15) 1107.809 (1,12) 978.5357 (1,13) 765.0412 (1,15) 579.5405 (1,16) 352.3990 (1,17) 180.0673 (1,16) 58.8429 (1,14) 974.4742 (1,13) 867.2772 (1,13) 673.0672 (1,15) 512.3773 (1,16) 312.5216 (1,17) 160.2819 (1,17) 53.6600 (1,14) 1000 140 100 800 80 60 40 inside stiffeners h1=0.03175m, h2=0.01m b1=b2=0.0127m, n1=n2=15 20 20 30 40 60 70 80 2: Orthogonal 400 200 50 600 10 R=1.27m L=2.54m h=0.0127m 2: Orthogonal Buckling load q (kPa) Buckling load P (MN) R=1.27m L=2.54m h=0.0127m 120 1: Unstiffened 1: Unstiffened inside stiffeners h1=0.03175m, h2=0.01 m b1=b2=0.0127m, n1=n2=15 10 20 30 40 50 60 70 80 α (degree) α (degree) Fig Effects of a on critical load Pcr (q = 0) Fig Effects of a on critical load qcr (P = 0) 4.2.6 Effect of volume fraction index k The effects of index volume k on the critical buckling loads and post-buckling behavior are given in Figs and for stiffened FGM truncated conical shell It is found that the critical loads Pcr and qcr of ES-FGM shells decrease with the increase of k In addition the buckling strength of FGM truncated conical shell is more than fully metal shell and less than that of fully ceramic shell This property appropriate to the real characteristic of material, because the higher value of k corre- sponds to a metal-richer shell which usually has less stiffness than a ceramic-richer one Conclusions This paper presents an analytical solution to investigate the linear buckling of eccentrically stiffened FGM truncated conical subjected to axial compressive load and uniform pressure load The change of spacing between stringers in the meridional direction 111 D Van Dung et al / Composite Structures 106 (2013) 104–113 200 250 α =100 Unstiffened, α =10 Orthogonal, α =30 Unstiffened, α =30 1: Orthogonal, 2: 3: 150 4: 100 50 200 inside stiffeners h1=0.03175m, h2=0.01m b1=b2=0.0127m n1=n2=15 150 200 250 300 350 R=1.27m L=2.54m h=0.0127m 150 100 inside stiffeners h1=0.03175m, h2=0.01m b1=b2=0.0127m, n1=n2=15 10-2 400 2: 50 100 α =100 α =200 3: α =30 4: α =40 1: Buckling load P (MN) Buckling load P (MN) R=1.27m L=2.54m 10-1 100 101 102 k R/h Fig Effects of k on critical load Pcr (q = 0) Fig Effects of R/h on critical load P (q = 0) 1600 α =100 α =200 3: α =30 4: α =40 1: 1500 1400 1: Orthogonal, α =100 Buckling load q (kPa) 2: Unstiffened, α =100 R=1.27m L=2.54m 3: Orthogonal, α =300 4: Unstiffened, α =30 Buckling load q (kPa) 1000 500 inside stiffeners h1=0.03175m, h2=0.01m b1=b2=0.0127m n1=n2=15 1200 2: 1000 800 600 400 200 inside stiffeners h1=0.03175m, h2=0.01m b1=b2=0.0127m, n1=n2=15 -2 10 -1 10 10 10 10 10 k 100 150 200 250 300 350 400 Fig Effects of k on critical load qcr (P = 0) R/h Fig Effects of R/h on critical load q (P = 0) 5000 R=1.27m h=0.0127m 4000 Buckling load q (kPa) R=1.27m L=2.54m h=0.0127m 3: Orthogonal, α =40 2: Unstiffened, α =20 4: Unstiffened, α =40 inside stiffeners h1=0.03175m, h2=0.01m b1=b2=0.0127m, n1=n2=15 3000 2000 1000 0.5 1: Orthogonal, α =20 closed-form expression for determining the critical buckling load is obtained Numerical results showing the effect of stiffener, volume fraction index and geometrical parameters on the buckling response of FGM truncated conical shells are obtained The present results show some remarks as 1.5 L/R Critical load increases when the number of stiffeners increases and inversely Presence of stiffeners enhances the stability of truncated conical shells in which the critical buckling loads of truncated conical shells with the orthogonal stiffeners are greatest Loading carrying capacity of ES-FGM truncated conical shell is reduced considerably when R/h ratio or volume fraction index k increases Critical loads Pcr and qcr of ES-FGM truncated conical shells decrease when the semi-vertex angle increases Buckling strength of ES-FGM truncated conical shell is more than fully metal shell and less than that of fully ceramic shell Critical pressure qcr decreases in the increase of pre-loaded axial compression P Fig Effects of L/R on critical load q (P = 0) Acknowledgements also is taken into account Equilibrium and stability equations based on the smeared stiffeners technique and classical shell theory, are derived The couple set of three variable coefficient partial differential equations is investigated by Galerkin method and the This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No 107.01-2012.02 The authors are grateful for this financial support 112 D Van Dung et al / Composite Structures 106 (2013) 104–113 Appendix A L32 ẳ L12 # x0 ỵ Lị3 x30 L3 L2 ẳ A12 ỵ A66 ị ỵ À 2m2 p2 4L n Er A2 A22 ỵ ỵ A66 ; m d2 L13 ¼ " p2 mn L33 ¼ " h i nL2 A12 ỵ A66 ị x0 ỵ Lị3 x30 ỵ A12 ỵ A66 ị 8m 12L 2 nL Er A2 ỵ A66 ; A22 ỵ 8m d2 Er A2 p L2x0 ỵ Lị A66 sin aịL2x0 þ LÞ 16 sin a d2 h i 3p p3 m 4 À A66 ðsin aÞ ðx0 þ LÞ À x0 þ A66 ðsin aÞLð2x0 þ LÞ; 8L2 pn2 A22 ỵ " pn3 B22 þ C p3 m2 n ðx0 þ LÞ À x30 L3 L23 ẳ Lỵ B12 ỵ 2B66 ị 2 16 sin a 4m2 p2 2L pn Er A2 pnL ỵ A22 ỵ B22 ỵ C ị; cot aịL2x0 ỵ Lị ỵ L31 ¼ À d2 p3 m2 L2 " B11 sin a L 3L4 x x0 ỵ Lị3 ỵ 2mp 4m3 p3 # n2 B22 ỵ C 2 B22 ỵ C L sin aÞL2 m 16 sin a 4m Er A2 L2 2x0 ỵ Lị p4 m3 ỵ A22 ỵ cot aị sin a ỵ sin aịB11 4m d2 L " # x0 ỵ Lị x0 L 3L5 3 ỵ x x0 ỵ Lị ỵ 10 2m2 p2 4m4 p4 " # p4 m3 x0 ỵ Lị4 x40 3L3 2x0 ỵ Lị ỵ sin aịC 01 8m2 p2 L " # p2 mn2 B12 ỵ 2B66 ị x0 ỵ Lị3 x30 L3 : ỵ sin a 4L 4m2 p2 " # p2 m x0 ỵ Lị4 x40 3L3 2x0 ỵ Lị cot aị sin a A12 ỵ L 8m2 p2 " # p2 m x0 ỵ Lị3 x30 L3 þ ðsin aÞðB22 þ C Þ À ; L 4m2 p2 # L 3L4 pL Er I ỵ x0 x0 ỵ Lị3 ỵ sin a D22 ỵ 3 2mp 4m p d2 p5 m4 B22 ỵ C ịL2x0 þ LÞ cot a sin a À sin a:D11 L " # x0 ỵ Lị5 x50 L2 3L5 ỵ x x0 þ LÞ þ 10 2m2 p2 4m4 p4 " # p5 m4 Es I1 x0 ỵ Lị4 x40 3L3 2x0 ỵ Lị sin a 8m2 p2 k0 L pn4 L Er I D ỵ 22 d2 32 sin a " # p3 m2 n2 D12 ỵ 2D66 x0 ỵ LÞ3 À x30 L3 À À sin a 4m2 p2 2L2 " # p3 m2 Er I2 x0 ỵ Lị3 x30 L3 sin a D22 ỵ d2 4m2 p2 L pn2 L Er I2 E r A2 ỵ p A22 ỵ D12 ỵ 2D66 ỵ D22 ỵ sin a d2 d2 " # x0 ỵ Lị3 À x30 L3 cot2 a sin a  À 4m2 p2 ỵ p2 mn L22 ẳ pn2 L D12 ỵ 2D66 p4 m3 ỵ D11 sin a sin a L  p2 mn2 p2 m ðB12 þ 2B66 Þð2x0 þ LÞ þ sin aðB22 þ C ị2x0 ỵ Lị 16 sin a " # 3 pm x0 ỵ Lị x0 L3 þ sin a cot a:A12 þ L 4m2 p2 " # 4 pm x0 ỵ Lị x0 3L 2x0 ỵ Lị ỵ sin a B ỵ 11 4m2 p2 2L3 " # 3 p4 m3 L3 x0 ỵ Lị x0 þ ðsin aÞC þ 2m2 p2 2L3 p2 m L2 Er A2 A22 ỵ B11 sin a2x0 ỵ Lị ỵ cot a sin a; 4m d2 L21 ẳ B22 ỵ C ịL2x0 ỵ Lị ỵ p3 m2 n B12 ỵ 2B66 ị 2L#2 " x0 ỵ Lị4 x40 3L3 2x0 ỵ Lị pn3 B22 ỵ C ỵ L2x0 ỵ Lị 2 8m p 32 sin2 a " # pn Er A2 ðx0 þ LÞ3 À x30 L3 A22 þ À cot a; þ d2 4m2 p2 " L11 # p3 m2 x0 ỵ Lị x40 3L3 2x0 ỵ Lị p m Es A ẳ sin a A ỵ 11 2 4m p 2L 2L2 k0 " # x0 ỵ Lị x30 L3 pn2 sin a ỵ A66 L2x0 þ LÞ À 2 2m p 16 sin a p Er A2 p A22 ỵ sin aL2x0 ỵ Lị ỵ A11 sin aịL2x0 ỵ Lị; d2 pn À # À p p3 m2 B12 4L pn2 sin a h i 3p L cos aị x0 ỵ Lị4 x40 ỵ B12 cos aị2x0 ỵ Lị B22 ỵ C ịcot aịL2x0 ỵ Lị; " # pn2 tan a x0 ỵ Lị4 x40 3L3 2x0 ỵ Lị p2 m L34 ¼ À À sin a 8m2 p2 2L " # 3L4 2x ỵ Lị L p3 m2 x x0 ỵ Lị ỵ tan a sin a ỵ 2mp 2m3 p3 2L2  tan a sin a (" ) # x0 ỵ Lị6 x60 5L2 15L5 2x0 þ LÞ Â þ x À ðx þ LÞ ; ỵ 12 8m2 p2 8m4 p4 ỵ L35 ẳ " # x0 ỵ Lị4 x40 3L3 2x0 ỵ Lị : 8m2 p2 2L2 cos a p2 m2 References [1] Seide P Axisymmetrical buckling of circular cones under axial compression J Appl Mech 1956;23:625–8 [2] Seide P Buckling of circular cones under axial compression J Appl Mech 1961;28:315–3126 [3] Singer J Buckling of circular conical shells under axisymmetrical external pressure J Mech Eng Sci 1961;3:330–9 [4] Lu SY, Chang LK Thermal buckling of conical shells AIAA J 1967;5:1877–82 [5] Lu SY, Chang LK Nonlinear 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