DSpace at VNU: Nonlinear buckling and post-buckling analysis of eccentrically stiffened functionally graded circular cylindrical shells under external pressure

Thin-Walled Structures 63 (2013) 117–124 Contents lists available at SciVerse ScienceDirect Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws Nonlinear buckling and post-buckling analysis of eccentrically stiffened functionally graded circular cylindrical shells under external pressure Dao Van Dung, Le Kha Hoa n Vietnam National University, Hanoi, Viet Nam a r t i c l e i n f o abstract Article history: Received May 2012 Received in revised form August 2012 Accepted 11 September 2012 Available online 27 November 2012 The nonlinear buckling and post-buckling behavior of functionally graded stiffened thin circular cylindrical shells subjected to external pressure are investigated by the analytical approach in this paper The shells are reinforced by eccentrically rings and stringers attached to the inside and material properties of shell and stiffeners are assumed to be continuously graded in the thickness direction Fundamental relations, equilibrium equations are derived based on the smeared stiffeners technique and the classical shell theory with the geometrical nonlinearity in von Karman sense Approximate three-terms solution of deflection is more correctly chosen and explicit expression to finding critical load and post-buckling pressure-deflection curves are given by using the Galerkin’s method The numerical results show the effectiveness of stiffeners in enhancing the stability of shells & 2012 Elsevier Ltd All rights reserved Keywords: Functionally graded material Stiffened cylindrical shells Post-buckling Nonlinear Introduction Stiffened cylindrical shells with the functionally graded material (FGM) properties more and more are widely used in modern engineering In most of these applications, shell is subjected to compressive loads and it may be buckled Therefore, the research on nonlinear stability of these structures has received considerable attentions by scientists Van der Neut [1] pointed out the importance of the eccentricity of stiffeners in the buckling of isotropic cylindrical shells under axial compressive load Baruch and Singer [2] showed the effect of eccentricity of stiffeners on the general instability of stiffened cylindrical shells under hydrostatic pressure They concluded that the behavior of eccentricity effect dependents very strongly on the geometry of the shell Shen et al [3] investigated the buckling and post-buckling behavior of perfect and imperfect stiffened cylindrical shells under combined external pressure and axial compression by using the boundary layer theory The singular perturbation technique to determine the buckling loads and the post-buckling equilibrium paths is applied in their work Bushnell [4] considered the nonlinear equilibrium of perfect locally deformed stringer-stiffened panels under combined in-plane loads Reddy and Starnes [5] studied the buckling of circumferentially or axially stiffened laminated cylindrical shells subjected to simply supported end condition by using the layerwise theory and the smeared stiffener approach Based on the Donnell equations and the perturbation technique, n Corresponding author E-mail address: lekhahoa@gmail.com (L.K Hoa) 0263-8231/$ - see front matter & 2012 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.tws.2012.09.010 the general solution for nonlinear buckling of non-homogeneous axial symmetric ring- and stringer-stiffened cylindrical shells is given by Ji and Yeh [6] The post-buckling analysis of stiffened braided thin shells subjected to combined loading of external pressure and axial compression by perturbation method is reported by Zeng and Wu [7] The post-buckling analysis of pressure-loaded functionally graded cylindrical shells without stiffeners based on the classical shell theory with von Karman– Donnell-type of kinetic nonlinearity is presented by Shen [8] Jiang et al [9] studied the buckling of stiffened circular cylindrical panels subjected to axial uniform compressive load by using the differential quadrature element method The cylindrical panel and the stiffeners are treated separately there Li and Shen [10] presented the investigation on a post-buckling analysis of 3D braided composite cylindrical shells under combined external pressure and axial compression in thermal environment They used the higher order shear deformation shell theory and the singular perturbation technique to determine interactive buckling loads and post-buckling equilibrium paths Huang and Han [11] presented the research on nonlinear post-buckling of un-stiffened FGM cylindrical shells under uniform radial pressure by using the nonlinear large deflection theory of cylindrical shell In that work, the nonlinear buckling shape observed in experiment is taken into account Sadeghifar et al [12] investigated the buckling of stringer-stiffened laminated cylindrical shells with non-uniform eccentricity based on the Love’s first-order shear deformation theory The critical loads are calculated using the Rayleigh–Ritz procedure Stamatelos et al [13] presented the results on the local buckling and post-buckling behavior of isotropic and orthotropic 118 D.V Dung, L.K Hoa / Thin-Walled Structures 63 (2013) 117–124 stiffened panels based on the classical lamination plate theory and two-dimensional Ritz displacement function for arbitrary edge supports Recently, Najafizadeh et al [14] with the stability equations given in terms of displacement investigated the mechanical buckling behavior of functionally graded stiffened cylindrical shells reinforced by rings and stringer subjected to axial compressive loading 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 [15] investigated the nonlinear static post-buckling of eccentrically stiffened functionally graded plates and shallow shells with homogeneous stiffeners The review of the literature signifies that there is no work on the analytical solution for externally pressurized stiffened FGM cylindrical shells In this paper, the nonlinear buckling and postbuckling behaviors of eccentrically stiffened functionally graded thin circular cylindrical shells under uniform external pressure are investigated by approximate three-terms solution of deflection The material properties of shell and stiffeners are assumed to be continuously graded in the thickness direction The expression À Á of deflection including the linear buckling shape sin mpx=L À Á Á 2À sin ny=R and the nonlinear buckling shape sin mpx=L are more correctly chosen The resulting equations are solved by the Galerkin’s method to obtain closed-form expressions to determine critical buckling loads and nonlinear post-buckling loads–deflection curves The influences of various parameters such as dimensional parameters, buckling modes, volume fraction index of materials and number of stiffeners on the stability of shell are considered in detail Eccentrically stiffened functionally graded cylindrical shells Consider a thin circular cylindrical shell with mean radius R, thickness h and length L subjected to uniform radial load of intensity q Assume that two butt-ends of shell are only deformed in their planes and they still are circular [11] The middle surface of the shells is referred to the coordinates x, y, z as shown in Fig 1a Further, assume that the shell is stiffened by closely spaced circular rings and longitudinal stringers attached to inside of the shell skin, and the stiffeners and skin are made of functionally graded materials varying continuously through the thickness direction of the shell with the power law as follows [14]  k 2z ỵ h h h Esh ẳ Em ỵ Ecm 1ị , Ecm ẳ Ec ÀEm , k Z0, À r z r : 2h 2 Es ẳ Ec ỵ Emc  k 2zh , 2hs Emc ¼ Em ÀEc , k2 Z 0, 2ị Er ẳ Ec ỵ Emc  k 2zh h h r z r ỵhr , , k3 Z0, 2hr 2 3ị nsh ẳ ns ẳ nr ¼ n ¼ const, where k, k2 and k3 are volume fractions indexes of shell, stringer and ring, respectively and subscripts c, m, sh, s and r denote ceramic, metal, shell, longitudinal stringers and circular ring, respectively It is evident that, from Eqs (1)–(3), a continuity between the shell and stiffeners is satisfied Note that the thickness of the stringer and the ring are respectively denoted by hs and hr , andEc , Em are Young’s modulus of the ceramic and metal, respectively The coefficient n is Poison’s ratio According to the non-linear strain–displacement relations of cylindrical shells, the mid-surface strain components are [19] e0x ẳ u;x ỵ w,x ị2 , e0y ẳ v;y w ỵ w,y ị2 , R g0xy ẳ u;y ỵv;x ỵ w;x w;y , 4ị in which u ẳ ux,yị, v ¼ vðx,yÞ, and w ¼ wðx,yÞ are the displacements of the middle surface points along x, y and z axes, respectively The strain components across the shell thickness at a distance z from the mid-plane are in the form ex ẳ e0x ỵzkx , ey ẳ e0y ỵzky , gxy ẳ g0xy ỵ2zkxy , kx ẳ w;xx , ky ẳ w;yy , kxy ẳ w;xy , 5ị where kx , ky and kxy are the change of curvatures and twist of shell, respectively Using Eq (5), the compatible equation is written as R e0x,yy ỵ e0y,xx g0xy,xy ẳ w;xx ỵw2;xy w;xx w;yy hs L ds es bs h x y h er z R h h r z r ỵ hs 2 br hr dr Fig Geometry and coordinate system of a stiffened FGM circular cylindrical shell ð6Þ D.V Dung, L.K Hoa / Thin-Walled Structures 63 (2013) 117–124 The Hooke’s stress–strain relations are applied for shell Esh ex ỵ ney , 1n2 Esh ssh ey ỵ nex , y ẳ 1n2 Esh ssh g , xy ẳ 21 ỵ nị xy 7aị and for stiffeners ssx ¼ Es ex , sry ¼ Er ey : Z h=2 ỵ hr zEr zịdz ẳ ð10Þ r ð7bÞ Taking into account the contribution of stiffeners by the smeared stiffeners technique and omitting the twist of stiffeners and integrating the stress–strain equations and their moments through the thickness of the shell, the expressions for force and moment resultants of an eccentrically stiffened FGM cylindrical shell are expressed by [14,19] Nx ẳ C 11 e0x ỵ C 12 e0y ỵC 14 kx ỵ C 15 ky , Ny ẳ C 12 e0x ỵC 22 e0y ỵC 24 kx ỵ C 25 ky , Nxy ẳ C 33 g0xy ỵ C 36 kxy ,   Ec hr hhr ỵ1 h h=2   hr þ , þEmc hr h k3 þ2 h 2k3 þ2 ! Z h=2 ỵ hr Ec 3 h 3h hr E3r ẳ z2 Er zịdz ẳ ỵ ỵ h2 hr h=2 r " # 1 h h , ỵEmc hr þ þ k3 þ3 k3 þ2 hr 4ðk3 þ 1Þ h2 E2r ẳ ssh x ẳ 119 8aị where bs and br denote widths of stiffeners, respectively Also, ds and dr are the distances between two stringers and rings, respectively, and the eccentricities es and er represent the distance from the shell middle surface to the centroid of the stiffeners cross section (Fig 1b) For using later, the reverse relations are deduced from Eq (8a) as e0x ¼ C n22 N x C n12 Ny ỵC n14 kx ỵ C n15 ky , e0y ẳ C n12 N x ỵ C n11 Ny ỵ C n24 kx ỵ C n25 ky , g0xy ¼ C n33 Nxy ÀC n36 kxy 11ị where M x ẳ C 14 e0x ỵC 24 e0y ỵ C 44 kx ỵ C 45 ky , D ¼ C 22 C 11 ÀC 212 , C n22 ¼ C 22 =D, C n12 ¼ C 12 =D, M y ẳ C 15 e0x ỵ C 25 e0y ỵ C 45 kx ỵC 55 ky , M xy ẳ C 63 g0xy ỵC 66 kxy , C n14 ¼ ðC 12 C 24 ÀC 22 C 14 Þ=D, C n15 ¼ ðC 12 C 25 ÀC 22 C 15 Þ=D, ð8bÞ where the stiffness parameter C ij is given by C 11 ¼ C 14 ¼ C 22 ¼ C 25 ¼ C 36 ¼ C 45 ẳ C 63 ẳ E1 E1s bs nE1 ỵ , C 12 ¼ , ds 1Àn2 1Àn2 E2 E2s bs nE2 ỵ , C 15 ẳ , ds 1n2 1n2 E1 E1r br nE2 ỵ , C 24 ẳ , dr 1n2 1n2 E2 E2r br E1 , ỵ , C 33 ẳ dr 21 ỵ nị 1n2 E2 E3 E3s bs , C 44 ẳ ỵ , 1ỵn ds 1n2 nE3 E3 E3r br , C 55 ẳ ỵ , dr 1Àn2 1Àn2 E2 E3 , C 66 ¼ , 21 ỵ nị 1ỵn in which Z h=2 Ecm h , Esh zịdz ẳ Em hỵ E1 ẳ kỵ1 Àh=2 Z h=2 kEcm h E2 ¼ , zEsh zịdz ẳ k ỵ1 ị kỵ 2ị Àh=2 ! Z h=2 Em h 1 ỵ , E3 ẳ ỵ Ecm h z2 Esh zịdz ẳ 4k ỵ 1ị k ỵ k þ3 12 Àh=2 Z h=2 þ hs hs , E1s ẳ Es zịdz ẳ Ec hs ỵ Emc k2 ỵ h=2   Z h=2 ỵ hs Ec hs hhs ỵ1 E2s ẳ zEs zịdz ẳ h h=2   hs ỵ , ỵ Emc hs h k2 ỵ2 h 2k2 ỵ2 ! Z h=2 ỵ hs Ec 3 h 3h hs z2 Es zịdz ẳ ỵ ỵ E3s ẳ h2 hs h=2 s " # 1 h h ỵ ỵ Emc hs ỵ , k2 þ k2 þ hs 4ðk2 þ 1Þ h2 s Z h=2 ỵ hr hr , Er zịdz ẳ Ec hr ỵ Emc E1r ẳ k3 ỵ h=2 C n11 ¼ C 11 =D, C n24 ¼ ðC 12 C 14 ÀC 11 C 24 Þ=D, C 36 , C n36 ¼ : C n25 ¼ ðC 12 C 15 ÀC 11 C 25 Þ=D, C n33 ¼ C 33 C 33 ð12Þ Substituting Eq (11) into Eq (8b) yields Mx ẳ Dn14 Nx ỵDn24 N y þ Dn44 kx þ Dn45 ky , My ¼ Dn15 Nx ỵDn25 N y ỵ Dn54 kx ỵ Dn55 ky , Mxy ẳ Dn63 Nxy ỵ Dn66 kxy , 13ị where Dn14 ¼ C 14 C n22 ÀC 24 C n12 , n Dn44 ẳ C 44 ỵ C 24 C n24 ỵ C 14 C n14 , D24 ẳ C 24 C 11 ÀC 14 C 12 , Dn45 ẳ C 14 C n15 ỵ C 24 C n25 þ C 45 , n n Dn15 ¼ C 15 C n22 ÀC 25 C n12 , n n n D25 ¼ C 25 C 11 ÀC 15 C 12 , n Dn55 ẳ C 15 C n15 ỵ C 25 C n25 ỵ C 55 , D63 ẳ C 63 C 33 , D66 ¼ C 66 ÀC 63 C n36 : n 9ị Dn54 ẳ C 15 C n14 ỵ C 25 C n24 ỵ C 45 , n ð14Þ The equilibrium equations of cylindrical shell based on the classical shell theory are given by [14,17] Nx,x ỵN xy,y ẳ 0, Nxy,x ỵ Ny,y ẳ 0, Mx,xx ỵ 2M xy,xy ỵM y,yy ỵ Ny ỵN x w;xx ỵ 2Nxy w;xy ỵ N y w;yy ỵ q ẳ R ð15Þ The first two of Eq (15) are identically satised by introducing a stress function jx,yị as Nx ẳ j;yy , Ny ¼ j;xx , Nxy ¼ Àj;xy : ð16Þ Introduction of Eqs (13) and (16) into the third of Eq (15), and taking into account Eq (5), gives the following equation a11 w;xxxx ỵ a12 w;xxyy ỵ a13 w;yyyy ỵ a14 j;xxxx ỵ a15 j;xxyy ỵ a16 j;yyyy þ j þ j;yy w;xx þ j;xx w;yy À2j;xy w;xy ỵ q ẳ R ;xx 17ị where a11 ẳ Dn44 , a12 ẳ Dn45 ỵ 2Dn66 ỵ Dn54 , a13 ẳ Dn55 , a14 ẳ Dn24 , a15 ẳ Dn14 2Dn63 ỵ Dn25 , a16 ẳ Dn15 : 18ị Eq (17) includes two dependent unknown functions wandj and to find a second equation relating to these two functions 120 D.V Dung, L.K Hoa / Thin-Walled Structures 63 (2013) 117–124 the geometrical compatibility Eq (6) is used For this aim, substituting Eq (11) into Eq (6), gives b11 j;xxxx ỵ b12 j;xxyy ỵ b13 j;yyyy ỵ b14 w;xxxx ỵ b15 w;xxyy ỵ b16 w;yyyy w2;xy ỵ w;xx w;yy ỵ w;xx ẳ R 19ị 28ị 29ị where 1h a11 a4 ỵ a12 a2 b2 ỵ a13 b4  ! 2 a , ỵ a5 a14 a4 ỵ a15 a2 b ỵ a16 b R D01 ẳ b11 ¼ C n11 , b12 ¼ C n33 À2C n12 , b13 ¼ C n22 , À Á b14 ¼ C n24 , b15 ẳ C n14 ỵC n25 ỵ C n36 , b16 ẳ C n15 : 20ị Eqs (17) and (19) are nonlinear governing equations used to investigate the nonlinear stability of eccentrically stiffened FGM cylindrical shells under uniform radial loads Buckling analysis Assume that the cylindrical shell is simply supported at the edges x ¼ and x ¼ L The deflection of radial loaded shell can be expressed by [11,16] w ẳ wx,yị ẳ f ỵ f sinaxsinby ỵf sin ax, 21ị in which a ẳ mp=L, b ẳ n=R and m, n are the half waves numbers along x-axis and waves numbers along y-axis, respectively The first term of w in Eq (21) represents the uniform deflection of points belonging to two butt-ends x¼0 and x ¼L, the second term-a linear buckling shape, and the third-a nonlinear buckling shape As can be seen that the simply supported boundary condition at x¼ and x¼L is fulfilled on the average sense Substituting Eq (21) into Eq (19) yields b11 j;xxxx þ b12 j;xxyy þ b13 j;yyyy ¼ B01 cos 2ax þ B02 cos 2by þ B03 sin axsin by þB04 sin 3axsin by ð22Þ where   ! 2 f ỵ a2 b f , 2 2 B02 ¼ a2 b f , 8b14 a4 À a2 R   ! 2 a f ỵ f f a2 b , B03 ¼ À b14 a4 þ b15 a2 b þ b16 b À R B04 ẳ f f a2 b : 23ị The general solution of this equation is given by j ẳ B1 cos 2ax ỵB2 cos 2by ỵ B3 sin axsin by ỵ B4 sin 3axsin by s0y hx2 , ð24Þ where s0y is the negative average circumferential stress and B1 ẳ a1 f ỵ a2 f , B2 ¼ a3 f , B3 ẳ a4 f f ỵ a5 f , B4 ¼ a6 f f , ð25Þ in which À Á À Á Á À a1 ¼ 4b14 a4 Àa2 =R = 8b11 a4 , a2 ¼ a2 b = 32b11 a4 ,     2 a3 ¼ a2 b = 32b13 b , a4 ¼ Àa2 b = b11 a4 ỵ b12 a2 b ỵ b13 b ,   a5 ẳ b14 a4 ỵ b15 a2 b ỵ b16 b a2 =R   = b11 a4 ỵ b12 a2 b ỵ b13 b ,   2 ð26Þ a6 ẳ a2 b = 81b11 a4 ỵ 9b12 a2 b þ b13 b In order to establish a load–deflection curve, first of all, introducing w and j into the left side of Eq (17), then applying Galerkin’s method in the ranges r y r2pR and r x r L, gives s0y ẳ D06 f ỵ D07 f ỵD08 f f ẳ 0, where B01 ¼   2 f ẳ D01 ỵ D04 f ỵD05 f À s0y hb =D03 , Rq , h ð27Þ a2 b2 D03 ẳ a2 b a2 ỵ a3 Þ, D04 ¼ À ða4 Àa6 Þ,   ! 2 a 2 ỵ a2 b a5 À2a2 b a1 , D05 ¼ À a4 a14 a4 ỵ a15 a2 b ỵ a16 b R   2a , D06 ¼ 8a2 4a11 a2 8a2 a14 a1 ỵ R   4a2 2 D07 ¼ a5 b À16a2 a14 a2 ỵ a , R D08 ẳ 4a4 a6 ịa2 b : ð30Þ In addition to three Eqs (27)–(29), the cylindrical shell must also satisfy the circumferential closed condition [11,16] as  Z 2pR Z L Z 2pR Z L  w v;y dxdy ẳ e0y ỵ w,y Þ2 dxdy ¼ R 0 0 Using Eqs (11), (16) and (24), this integral becomes À8C n11 s0y hỵ 2 2f ỵ f b f ẳ R 31ị Substituting Eq (28) into Eq (29) with s0y ¼ Rq=h, leads to " # D03 D06 f 2 Á D01 ỵ D04 f ỵ D05 f qẳ 32ị D07 ỵD08 f Rb Expression (32) is used to determine the critical loads and to analyze the post-buckling load–deflection curves of nonlinear buckling shape of stiffened FGM cylindrical shells If f ¼ 0, i.e the nonlinear buckling shape is ignored, Eq (32) becomes 2D01 h ẳ a11 a4 ỵ a12 a2 b2 ỵ a13 b4 q¼ 2 Rb Rb   b14 a4 ỵ b15 a2 b2 ỵ b16 b4 a2 =R   ỵ b11 a4 ỵ b12 a2 b2 ỵ b13 b4  i n a14 a4 ỵ a15 a2 b ỵ a16 b a2 =R 33ị Eq (33) is used to find critical loads in case linear buckling shape From Eq (21), it is obvious that the maximal deflection of the shells W max ¼ f ỵf ỵf 34ị locates at x ẳ iL=2mị, y ẳ jpR=2nị, where i, j are odd integer numbers Solving f and f from Eqs (28) and (31) with respect to f , then substituting them into Eq (34), we obtain   f Rb 2 W max ẳ ỵ C n11 Rs0y h D01 ỵ D04 f ỵ D05 f À s0y hb 2 8D03 !1=2 2 b s0y hÀ2D01 À2D04 f À2D05 f ð35Þ þ 2D03 Combining Eq (32) with Eq (35), the effects of inhomogeneous and dimensional parameters on the post-buckling loadmaximal deflection curves of shells can be analyzed D.V Dung, L.K Hoa / Thin-Walled Structures 63 (2013) 117–124 Numerical results 4.1 Validation of the present approach To verify the present study, an isotropic cylindrical shell under external pressure q is considered with the following geometric and material properties as [2,5,18] E ¼ 30 Â 106 Psi, n ¼ 0:3, h ¼ in, R ¼ 82:1693 in, L ¼ 372:9745 in, À Á es ¼ er ¼ 1:653 in, hs ¼ hr ¼ er Àh=2 , 121 It is clear that the present results coincide with the ones of the work [11] In each subsection below, to illustrate the present approach for nonlinear buckling and post-buckling analysis of stiffened FGM cylindrical shells under external pressure, consider a ceramicmetal shell consisting of zirconia and aluminum with the following properties [14] Ec ¼ 151 GPa, Em ¼ 70 GPa and Poisson’s ratio n is assumed to be 0.3 Also assume that k2 ¼ k3 ¼ 1=k for all of the examples considered hereafter 4.2 Effects of buckling modes ds ¼ 2pR=ns ¼ 2pR=516, dr ¼ L=nr ¼ L=373, bs ¼ 0:1471ds h=hs , br ¼ 0:1471dr h=hr , where ns , nr are the number of stringer and ring of shell, respectively The critical buckling loads calculated for stiffened and unstiffened shells are listed in Table to compare with the results given by Shen [18] using asymptotic perturbation technique and with the results reported by Reddy [5] based on Donnell’s shell theory As can be seen, good agreement is obtained in this comparison Fig 2a and b shows the comparisons of the present postbuckling paths with the results which was also analyzed by Huang and Han [11] using the nonlinear large deflection theory and the Ritz energy method for un-stiffened FGM cylindrical shells under external pressure Assume that the material and geometrical parameters of shell are taken by k¼k2 ¼k3 ¼1, h ¼ 0:305 Â 10À3 m, R ¼ 60:643 Â 10À3 m, L ¼ 387:35 Â 10À3 m, hs ¼ 0:076 Â 10À3 m, bs ¼ 21:155 Â 10À3 m, À3 hr ¼ 0:127 Â 10 m, br ¼ 1:27 Â 10À3 m The number of stringers as well as rings are equal to 15 Using Eq (32) and programming the Matlab software, the results are given in Table and Fig Table shows that the buckling load is minimum qcr ¼ 25:1349 KPa corresponding to the buckling mode (m, n)¼(1, 4) Also, the same result qcr is represented on Fig As can be seen the lowest point of the envelope curves is regarded as the nonlinear buckling mode (m, n)¼(1, 4) Further, during the first stage, the buckling load reduces when the ratio f2/h increases At the second stage, when the ratio f2/h reaches a determined value, the buckling loads increases with the increase of this ratio 4.3 Effects of inhomogeneous and geometric parameters Table Comparisons of buckling load q for isotropic cylindrical shell under external pressure Shell Baruch and Singer [2] Un-stiffened Stringer stiffened (inside) Ring stiffened (inside) Orthogonal stiffened (inside) 102 103 93.5 94.7 370 357.5 368.3 (1,3) 379.6944 (1,3) 377 365 374.1 (1,3) 387.1920 (1,3) Shen [18] Present calculate by Eq (33) 100.7 (1,4)a 103.3271 (1,4) 102.2 (1,4) 104.4937 (1,4) The numbers in the parenthesis denote the buckling modes (m, n) 300 70 present Ref [11] 250 present Ref [11] 60 50 q (KPa) 200 q (KPa) a Reddy and Starnes [5] Based on Eqs (32) and (35), with the database given in the Subsection 4.2, the effects of the volume fraction indexes k, k2 , k3 , of the radius-to-thickness ratios R=h and of the À Á length-to-radius ratios L=R on qÀW max =h relation curves of stiffened FG cylindrical shell are considered Fig shows the post-buckling equilibrium paths under various values of k ¼ , 0:2, 0:6, , 5, 10 ỵ1, and R ẳ L=2, h ẳ R=500 As expected, the critical load qcr decreases with the increase of k This property corresponds to the real property of material, because the higher value of k corresponds to a metalricher shell which usually has less stiffness than a ceramicricher one À Á Fig plots the qÀW max =h post-buckling curves versus R=h which is chosen to be 200, 300, 400, 500 It is observed that the 150 100 50 30 20 Ec=168.0421 GPa, Em=105.6835 GPa, ν=0.3 k=1, L/R=2, R/h=200, m=1, n=7 40 k=1, R/h=500, L/R=1, m=1, n=13 10 Ec=168.0421 GPa, Em=105.6835 GPa, ν=0.3 0 Wmax/h 10 11 Fig Comparison with results of [11] for un-stiffened FGM shells Wmax/h 10 122 D.V Dung, L.K Hoa / Thin-Walled Structures 63 (2013) 117–124 Table Effects of buckling modes on critical load qcr (KPa) m n ¼1 10 1.9876e ỵ3 4.1555e þ4 1.5733e þ5 3.0620e þ5 109.9864 506.0391 3.0260e þ 1.0439e ỵ 27.2791 100.1864 241.8345 797.6924 25.1349 49.9465 88.3140 161.3442 34.9210 45.0970 64.6095 90.0666 49.0235 54.2510 65.5668 81.1217 66.1908 69.5581 76.9213 87.9760 86.1463 88.7385 94.1461 102.6104 108.8142 111.0518 115.4555 122.3772 134.1695 136.2311 140.0958 146.0874 120 110 100 n=8 100 L/R=2, m=1, k=k2=k3=1, hs=0.076mm, bs=21.155mm, hr=0.127mm, br= 1.27mm 80 90 n=7 q (KPa) q (kPa) 80 70 n=6 60 n=5 50 40 n=4 40 30 1: R/h=200, n=7 2: R/h=300, n=8 3: R/h=400, n=9 4: R/h=500, n=9 60 n=3 20 qcr=25.1349 KPa 20 10 f2/h 12 14 16 18 20 Fig Effects of buckling modes n (m¼ 1) on (qÀf =h) curves Wmax/h 10 12 Fig Effects of R/h ratio on post-buckling curves of shell 25 1: L/R=1, n=13 2: L/R=2, n=9 3: L/R=3, n=7 4: L/R=4, n=7 20 1: k =0, 2: k =0.2, 3: k = 0.6, 4: k = 1, 5: k =5, 6: k =10, 7: k =∞ 18 16 20 12 10 q (KPa) q (KPa) 1 14 k=k2=k3=1, m=1, R/h=500, hs=0.076mm, bs=21.155mm, hr=0.127mm, br= 1.27mm 15 10 4 L/R=2, R/h=500 (m, n) = (1, 9) 0 0 10 12 Wmax/h 14 16 18 20 10 15 Wmax/h Fig Effects of L/R ratio on post-buckling curves of shell Fig Effects of k on (qÀW max =h) curves buckling load q decreases markedly with the increase of R=h ratio, i.e the more the shell is thin the more the value of critical load is small Effects of ratio L=R on the post-buckling curves of shell are represented on Fig As can be observed, the capacity of mechanical load q bearing of the FGM shells is considerably reduced with the increase of L=R Besides, Figs and also show the wave numbers n increases with the increase of R=h, but decrease with the increase of L=R Thus, both R=handL=R considerably affect the nonlinear buckling mode of shells 4.4 Comparing the critical buckling loads of stiffened and un-stiffened FGM cylindrical shells The effects of stiffeners on the critical buckling loads and postbuckling behavior of FGM cylindrical shells are considered in this subsection Using the database in Subsection 4.2, Table and Fig compare the critical buckling loads qcr of stiffened FGM shell with the ones of un-stiffened FGM shell when m¼1, and k, k2 ¼ k3 ¼ 1=kandn vary As can be seen, the critical buckling loads for the FGM stiffened cylindrical shells are generally upper than the corresponding values for the FGM un-stiffened D.V Dung, L.K Hoa / Thin-Walled Structures 63 (2013) 117–124 50 Table Comparing the critical buckling loads of stiffened and un-stiffened FGM shells vs k Shell k¼ 0, n¼ k¼0.5, n¼4 k¼ 1, n¼ k¼1, n¼ Un-stiffened Stringer stiffened Ring stiffened Orthogonal stiffened 32.7108 33.2200 35.8205 36.3245 25.7267 26.1480 27.8415 28.2585 22.9867 23.3630 24.7623 25.1349 15.1639 15.4000 16.6055 16.8392 123 1: ns=nr=10 2: ns=nr=20 3: ns=nr=30 4: ns=nr=40 5: ns=nr=50 Orthogonal Stiffened shell hs=0.076mm, bs=21.155mm hr=0.127mm, br=1.27mm 45 40 q (KPa) 35 70 30 60 L=387.35mm R=60.643mm h=0.305mm 25 q (KPa) 50 k=k2=k3=1, m=1, n=4 20 40 30 10 15 20 Wmax/h 25 30 35 Fig Effects of number of stiffeners on (qÀW max =h) curves 20 Table Effects of number of stiffeners on buckling loads q (KPa), (m, n) ¼ (1, 4) 10 0 10 15 20 25 Wmax/h 30 35 40 Number of stiffeners 10 20 30 40 50 Stringer stiffened shell Ring stiffened shell Orthogonal stiffened shell 23.2418 23.4803 23.7038 23.9134 24.1104 24.1355 25.3813 26.5962 27.6673(n) 28.3651(n) 24.3889 25.8684 27.2992 28.6843 30.0264 Fig Effects of k on (qÀW max =h) curves where (n) indicates m ¼1, n¼3 Table Comparing the critical buckling loads of stiffened and un-stiffened FGM shells vs h h (n) 0.305 (4) 0.533 (3) 0.610e (3) 0.686 (3) 0.762 (3) Un-stiffened shell Stringer stiffened shell Ring stiffened shell Orthogonal stiffened shell 22.9867 23.3630 24.7623 25.1349 85.2212 86.6765 87.5988 89.0446 119.6377 121.0893 122.5735 124.0163 162.3639 163.8110 165.9090 167.3479 200 1: h=0.305mm, n=4 Stiffened 2: h=0.381mm, n=3 Unstiffened 3: h=0.457mm, n=3 180 4.5 Effects of number of stiffeners To investigate the effects of number of stiffeners, the database given in the Subsection 4.2 is used here Fig and Table illustrate the effects of number of stiffener (ns ¼nr ¼10, 20, 30, 40 and 50) on buckling loads and qÀW max =h curves As can be seen, these curves become lower when the number of stiffeners decreases and buckling loads increases when the number of stiffeners increases That means the percentage increase in the buckling load rises continuously with the increment of the number of stiffeners This increase is about 19% for orthogonal stiffened shell, in comparison ns ¼nr ¼10 with ns ¼nr ¼50 (in Table 5) 4: h=0.533mm, n=3 160 L=387.35mm; R=60.643mm; k=k2=k3=1 140 q (KPa) 214.8787 216.3212 219.0910 220.5256 120 100 80 60 40 Stiffened: hs=0.076mm; bs=21.155mm hr=0.127mm; br=1.27mm 20 0 10 20 30 cylindrical shells The prime reason is that the presence of stiffeners makes the shells to become stiffer The same results are presented in Table and Fig for FGM un-stiffened and stiffened shells Herein, the thickness of shell varies from 0.305 mm to 0.762, the wave numbers n are chosen to be 2, 3, 4, and m¼1, k ¼ k2 ¼ k3 ¼ Also as can be seen that the critical buckling loads are increased by increasing the thickness of shell Tables and show that the critical loads of FGM un-stiffened shells are the smallest Critical loads of stringer stiffened shell are smaller than ring stiffened shell Finally, the critical loads of FGM orthogonal stiffened shell are the greatest 40 Wmax/h Fig Effects of thickness h on (qÀW max =h) curves 50 Concluding remarks In this paper, the thin FGM cylindrical shells reinforced by eccentrically rings and stringers attached to the inside and material properties of shell and stiffeners varying continuously graded in the thickness direction are considered 124 D.V Dung, L.K Hoa / Thin-Walled Structures 63 (2013) 117–124 Analytical approach to investigate the nonlinear buckling and post-buckling behavior of eccentrically stiffened FGM cylindrical shells under external pressure based on the smeared stiffeners technique and the classical shell theory with geometrical nonlinearity in von Karman sense is studied Approximate three-term solution of deflection including the linear and nonlinear buckling shape is more correctly chosen and the close-form expressions to determine critical buckling loads and nonlinear post-buckling load–deflection curves are obtained by using Galerkin’ method Effects of various parameters such as dimensional parameters, buckling modes, volume fraction index of materials and number of stiffeners on the stability of shell are considered in detail Results show the effectiveness of stiffeners in enhancing the stability of shells Major purpose of this study is to analyze the global buckling and post-buckling behavior of FGM stiffened cylindrical shells For local buckling analysis, the approach of Stamatelos et al [13] may be used Acknowledgments The research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number 107.01-2012.02 The authors are grateful for this financial support References [1] Van der Neut A The general instability of stiffened cylindrical shells under axial compression Rep S314 Amsterdam: National Aeronautical Research Institude; 1947 [2] Baruch M, Singer J Effect of eccentricity of stiffeners on the general instability of stiffened cylindrical shells under hydro-static pressure Journal of Mechanical Engineering Science 1963;5:23–7 [3] Shen HS, Zhou P, Chen TY Post-buckling analysis of stiffened cylindrical shells under combined external pressure and axial compression Thin-Walled Structures 1993;15:43–63 [4] Bushnell D Nonlinear equilibrium of imperfect locally deformed stringerstiffened panels under combined in-plane loads Computers & Structures 1987;27:519–39 [5] Reddy JN, Starnes JH General buckling of stiffened circular cylindrical shells according to a Layerwise theory Computers & Structures 1993;49:605–16 [6] Ji ZY, Yeh KY General solution 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Post-buckling analysis of imperfect stiffened laminated cylindrical shells under combined external pressure and thermal loading International Journal of Mechanics 1998;40(4):339–55 [19] Brush DD, Almroth BO Buckling of bars, plates and shells New York: Mc Graw-Hill; 1975  ... for externally pressurized stiffened FGM cylindrical shells In this paper, the nonlinear buckling and postbuckling behaviors of eccentrically stiffened functionally graded thin circular cylindrical. .. to investigate the nonlinear buckling and post -buckling behavior of eccentrically stiffened FGM cylindrical shells under external pressure based on the smeared stiffeners technique and the classical... mode of shells 4.4 Comparing the critical buckling loads of stiffened and un -stiffened FGM cylindrical shells The effects of stiffeners on the critical buckling loads and postbuckling behavior of