DSpace at VNU: Research on nonlinear torsional buckling and post-buckling of eccentrically stiffened functionally graded...
Composites: Part B 51 (2013) 300–309 Contents lists available at SciVerse ScienceDirect Composites: Part B journal homepage: www.elsevier.com/locate/compositesb Research on nonlinear torsional buckling and post-buckling of eccentrically stiffened functionally graded thin circular cylindrical shells Dao Van Dung, Le Kha Hoa ⇑ Vietnam National University, Hanoi, Viet Nam a r t i c l e i n f o Article history: Received November 2012 Received in revised form 28 February 2013 Accepted 10 March 2013 Available online 22 March 2013 Keywords: Functionally graded material A Discontinuous reinforcement B Buckling B Elasticity C Analytical modelling a b s t r a c t This paper is presented to solve the nonlinear buckling and post-buckling problem of functionally graded stiffened thin circular cylindrical shells only under torsion by the analytical approach The shells are reinforced by rings and stringers attached to their inside and the material properties of shell and the stiffeners are assumed to be continuously graded in the thickness direction Theoretical formulations based on the smeared stiffeners technique and the classical shell theory with the geometrical nonlinearity in von Karman sense are derived Approximate three-term solution of deflection is chosen more correctly and the explicit expression to finding critical load and post-buckling torsional load–deflection curves are given The effects of various parameters and the effectiveness of stiffeners on the stability of shell are shown Ó 2013 Elsevier Ltd All rights reserved Introduction Cylindrical shell is one of the important structures used widely in engineering applications When shells are subjected to compressive loads, they may be buckled As a result, an investigation of buckling and post-buckling of these shells is a necessary fundamental problem and has been attracted attention of many researchers Concerning the buckling problem of thin-walled tubes under torsion, pioneer approximate solutions were obtained by Donnell [1] The post-buckling of cylinders under torsion and axial compression was studied by Loo [2] Nash [3] presented the approximate solutions on the buckling of initially imperfect torsion-loaded cylindrical shells by applying the Ritz method Yamaki [4] obtained the approximate solutions on the post-buckling behavior of shells under torsion that the results were found to be in reasonable agreement with experiment ones Shaw et al [5] solved the problem on the imperfect laminated cylindrical shells in torsion and axial compression Their analysis were based on Donnell-type nonlinear kinematic relations and laminated cylindrical shell theory Lennon and Das [6] analyzed the torsional buckling behavior of stiffened cylinders under combined loading The effects of stiffeners on post-buckling behavior in torsion was investigated in that paper Mao and Lu [7] studied an elastic plastic buckling of cylindrical shells under torsion with a deep thick-shell model in which the effect of the factor (1 + z/R) and the effect of the mechanical boundary conditions are considered Using singular perturbation technique, ⇑ Corresponding author Tel.: +84 989358315 E-mail address: lekhahoa@gmail.com (L.K Hoa) 1359-8368/$ - see front matter Ó 2013 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.compositesb.2013.03.030 Zhang and Han [8] investigated the buckling and post-buckling of imperfect cylindrical shells subjected to torsion based on the von Karman–Donnell-type nonlinear differential equations By the above same method, Shen and Xiang [9] analyzed the buckling and post-buckling of an anisotropic laminated cylindrical shells under torsion or under combined axial compression and torsion based on the classical shell theory with von Karman–Donnell-type of kinematic nonlinearity the extension–twist, extension–flexual and flexual–twist couplings are considered Paimushin [10] reported details of local and global buckling of cylindrical shells under combined loading He showed the existence of previously unknown torsional, flexural, and torsional–flexural buckling modes for cylindrical shells which were subjected to simultaneous compression and external pressure Takano [11] studied the buckling of thin and moderately thick anisotropic cylinders under combined torsion and axial compression His investigation showed that the buckling loads of a cylindrical shell are affected not only by anisotropy and transverse shear stiffness but also by shell length Some significant results on the mechanics of composite shells and curved beams have been obtained Fraternali and Reddy [12] presented the penalty model for the analysis of laminated composite shells Their method offers the possibility to easily obtain accurate interlaminar stresses By the same method, a one-dimensional theory and a finite element model for the stress analysis of curved composite beams are investigated by Ascione and Fraternali [13], Fraternali and Bilotti [14] and Fraternali and Feo [15] A new class of composite material known as functionally graded materials (FGMs) has been received considerable attention recently Shen [16], based on the higher order shear deformation 301 D.V Dung, L.K Hoa / Composites: Part B 51 (2013) 300–309 theory, obtained the results of stability problem of torsion-loaded functionally graded shells in thermal environments A singular perturbation technique is employed to determine buckling shear load and post-buckling equilibrium paths Huang and Han [17] studied the nonlinear buckling of torsion-loaded FGM un-stiffened cylindrical shells by using the nonlinear large deflection shell theory and Ritz method The nonlinear buckling shape observed in experiment is taken into account in their work Sofiyev and Kuruoglu [18] investigated the torsional vibration and buckling of unstiffened cylindrical shell with functionally graded coatings surrounded by an elastic medium The modified Donnell type dynamic stability and compatibility equations with linear strain–displacement relation of three-layered of cylindrical shell and Galerkin method are used to determine the expressions for torsional buckling load and torsional frequency parameter Li and Wang [19] investigated an elastic stability of a simply supported FGM sandwich circular cylindrical shell under torsion loading by semi-analytical method The governing equations for static buckling of the structure in terms of displacements were formulated using the Flugge thin shell theory in which the strain–displacement relation is linear Bich et al [20] presented the buckling of un-stiffened FGM conical panels subjected to mechanical loads by using the equilibrium and linear stability equations in terms of displacement components Galerkin method was applied to obtain closed-form relations of bifurcation type buckling loads For dynamic analysis of FGM shells, many studies have been focused on the characters of vibration and behavior of buckling of un-stiffened shells Sofiyev and Schnack [21] studied the stability of FGM cylindrical shells under linearly increasing dynamic torsional loading The modified Donnell type dynamic stability equation and Galerkin method were used However, the geometrical relation is linear and the approximate solution was chosen by one-term Bich et al [22] presented an analytical approach to investigate the nonlinear static and dynamic unsymmetrical responses of un-stiffened FGM shallow spherical shells under external pressure incorporating the effect of temperature The classical shell theory is used and Galerkin method is applied Bich and Nguyen [23] studied the nonlinear vibration of FGM un-stiffened cylindrical shells based on improved Donnell equations ignoring the shallowness of shell Their results shown that the Volmir’s assumption can be used for nonlinear dynamic analysis with an acceptable accuracy For stiffened cylindrical shell, the stability problem is also very interest subject Van der Neut [24] pointed out the importance of the eccentricity of stiffeners in the buckling of isotropic cylindrical shells under axial compressive load Barush and Singer [25] 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 The researches on this problem have been continued for many year to obtain more precise and reasonable solution Recently, Najafizadeh et al [26] with the linear 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 [27] presented an analytical approach to investigated the nonlinear post-buckling 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 [28] obtained the results on the nonlinear buckling and post-buckling analysis of eccentrically stiffened FGM circular cylindrical shells under external pressure The material properties of shell and stiffeners are assumed 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 [29] 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 [30] 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 there are very little researches on nonlinear stability of eccentrically stiffened FGM shells and there is no work on the analytical solution for torsion-loaded stiffened FGM cylindrical shells Following the idea of works [26,28], in this paper the nonlinear buckling and post-buckling behaviors of eccentrically stiffened functionally graded thin circular cylindrical shells subjected to uniform torsional load are investigated The present novelty is that the shells under torsional load are reinforced by rings and stringers attached to their inside and the material properties of shell and the stiffeners continuously are graded in the thickness direction The theoretical formulations based on the smeared stiffeners technique and the classical shell theory with the geometrical nonlinearity in von Karman sense, are derived In addition, an approximate three-term solution of deflection including the linear buckling shape sin (mpx/L) sin n(y À kx)/R and the nonlinear buckling shape sin2(mpx/L) are more correctly chosen The resulting equations are solved by 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 stiffener, twist angle, dimensional parameters, buckling modes, and volume fraction index of materials on the stability of shell are clarified in detail Eccentrically stiffened functionally graded cylindrical shells Consider a thin circular cylindrical shell with mean radius R, thickness h and length L only subjected to uniform torsional loads Assume that two butt-ends of shell are only deformed in their planes and they still are circular [32] The middle surface of the shells is referred to the coordinates (x, h, z), y = Rh 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 [26,28]: k 2z ỵ h ; 2h h h k P 0; À z ; 2 Esh ¼ Em ỵ Ecm k 2z h ; 2hs h h z ỵ hs ; k2 P 0; 2 Es ẳ Ec ỵ Emc Er ẳ Ec ỵ Emc k 2z h ; 2hr Ecm ẳ Ec Em ; 1ị Emc ¼ Em À Ec ; ð2Þ k3 P 0; h h z ỵ hr ; 2 3ị msh ¼ ms ¼ mr ¼ m 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 302 D.V Dung, L.K Hoa / Composites: Part B 51 (2013) 300–309 (a) (b) Fig Geometry and coordinate system of a stiffened FGM circular cylindrical shell 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; and Ec, Em are Young’s modulus of the ceramic and metal, respectively The coefficient m is Poison’s ratio To account for the effect of large deflection, the von Karman type nonlinear kinematic relation for the strain components across the shell thickness at a distance z from the middle surface are of the form [31] x y xy ex ẳ e ỵ zkx ; ey ẳ e ỵ zky ; cxy ẳ c ỵ 2zkxy ; kx ¼ Àw;xx ; ky ¼ Àw;yy ; kxy ¼ w;xy ; 4ị c0xy ẳ u;y ỵ v ;x ỵ w;x w;y ; e0x ẳ u;x ỵ w2;x ; e0y ẳ v ;y w ỵ w ; R ;y Ny ¼ C 12 e0x þ C 22 e0y þ C 24 kx þ C 25 ky ; R e0x;yy ỵ e0y;xx c0xy;xy ẳ w;xx ỵ w2;xy w;xx w;yy : 5ị 6ị M x ẳ C 14 e0x ỵ C 24 e0y ỵ C 44 kx ỵ C 45 ky ; M y ẳ C 15 e0x ỵ C 25 e0y þ C 45 kx þ C 55 ky ; ð10Þ M xy ẳ C 63 c0xy ỵ C 66 kxy ; Esh ex ỵ mey ị; m2 Esh ẳ ey ỵ mex ị; m2 Esh ẳ c ; 21 ỵ mị xy E1 E1s bs mE1 E2 E2s bs mE2 ỵ ; C 12 ẳ ; C 14 ẳ ỵ ; C 15 ẳ ; À m2 ds À m2 À m2 ds À m2 E1 E1r br mE2 E2 E2r br E1 ỵ ; C 24 ẳ ; C 25 ẳ þ ; C 33 ¼ ; C 22 ¼ À m2 dr À m2 À m2 dr 21 ỵ mị E2 E3 E3s bs mE3 E3 E3r br ; C 44 ẳ ỵ ; C 45 ẳ ; C 55 ẳ ỵ ; C 36 ẳ 1ỵm À m2 ds À m2 À m2 dr E2 E3 ; ; C 66 ¼ C 63 ẳ 21 ỵ mị 1ỵm 11ị in which Z E1 ¼ Hooke’s law for cylindrical shell is defined as h=2 Esh zịdz ẳ Em h ỵ h=2 rsh x ẳ Z E2 ẳ zEsh zịdz ẳ Z E3 ẳ z2 Esh zịdz ẳ h=2 E1s ẳ kEcm h ; 2k þ 1Þðk þ 2Þ h=2 Z ! Em h 1 ; ỵ ỵ Ecm h 4k ỵ 1ị k ỵ k ỵ 12 h=2ỵhs Es zịdz ẳ Ec hs ỵ Emc h=2 8ị Taking into account the contribution of stiffeners by the smeared stiffeners technique and omitting the twist of stiffeners and integrating the stressstrain equations and their moments Ecm h ; kỵ1 h=2 h=2 7ị and for stiffeners rsx ẳ Es ex ; rry ẳ Er ey : 9ị Nxy ẳ C 33 c0xy ỵ C 36 kxy ; C 11 ¼ where u = u(x, y), v = v(x, y) and w = w(x, y) are the displacements of the middle surface points along x, y and z axes, respectively, and kx, ky and kxy are the change of curvatures and twist of shell, respectively The compatible equation deduced from Eqs (5) is written by rsh xy Nx ¼ C 11 e0x ỵ C 12 e0y ỵ C 14 kx þ C 15 ky ; where the stiffness parameters Cij are given by in which rsh y through the thickness of the shell, the expressions for force and moment resultants of an eccentrically stiffened FGM cylindrical shell are expressed by [26,31] hs ; k2 ỵ Ec hs hhs ỵ1 h h=2 hs þ þ Emc hs h k2 þ h 2k2 ỵ E2s ẳ Z h=2ỵhs zEs zịdz ẳ 303 D.V Dung, L.K Hoa / Composites: Part B 51 (2013) 300309 Z h=2ỵhs ! Ec 3 h h ỵ ỵ1 hs h=2 hs hs " # 1 h h þ þ þ Emc hs ; k2 þ k2 þ hs 4ðk2 þ 1Þ h2s Z h=2þhr hr Er zịdz ẳ Ec hr ỵ Emc ; E1r ẳ k ỵ1 h=2 Z h=2ỵhr Ec hr zEr zịdz ẳ hhr E2r ẳ ỵ1 h h=2 hr ; ỵ Emc hr h þ k3 þ h 2k3 þ ! Z h=2ỵhr Ec 3 h h E3r ẳ z Er zịdz ẳ hr ỵ ỵ1 h2r hr h=2 " # 1 h h ỵ ; ỵ ỵ Emc hr k3 þ k3 þ hr 4ðk3 þ 1Þ h2r E3s ¼ Introduction of Eqs (15) and (18) into the third of Eqs (17), taking into account Eq (4), yields the following equation: z2 Es zịdz ẳ a11 w;xxxx ỵ a12 w;xxyy ỵ a13 w;yyyy ỵ a14 u;xxxx ỵ a15 u;xxyy ỵ a16 u;yyyy ỵ u;xx ỵ u;yy w;xx þ u;xx w;yy À 2u;xy w;xy ¼ 0; R in which b11 u;xxxx ỵ b12 u;xxyy ỵ b13 u;yyyy þ b14 w;xxxx þ b15 w;xxyy þ b16 w;yyyy ð12Þ My ẳ D15 Nx ỵ D25 Ny ỵ D54 kx þ DÃ55 ky ; ð14Þ ð15Þ DÃ66 kxy ; DÃ44 ẳ C 44 ỵ C 24 C 24 ỵ C 14 C Ã14 ; DÃ24 ¼ C 24 C Ã11 À C 14 C Ã12 ; DÃ45 ¼ C 14 C 15 ỵ C 24 C 25 ỵ C 45 ; DÃ15 ¼ C 15 C Ã22 À C 25 C 12 ; D54 ẳ C 15 C 14 ỵ C 25 C 24 ỵ C 45 ; ẳ C 25 C Ã11 À C 15 C Ã12 ; DÃ55 ẳ C 15 C 15 ỵ C 25 C 25 16ị ỵ C 55 ; D63 ẳ C 63 C Ã33 ; DÃ66 ¼ C 66 À C 63 C Ã36 : The equilibrium equations of cylindrical shell based on the classical shell theory are given by [31,32] Nx;x ỵ Nxy;y ẳ 0; Nxy;x ỵ Ny;y ẳ 0; Mx;xx ỵ 2Mxy;xy ỵ My;yy ỵ Ny ỵ Nx w;xx ỵ 2Nxy w;xy ỵ Ny w;yy ẳ 0: R 17ị The rst two of Eqs (17) are identically satisfied by introducing a stress function u(x, y) as Nx ¼ u;yy ; Ny ¼ u;xx ; Nxy ¼ Àu;xy : b11 ¼ C Ã11 ; b14 ¼ ÀC Ã24 ; b12 ¼ C Ã33 À 2C Ã12 ; b13 ¼ C Ã22 ; b15 ẳ C 14 ỵ C 25 ỵ C 36 ; b16 ẳ C 15 : 22ị Eqs (19) and (21) are the nonlinear governing equations used to investigate the nonlinear stability of eccentrically stiffened FGM cylindrical shells under uniform torsion loads w ẳ wx; yị ẳ f0 ỵ f1 sin ax sin by kxị þ f2 sin ax; ð23Þ in which a = mp/L, b = n/R and m is the number of axis half waves and n is the number of circumferential waves The first term of w in Eq (23) represents the uniform deflection of points belonging to two butt-ends x = and x = L, 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 x = and x = L is fulfilled on the average sense Substituting Eq (23) into Eq (21) obtains b11 u;xxxx þ b12 u;xxyy þ b13 u;yyyy ¼ B01 cos 2ax þ B02 cos 2bðy À kxÞ ! ! a a ỵ B03 cos b y ỵ k x ỵ B04 cos b y ỵk x b b & ! !' a a ỵ B05 cos b y ỵ k x cos b y ỵ k x ; ð24Þ b b where DÃ25 where Consider a torsion-loaded cylindrical shell and it is simply supported at two butt-ends x = and x = L The deflection of shell in this case can be expressed by [17,32] Mx ẳ D14 Nx ỵ D24 Ny ỵ D44 kx þ DÃ45 ky ; DÃ14 ¼ C 14 C Ã22 À C 24 C Ã12 ; ð21Þ Solution of the problem Substituting Eqs (13) into Eqs (10), the moment resultants become ỵ w2;xy ỵ w;xx w;yy ỵ w;xx ẳ 0; R 13ị D ẳ C 22 C 11 À C 212 ; C Ã22 ¼ C 22 =D; C Ã12 ¼ C 12 =D; C Ã14 ¼ ðC 12 C 24 À C 22 C 14 Þ=D; C Ã15 ¼ ðC 12 C 25 À C 22 C 15 ị=D; C 11 ẳ C 11 =D; C Ã24 ¼ ðC 12 C 14 À C 11 C 24 ị=D; C 36 ; C 36 ẳ : C Ã25 ¼ ðC 12 C 15 À C 11 C 25 ị=D; C 33 ẳ C 33 C 33 Mxy ẳ 20ị Eq (19) includes two dependent unknown functions w and u and to find a second equation relating to these two functions the geometrical compatibility Eq (6) is used For this aim, substituting Eq (13) into Eq (6), obtains where DÃ63 Nxy Á a11 ¼ ÀDÃ44 ; a12 ẳ D45 ỵ 2D66 ỵ D54 ; a13 ẳ ÀDÃ55 ; À Á a14 ¼ DÃ24 ; a15 ¼ D14 2D63 ỵ D25 ; a16 ẳ D15 : where the 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 later use, the reverse relations obtained from Eqs (9) are as e0x ẳ C 22 Nx C 12 Ny ỵ C 14 kx ỵ C 15 ky ; e0y ẳ C 12 Nx ỵ C 11 Ny ỵ C 24 kx þ C Ã25 ky ; c0xy ¼ C Ã33 Nxy À C Ã36 kxy ; ð19Þ ð18Þ where ! 1 B01 ¼ 2f a2 4b14 a2 ỵ f12 a2 b2 ; B02 ẳ f12 a2 b2 ; R 2 & 1 1 2 2 B03 ¼ f1 b14 ẵa2 ỵ b k ị ỵ 2abkị b15 b2 a2 ỵ b2 k2 ị ỵ b16 b4 2 R !' 1 2 2 2 ỵabk 2b14 a ỵ b k ị ỵ b15 b ỵ f1 f2 a b ; R & 1 2 2 b15 b2 a2 ỵ b2 k2 ị B04 ẳ f1 b14 ẵa ỵ b k ị þ ð2abkÞ þ R !' 1 2 b16 b ỵ 2abk 2b14 a2 ỵ b k ị ỵ b15 b2 f1 f2 a2 b2 ; R 2 B05 ¼ f1 f2 a b ; ð25Þ The general solution of Eq (24) for torsion-loaded shell is of the form 304 D.V Dung, L.K Hoa / Composites: Part B 51 (2013) 300–309 ! Àk x b ! ! a a ỵ k x ỵ B5 cos b y ỵ k x ỵ B4 cos b y À b b ! a ð26Þ þ B6 cos b y þ À k x shxy; b u ẳ B1 cos 2ax ỵ B2 cos 2by kxị ỵ B3 cos b y ỵ a where s is torsional load intensity and the coefficients Bi are defined by B1 ¼ B01 ¼ A11 f2 ỵ A12 f12 ; 16b11 a4 B02 ỵ a2 b2 A31 ỵ A41 2A11 ị ỵ A42 b4 " # 4 2 a a a14 ỵ k ỵ a15 ỵ k þ a16 ; b b Rb ¼ A21 f12 ; 16b4 ẵb11 k4 ỵ b12 k2 ỵ b13 B03 ! ẳ A31 f1 ỵ A32 f1 f2 ; B3 ¼ 4 2 a b b11 b k ỵ b12 ab k ỵ b13 B2 ẳ B4 ẳ b b11 B5 ẳ aỵk 4 b B04 ỵ b12 2 aỵk b ! ẳ A41 f1 ỵ A42 f1 f2 ; 27ị ỵ b13 B05 ! ¼ A5 f f ; 4 2 a b b11 b ỵ k ỵ b12 ab ỵ k ỵ b13 31ị In addition to Eqs (29) and (30), the cylindrical shell must also satisfy the circumferential closed condition [17,32] as ÀB05 ! ¼ A6 f f ; 4 2 b4 b11 ab k ỵ b12 ab k ỵ b13 Z 2pR Z A ẳ a2 ỵ b2 k2 ; A11 ẳ ; 32b2 ẵb11 k4 ỵ b12 k2 ỵ b13 n o h i 12 b14 ẵA2 ỵ 2abkị2 ỵ 1R b15 b2 A b16 b4 ỵ abk 2b14 A ỵ 1R b15 b2 ! A31 ¼ ; 4 2 b4 b11 ab k ỵ b12 ab k ỵ b13 ak 4 b a2 ỵ b12 ak 2 b !; ỵ b13 h i b14 ẵA2 þ ð2abkÞ2 þ 1R À b15 b2 A À b16 b4 ỵ 2abk 2b14 A ỵ 1R b15 b2 ! A41 ¼ ; 4 2 2b4 b11 ab ỵ k ỵ b12 ab ỵ k ỵ b13 A42 ¼ 2b2 b11 A5 ¼ A6 ¼ v ;y dx dy ¼ Z 2pR Z 0 L e0y ỵ w w;y dx dy ¼ 0: R 2 2f þ f2 À Rf b2 ¼ 0: 4b14 a2 À 1=R b2 ; A12 ¼ ; 32b11 a2 8b11 a2 a2 A21 ẳ 2b2 b11 L 32ị Using Eqs (13), (18), (23) and (26), this integral becomes in which A32 ẳ D3 ẳ 2A21 ỵ A12 ịa2 b2 ; D4 ẳ A32 ỵ A42 A5 ỵ A6 Þa2 b2 ; ! D5 ¼ À8a2 2a11 a2 ỵ 4a14 a2 A11 ; R ỵ 2A31 A41 ịa2 b2 ; D6 ¼ 8a2 A12 4a14 a2 À R D7 ¼ 2a2 b2 A32 ỵ A42 A5 ỵ A6 ị: B6 ẳ D1 ẳ a11 ẵa2 ỵ b2 k2 ị2 þ ð2abkÞ2 þ a12 b2 ða2 þ b2 k2 ị ỵ a13 b4 " # 4 2 a a À A31 b4 a14 À k þ a15 À À k þ a16 b b Rb " # 4 2 a a ỵ A41 b a14 ỵ k ỵ a15 ỵ k ỵ a16 ; b b Rb " # 4 2 a a k ỵ a15 k ỵ a16 D2 ¼ ÀA32 b a14 b b Rb aỵk b 4 a2 ỵ b12 aỵk b 2 !; ð33Þ Eliminating f2 from Eqs (29) and (30) leads to the equation representing s–f1 relation as " s ¼ D1 ỵ #, D4 D6 f12 D2 D6 f12 2hb2 kị: ỵ D3 f ỵ D5 ỵ D7 f12 D5 ỵ D7 f Eq (34) is used to analyze the post-buckling s–f1 curves of stiffened FGM cylindrical shells When f1 ? 0, Eq (34) becomes s ẳ D1 =2hb2 kị: ð35Þ Eq (35) is used to find upper critical loads in case linear buckling shape From Eq (23), it is obvious that the maximal deflection of the shells W max ẳ f0 ỵ f1 ỵ f2 ; ỵ b13 34ị ð36Þ !; 4 2 2b2 b11 ab ỵ k ỵ b12 ab ỵ k ỵ b13 locates at x = iL/(2m), y = jpR/(2n) + ikL/(2m), where i, j are odd integer numbers Solving f2 and f0 from Eqs (30) and (33) with respect to f1, then substituting them into Eq (36), obtains Àa2 !: 4 2 2b b11 ab À k þ b12 ab À k þ b13 D6 f12 ỵ Rf 21 b2 ỵ f1 : W max ẳ D5 ỵ D7 f12 a2 ð28Þ In order to establish a torsional load–deflection curve, first of all, introducing w and u into the left side of Eq (19), then applying Galerkin’s method in the ranges y 2pR and x L, lead to 2shb2 k ỵ D1 ỵ D2 f2 ỵ D3 f12 ỵ D4 f22 ẳ 0; 29ị D5 f2 D6 f12 ỵ D7 f12 f2 ẳ 0; ð30Þ where ð37Þ Combining Eq (34) with Eq (37), the effects of inhomogeneous and dimensional parameters on the post-buckling load–maximal deflection curves of shells can be analyzed The angle of twist is defined [8,32] as w¼ 2pRL ¼ 2pRL @u @ v ỵ dx dy @y @x 0 Z 2pR Z L c0xy À w;x w;y dx dy: Z 2pR Z L Using Eqs (13) and (18), this integral becomes ð38Þ 305 D.V Dung, L.K Hoa / Composites: Part B 51 (2013) 300–309 w¼ 2pRL Z 2pR Z L C Ã36 w;xy À C Ã33 u;xy À w;x w;y dx dy: present results coincide with the ones of the work [17] In addition, the present critical value scr = 204.12 MPa corresponding to the lowest point of the envelope curve (see Fig 2) is much close to the one of Ref [17] scr = 204.15 MPa obtained by using the Ritz energy method In the following subsections, the materials used [26] are Zirconia with Ec = 151 GPa and Aluminum with Em = 70 GPa Also assume that k2 = k3 = 1/k and m = 0.3 Substituting w and u from Eqs (23) and (26) into this equation obtains w ¼ C Ã33 sh ỵ b2 kf12 : 39ị When f1 = 0, Eq (39) shows that the relation between twist angle and shear stress is linear When f1 – 0, combining Eq (34) with Eq (39), the s–w relation of shells will be studied 4.2 Nonlinear critical torsional load finding procedure Consider a stiffened FGM shell with the material and geometrical parameters: k = 1, k2 = k3 = 1, L = 387.35 Â 10À3 m, L/R = 1, R/ h = 100, hs/h = hr/h = 1/2, bs = hs, br = hr The number of stringers as well as rings is equal to 20 Based on Eq (34) with various combinations of the modes (m, n, k) the critical load scr of stiffened FGM shell may be found As can be seen, from Table 3, the critical load scr = 265.0121 MPa corresponding to m = 1, n = and k = 0.55 Graphically, according to Ref [17], one also can define the critical condition as the possible lowest point of s–Wmax/h curves (Fig 3) Thus, the specific solution procedures are exhibited as follows: by using Eqs (34) and (37), a series of s versus Wmax/h, the curves can be drawn under various combinations of (m, n, k) From the lowest of these curves, an envelope curve is obtained The lowest point of the envelope curve is regarded as the critical condition By mentioned procedure, in this problem obtained the minimum value scr = 265.0121 MPa corresponding to the buckling mode (m, n, k) = (1, 9, 0.55) Numerical results and discussion 4.1 Comparison results To validate the present study, three comparisons on critical torsion load are made with results from open literatures Tables and compare the results of this paper for un-stiffened isotropic cylindrical shell under torsion load with the results given by Shen [16] using the higher order shear deformation shell theory and with experimental results of Nash [33] and Ekstrom [34] As can be seen that good agreements are obtained in these comparisons Fig shows the comparisons of the present post-buckling paths with the results which they were analyzed by Huang and Han [17] using the nonlinear large deflection theory for un-stiffened FGM cylindrical shells under torsion load As can be observed, the Table Comparisons of critical torsion load scr (psi) for un-stiffened isotropic cylindrical shell scr (psi) Exp of Nash Shen Present by Eq (34) (k = 0.25) Error (%) E = 27e+6 psi, m = 0.3 L = 38 in., R = in h = 0.0172 in 6590 6835 (m, n) = (1, 2) 6767 (m, n) = (1, 6) 2.68 (exp) 0.99 (Shen) Table Comparisons of critical torsion load scr (psi) for un-stiffened isotropic cylindrical shell scr (psi) Exp of Ekstrom Shen Present by Eq (35) (k = 0.1) Error (%) E = 29e+6 psi, m = 0.3 L = 19.85 in., R = in h = 0.0075 in 4800 4997 (m, n) = (1, 3) 5.0335e+003 (m, n) = (1, 6) 4.64 (exp) 0.73 (Shen) 250 50 1: (n, λ)=(7,0.3) 2: (n, λ)=(8,0.4) 3: (n, λ)=(9,0.5) 240 Upper (35) Nonlinear of Huang 40 Mcr (Nm) 230 τ (MPa) Lower (34) 220 Linear of Huang 30 M cr = 2π R hτ cr 20 Zirconia/Ti-6Al-4V T=300K, k=1, R/h=100, L/R=2 210 200 Z = −ν present τcr=204.12 MPa Huang L2 Rh 10 300 600 900 f1 / h Z (a) (b) Fig Comparison with results of [13] for un-stiffened FGM cylindrical shells 1200 1500 306 D.V Dung, L.K Hoa / Composites: Part B 51 (2013) 300–309 Table Critical buckling load versus (m, n, k) 350 300 m=1 10 11 12 13 14 321.2948(0.36)a 278.0196(0.53) 265.0121(0.55) 271.2489(0.56) 289.8609(0.57) 317.1922(0.58) 351.2284(0.59) 390.8035(0.59) 606.3897(1.12) 532.2178(0.85) 485.9032(0.72) 464.4975(0.66) 462.1628(0.64) 474.0187(0.63) 496.6100(0.62) 527.6847(0.61) 936.3017(1.10) 839.2876(0.94) 775.4175(0.84) 737.0675(0.76) 718.7535(0.71) 716.2710(0.68) 726.6889(0.66) 747.6301(0.64) 1445.007(1.24) 1301.552(1.08) 1201.095(0.95) 1133.255(0.86) 1091.230(0.79) 1069.487(0.74) 1063.989(0.70) 1071.617(0.68) 250 τ (MPa) n 1: R/h=100, (5, 0.26)a 2: R/h=200, (6, 0.21) 3: R/h=300, (7, 0.21) 4: R/h=400, (7, 0.18) 5: R/h=500, (8, 0.19) 200 150 100 50 a The number of k 0 10 12 14 16 18 20 Wmax / h 400 1: (n, λ)=(7, 0.36) 2: (n, λ)=(8, 0.53) 3: (n, λ)=(9, 0.55) 4: (n, λ)=(10, 0.56) 5: (n, λ)=(11, 0.57) τ (MPa) 370 340 Fig Effects of R/h ratio on post-buckling curves of shell, m = 1, L/R = aBulking mode (n, k) 350 310 1: L/R=1, (9,0.5) 2: L/R=1.5, (8,0.43) 3: L/R=2, (8,0.42) 4: L/R=2.5, (7,0.37) 5: L/R=3, (7,0.37) 300 τcr=265.0121 MPa 250 τ (MPa) 280 250 200 Wmax / h 150 Fig Critical buckling load (m = 1) 4.3 Effect of the mode (m, n, k) on the critical torsional load 100 10 12 14 16 18 20 Wmax / h In this subsection, by using Eqs (34) and (35), the effect of mode on the critical loads of stiffened FGM is presented in Table It is seen that, the lower and upper critical loads depend clearly on the mode In addition, the circumferential wave number n increases with increases of R/h ratio or L/R ratio decreasing 4.4 Effects of geometric parameters Based on Eqs (34) and (37) with the database given in Section 4.2, the effects of the radius-to-thickness ratios R/h and of the length-to-radius ratios L/R on s–Wmax/h post-buckling curves of stiffened FGM cylindrical shell are considered Fig plots the post-buckling curves versus R/h = (100; 200; 300; 400 and 500) It is observed that the torsional buckling load s decreases markedly with the increase of R/h ratio This result agrees with the actual property of structure, i.e the shell is thinner Fig Effects of L/R ratio on s–Wmax/h curves (m = 1, R/h = 100) the value of critical load is smaller This remark is also illustrated in Table Effect of L/R ratio also is analyzed in Table and Fig As can be seen the critical torsional loads of shells decreases considerably when L/R ratio increases Thus, both the cases, the critical torsion load is very sensitive with the change of R/h or L/R 4.5 Effects of volume fraction index Using the database in Section 4.2, the effects of index volume k on the critical buckling loads and post-buckling behavior are given Table Effect of mode on the critical buckling load (m = 1) R/h 100 200 300 400 500 600 700 800 900 1000 a L/R = L/R = Lower critical load calculated by Eq (34) Upper critical load calculated by Eq (35) Lower critical load calculated by Eq (34) Upper critical load calculated by Eq (35) 265.0122(9, 0.55)a 103.0009(12, 0.47) 60.0410(14, 0.44) 41.0507(15, 0.40) 30.6812(16, 0.38) 24.2046(17, 0.37) 19.8130(18, 0.36) 16.6709(19, 0.35) 14.3296(20, 0.35) 12.5235(20, 0.33) 292.8112(9, 0.50) 117.1717(12, 0.41) 69.2888(14, 0.37) 48.0152(16, 0.36) 36.0603(17, 0.33) 28.5939(18, 0.32) 23.5282(19, 0.31) 19.8365(19, 0.28) 17.0855(20, 0.28) 14.9701(21, 0.28) 170.2449(7, 0.38) 68.0410(9, 0.33) 40.0498(11, 0.34) 27.6813(12, 0.31) 20.8385(13, 0.30) 16.5655(14, 0.30) 13.5619(14, 0.27) 11.4799(14, 0.25) 9.8666(15, 0.26) 8.6516(15, 0.24) 197.2511(8, 0.37) 80.8716(10, 0.30) 49.0452(12, 0.29) 33.4107(12, 0.25) 25.7200(14, 0.26) 20.1268(14, 0.23) 16.5002(14, 0.22) 13.9804(15, 0.21) 12.4023(17, 0.23) 10.5252(15, 0.19) The numbers in the parentheses denote the buckling mode (n, k) 307 D.V Dung, L.K Hoa / Composites: Part B 51 (2013) 300–309 Table Effects of L/R and R/h on critical torsional load for FGM stiffened cylindrical shells R/h L/R = 1.5 2.5 100 200 300 400 500 265.0122(9, 0.55) 103.0009(12, 0.47) 60.0410(14, 0.44) 41.0507(15, 0.40) 30.6812(16, 0.38) 201.5580(8, 0.46) 80.4188(10, 0.38) 47.0817(12, 0.37) 32.4441(13, 0.34) 24.3378(14, 0.33) 169.6110(8, 0.44) 68.0410(9, 0.33) 40.0498(11, 0.34) 27.6813(12, 0.31) 20.8385(13, 0.30) 146.9519(7, 0.38) 59.7170(9, 0.34) 35.4203(10, 0.30) 24.6048(11, 0.27) 18.5277(12, 0.28) 133.4250(7, 0.38) 53.9648(8, 0.29) 32.1561(9, 0.26) 22.2854(10, 0.26) 16.8219(11, 0.25) 500 480 Stiffened 450 350 τ (MPa) τ (MPa) 400 300 250 150 440 420 400 380 360 340 1: k=0, (0.53s,0.50u) 2: k=0.5, (0.54s,0.5u) 3: k=1, (0.55s,0.50u) 4: k=∞, (0.6s,0.51u) 200 100 320 4: ns=nr=40, (8,0.73) 5: ns=nr=50, (8,0.75) 300 280 Wmax / h Wmax / h Fig Effects of k on s–Wmax/h curves (m = 1, n = 9, sStiffened, uUnstiffened) Fig Effects of number of orthogonal stiffeners (m = 1) increase is about 35.8% for orthogonal stiffened shell, in comparison ns = nr = 10 with ns = nr = 50 400 350 1: ns=nr=10, (9,0.62) 2: ns=nr=20, (8,0.68) 3: ns=nr=30, (8,0.71) 460 Unstiffened 1: R/h=100, (9,0.55) 2: R/h=200, (12,0.47) 3: R/h=300, (14,0.44) 300 4.7 Comparison of critical torsion loads of stiffened and un-stiffened FGM cylindrical shells τ (MPa) 250 200 150 100 50 -2 10 -1 10 10 10 10 k Fig Effects of k on torsional load (m = 1) in Figs and for stiffened FGM shell and un-stiffened FGM shell with m = 1, k2 = k3 = 1/k but n and k vary It can be observed, the critical torsional loads of shells with or without stiffener decrease with the increase of k This property appropriate to the real characteristic of material, because the higher value of k corresponds to a metal-richer shell which usually has less stiffness than a ceramic-richer one 4.6 Effects of number of stiffeners To investigate the effects of number of stiffeners, the database is used here taken from database in Section 4.2 with hs = hr = h, bs = hs, br = hr Fig and Table illustrate the effects of number of stiffener (ns = nr = 10, 20, 30, 40 and 50) on the critical torsional loads As expected, these curves become higher when the number of stiffeners increases and critical torsion loads decrease when the number of stiffeners decreases The prime reason is that the presence of stiffeners makes the shells to become stiffer Table also shows that the percentage increase in the buckling load rises continuously with the increment of the number of stiffeners This Using the database in Section 4.6, the comparison between the critical torsional loads scr of stiffened FGM and un-stiffened FGM shell is given Table shows that the critical torsional loads of FGM stiffened cylindrical shells are generally upper than the corresponding values of the FGM un-stiffened cylindrical shells In addition, the critical torsional loads of FGM un-stiffened shells are the smallest, the critical torsional loads of stringer stiffened shell are smaller than ring stiffened shell, and finally the critical loads of FGM ring-stringer stiffened shell are the greatest Thus a presence of stiffener enhances the stability of shell 4.8 Effects of k and Z on s–w post-buckling curves With the database in Section 4.6 and Z ẳ L2 =Rhị ẳ 300 given by [16], Fig shows the effects of volume fraction k on post-buckling s–w curves for un-stiffened and stiffened FGM cylindrical shells Comparing these curves, it can be seen that the, they become to be more down in the increase of k.pffiffiffiffiffiffiffiffiffiffiffiffiffi Fig 10 ffi illustrates the effects L2 of Batdorf shell parameter Z ¼ Rh À m2 on the post-buckling s–w curves of shells Similar to above case, the critical torsional loads of shells in this case also decrease when Z increases In addition, the post-buckling s–w curves are nonlinear and slope downward immediately after buckling With further increase in twist angle, the torsional load exhibits an increase after reaching the minimum post-buckling load Conclusions The shells stiffened by eccentrically rings and stringers attached to the inside and material properties of shell and stiffeners varying continuously graded in the thickness direction are investigated in 308 D.V Dung, L.K Hoa / Composites: Part B 51 (2013) 300–309 Table (m = 1, k = 1) Effects of number of stiffeners on the critical torsional loads scr (MPa) Shell 10 20 30 40 50 Stringer stiffened Ring stiffened Orthogonal Stiffened 252.7497(9, 0.49) 296.9082(9, 0.63) 300.7408(9, 0.62) 254.9822(9, 0.48) 327.7061(8, 0.71) 336.3076(8, 0.68) 257.0872(9, 0.47) 348.5922(8, 0.75) 363.0797(8, 0.71) 259.0760(9, 0.47) 365.9971(8, 0.79) 386.8793(8, 0.73) 260.9513(9, 0.46) 380.9263(8, 0.82) 408.4966(8, 0.75) Table Comparison of critical torsional loads for stiffened and un-stiffened FGM shells a Shell k = (ceramic) k = 0.5 k=1 k=5 k = (metal) Un-stiffened Stringer stiffened Ring stiffened Orthogonal Stiffened 352.3774(9, 0.50)a 356.3673(9, 0.49) 431.0026(9, 0.66) 437.7195(9, 0.64) 280.4184(9, 0.50) 284.7640(9, 0.48) 358.8464(8, 0.69) 366.4336(8, 0.67) 250.3686(9, 0.50) 254.9823(9, 0.48) 327.7061(8, 0.71) 336.3077(8, 0.68) 204.3720(9, 0.51) 210.4026(9, 0.48) 283.1657(8, 0.76) 296.1090(8, 0.72) 163.3540(9, 0.51) 171.0438(9, 0.46) 253.7053(8, 0.83) 272.3063(7, 0.77) The numbers in the parentheses denote the buckling mode (n, k) 450 300 400 250 350 τ (MPa) τ (MPa) 200 150 1: k=0, (8,0.43) 2: k=0.5, (8,0.42) 3: k=1, (8,0.42) 4: k=5, (8,0.42) 100 Z = L2 / ( Rh ) =300 50 R/h=100 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 300 250 200 150 R/h=100 50 1: k=0, (7,0.52) 2: k=0.5, (7,0.50) 3: k=1, (7,0.49) 4: k=5, (7,0.49) Z = L2 / ( Rh ) =300 100 0.5 1.5 ψ (deg) ψ (deg) (a) for un-stiffened shell (b) for stiffened shell 2.5 Fig Effects of k on s–w curves FGM cylindrical shell 250 300 250 200 200 150 τ (MPa) τ (MPa) 100 1: Z=300, (8,0.42) 2: Z=500, (7,0.36) 3: Z=1000, (7,0.36) 4: Z=1500, (6,0.31) 5: Z=2000, (6,0.31) k=1 R/h=100 50 150 100 0.5 1.5 2.5 3.5 1: Z=300, (7,0.49) 2: Z=500, (7,0.46) 3: Z=1000, (6,0.36) 4: Z=1500, (6,0.36) 5: Z=2000, (6,0.36) k=1 R/h=100 50 0 0.5 1.5 2.5 ψ (deg) ψ (deg) (a) unstiffened shell (b) stiffened shell 3.5 Fig 10 Effects of Z on s–w curves for cylindrical shell this paper An analytical approach to analyze the nonlinear buckling and post-buckling behavior of eccentrically stiffened FGM cylindrical shells under torsion based on the classical shell theory and the smeared stiffeners technique with geometrical nonlinearity in von Karman sense is presented The results obtained show some remarks as: i The expression of deflection with three-term including the linear and nonlinear buckling shape is more correctly chosen ii The close-form expressions to determine critical buckling loads and nonlinear post-buckling load–deflection curves are obtained D.V Dung, L.K Hoa / Composites: Part B 51 (2013) 300–309 iii Both the post-buckling mode and the post-buckling paths of torsion-loaded stiffened FGM cylindrical shells can be well predicted by using the nonlinear large deflection theory iv The stiffener system strongly enhances on the stability and load-carrying capacity of FGM cylindrical shells v The critical torsion load is affected significantly when material distribution was varied by changing the values of the power law exponent k Both the critical torsional load and the post-buckling carrying capacity decrease greatly when the radius-to-thickness or length-to-radius ratio increase vi The critical torsional load decreases with the increase of twist angle Acknowledgements 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 References [1] Donnell LH Stability of thin-walled tubes under torsion NACA Report No 479; 1933 [2] Loo TT Effects of large deflection and imperfections on the elastic buckling of cylinders under torsion and axial compression Proc Second US Natl Congr Appl Mech 1954;5:345–57 [3] Nash WA Buckling of initially imperfect cylindrical shells subjected to torsion J Appl Mech 1957:125–30 [4] Yamaki N Postbuckling behavior of circular cylindrical shells under torsion Ing Arch 1975:79–89 [5] Shaw D, Simitses GJ, Sheinman I Imperfect laminated cylindrical shells in torsion and axial compression Acta Astronaut 1983;10(5–6):395–400 [6] Lennon RF, Das PK Torsional buckling behavior of stiffened cylinders under combined loading Thin-Walled Struct 2000;38:229–45 [7] Mao R, Lu G A study of elastic–plastic buckling of cylindrical shells under torsion Thin-walled Struct 2002;40:1051–71 [8] Zhang X, Han Q Buckling and post-buckling behaviors of imperfect cylindrical shells subjected to torsion Thin-walled Struct 2007;45:1035–43 [9] Shen HS, Xiang Y Buckling and post-buckling of anisotropic laminated cylindrical shells under combined axial compression and torsion Compos Struct 2008;84:375–86 [10] Paimushin VN Torsional, flexural, and torsional–flexural buckling modes of a cylindrical shell under combined loading Mech Solid 2007;42:437–46 [11] Takano A Buckling of thin and moderately thick anisotropic cylinders under combined torsion and axial compression Thin-Walled Struct 2011;49:304–16 [12] Fraternali F, Reddy JN A penalty model for the analysis of laminated composite shells Int J Solids Struct 1993;30:3337–55 [13] Ascione L, Fraternali F A penalty model for the analysis of composite curved beams Comput Struct 1992;45:985–99 309 [14] Fraternali F, Bilotti G Non-linear elastic stress analysis in curved composite beams Comput Struct 1997;62:837–69 [15] Fraternali F, Feo L On a moderate rotation theory of thin-walled composite beams Compos Part B Eng 2000;31:141–58 [16] Shen HS Torsional buckling and postbuckling of FGM cylindrical shells in thermal environments Int J Non-Linear Mech 2009;44:644–57 [17] Huang H, Han Q Nonlinear buckling of torsion-loaded functionally graded cylindrical shells in thermal environment Euro J Mech A/Solids 2010;29: 42–8 [18] Sofiyev AH, Kuruoglu N Torsional vibration and buckling of the cylindrical shell with functionally graded coatings surrounded by an elastic medium Composites Part B 2013;45:1133–42 [19] Li S, Wang S Elastic stability of circular cylindrical shells with FGMS layers under end torsion force Lixue Xuebao/Chin J Theoret Appl Mech 2010;42(6):1172–9 [20] Bich DH, Phuong NT, Tung HV Buckling of functionally graded conical panels under mechanical loads Compos Struct 2012;94(4):1379–84 [21] Sofiyev AH, Schnack E The stability of functionally graded cylindrical shells under linearly increasing dynamic torsional loading Eng Struct 2004;26(10):1321–31 [22] Bich DH, Dung DV, Hoa LK Nonlinear static and dynamic buckling analysis of functionally graded shallow spherical shells including temperature effects Compos Struct 2012;94(9):2952–60 [23] Bich DH, Nguyen NX Nonlinear vibration of functionally graded circular cylindrical shells based on improved Donnell equations J Sound Vib 2012;331(25):5488–501 [24] Van der Neut A The general instability of stiffened cylindrical shells under axial compression Rep S314 National Aeronautical Research Institute Amsterdam; 1947 [25] Baruch M, Singer J Effect of eccentricity of stiffeners on the general instability of stiffened cylindrical shells under hydro-static pressure J Mech Eng Sci 1963;5:23–7 [26] Najafizadeh MM, Hasani A, Khazaeinejad P Mechanical stability of functionally graded stiffened cylindrical shells Appl Math Model 2009;33:1151–7 [27] Bich DH, Nam VH, Phuong NT Nonlinear post-buckling of eccentrically stiffened functionally graded plates and shallow shells Vietnam J Mech 2011;3:131–47 [28] Dung DV, Hoa LK Nonlinear buckling and post-buckling analysis of eccentrically stiffened functionally graded circular cylindrical shells under external pressure Thin-Walled Struct 2013;63:117–24 [29] Bich DH, Dung DV, Nam VH Nonlinear dynamical analysis of eccentrically stiffened functionally graded cylindrical panels Compos Struct 2012;94(8):2465–73 [30] Bich DH, Dung DV, Nam VH Nonlinear dynamic analysis of eccentrically stiffened imperfect functionally graded doubly curved thin shallow shells Compos Struct 2013;96:384–95 [31] Brush DO, Almroth BO Buckling of bars, plates and shells New York: Mc Graw-Hill; 1975 [32] Volmir AS Stability of elastic systems Science ed Moscow; 1963 [in Russian] [33] Nash WA An experimental analysis of the buckling of thin initially imperfect cylindrical shells subject to torsion Proc Soc Exp Stress Anal 1959;16(2):55–68 [34] Ekstrom RE Buckling of cylindrical shells under combined torsion and hydrostatic pressure Exp Mech 1963;3:192–7 ... idea of works [26,28], in this paper the nonlinear buckling and post -buckling behaviors of eccentrically stiffened functionally graded thin circular cylindrical shells subjected to uniform torsional. .. NX Nonlinear vibration of functionally graded circular cylindrical shells based on improved Donnell equations J Sound Vib 2012;331(25):5488–501 [24] Van der Neut A The general instability of stiffened. .. equilibrium paths Huang and Han [17] studied the nonlinear buckling of torsion-loaded FGM un -stiffened cylindrical shells by using the nonlinear large deflection shell theory and Ritz method The nonlinear