European Journal of Mechanics A/Solids 46 (2014) 42e53 Contents lists available at ScienceDirect European Journal of Mechanics A/Solids journal homepage: www.elsevier.com/locate/ejmsol Nonlinear dynamic analysis of eccentrically stiffened functionally graded circular cylindrical thin shells under external pressure and surrounded by an elastic medium Dao Van Dung a, Vu Hoai Nam b, * a b Vietnam National University, Ha Noi, Viet Nam University of Transport Technology, Ha Noi, Viet Nam a r t i c l e i n f o a b s t r a c t Article history: Received 17 July 2013 Accepted February 2014 Available online 18 February 2014 A semi-analytical approach eccentrically stiffened functionally graded circular cylindrical shells surrounded by an elastic medium subjected to external pressure is presented The elastic medium is assumed as two-parameter elastic foundation model proposed by Pasternak Based on the classical thin shell theory with the geometrical nonlinearity in von KarmaneDonnell sense, the smeared stiffeners technique and Galerkin method, this paper deals the nonlinear dynamic problem The approximate three-term solution of deflection shape is chosen and the frequencyeamplitude relation of nonlinear vibration is obtained in explicit form The nonlinear dynamic responses are analyzed by using fourth order RungeeKutta method and the nonlinear dynamic buckling behavior of stiffened functionally graded shells is investigated according to BudianskyeRoth criterion Results are given to evaluate effects of stiffener, elastic foundation and input factors on the frequencyeamplitude curves, natural frequencies, nonlinear responses and nonlinear dynamic buckling loads of functionally graded cylindrical shells Ó 2014 Elsevier Masson SAS All rights reserved Keywords: Functionally graded material Nonlinear dynamic analysis Stiffened circular cylindrical shell Introduction Functionally graded material (FGM) cylindrical shell has become popular in engineering designs of coating of nuclear reactors and space shuttle The static and dynamic behavior of FGM cylindrical shell attracts special attention of a lot of researchers in the world In static analysis of FGM cylindrical shells, many studies have been focused on the buckling and postbuckling of shells under mechanic and thermal loading Shen (2003) presented the nonlinear postbuckling of perfect and imperfect FGM cylindrical thin shells in thermal environments under lateral pressure by using the classical shell theory with the geometrical nonlinearity in von KarmaneDonnell sense By using higher order shear deformation theory; this author (Shen, 2005) continued to investigate the postbuckling of FGM hybrid cylindrical shells in thermal environments under axial loading Huang and Han (2008, 2009a, 2009b, 2010a, 2010b) studied the buckling and postbuckling of unstiffened FGM cylindrical shells under torsion load, axial compression, radial pressure, combined axial compression and * Corresponding author Tel.: ỵ84 983843387 E-mail address: hoainam.vu@utt.edu.vn (V.H Nam) http://dx.doi.org/10.1016/j.euromechsol.2014.02.008 0997-7538/Ó 2014 Elsevier Masson SAS All rights reserved radial pressure based on the Donnell shell theory and the nonlinear strainedisplacement relations of large deformation Shen (2009b) investigated the torsional buckling and postbuckling of FGM cylindrical shells in thermal environments The non-linear static buckling of FGM conical shells which is more general than cylindrical shells, were studied by Sofiyev (2011a,b) Zozulya and Zhang (2012) studied the behavior of functionally graded axisymmetric cylindrical shells based on the high order theory For dynamic analysis of FGM cylindrical shells, Darabi et al (2008) presented respectively linear and nonlinear parametric resonance analyses for un-stiffened FGM cylindrical shells Sofiyev and Schnack (2004) and Sofiyev (2005) obtained critical parameters for un-stiffened cylindrical thin shells under linearly increasing dynamic torsional loading and under a periodic axial impulsive loading by using the Galerkin technique together with Ritz type variation method Sheng and Wang (2008) presented the thermomechanical vibration analysis of FGM shell with flowing fluid Sofiyev (2003, 2004, 2009, 2012) and Deniz and Sofiyev (2013) were investigated the vibration and dynamic instability of FGM conical shells Hong (2013) studied thermal vibration of magnetostrictive FGM cylindrical shells Huang and Han (2010c) presented the nonlinear dynamic buckling problems of un-stiffened functionally graded cylindrical shells subjected to time-dependent axial D.V Dung, V.H Nam / European Journal of Mechanics A/Solids 46 (2014) 42e53 load by using the BudianskyeRoth dynamic buckling criterion (Budiansky and Roth, 1962) Various effects of the inhomogeneous parameter, loading speed, dimension parameters; environmental temperature rise and initial geometrical imperfection on nonlinear dynamic buckling were discussed For FGM cylindrical shell surrounded by an elastic foundation, the postbuckling of shear deformable FGM cylindrical shells surrounded by an elastic medium was studied by Shen (2009a) Shen et al (2010) investigated postbuckling of internal pressure loaded FGM cylindrical shells surrounded by an elastic medium Bagherizadeh et al (2011) investigated mechanical buckling of FGM cylindrical shells surrounded by Pasternak elastic foundation Sofiyev (2010) analyzed the buckling of FGM circular shells under combined loads and resting on the Pasternak type elastic foundation Torsional vibration and stability of functionally graded orthotropic cylindrical shells on elastic foundations is presented by Najafov et al (2013) For the FGM conical shell e general case of FGM cylindrical shells, mechanic behavior of shell on elastic foundation was studied by Sofiyev (2011c), Najafov and Sofiyev (2013), Sofiyev and Kuruoglu (2013) In practice, FGM plates and shells, as other composite structures, usually reinforced by stiffeners system to provide the benefit of added load carrying capability with a relatively small additional weight Thus study on nonlinear static and dynamic behavior of theses structures are significant practical problem However, up to date, the investigation on this field has received comparatively little attention Recently, Najafizadeh et al (2009) have studied linear static buckling of FGM axially loaded cylindrical shell reinforced by ring and stringer FGM stiffeners Bich et al (2011, 2012, 2013) have investigated the nonlinear static and dynamic analysis of FGM plates, cylindrical panels and shallow shells with eccentrically homogeneous stiffener system Dung and Hoa (2013a, 2013b) presented an analytical study of nonlinear static buckling and post-buckling analysis of eccentrically stiffened functionally graded circular cylindrical shells under external pressure and torsional load with FGM stiffeners and approximate three-term solution of deflection taking into account the nonlinear buckling shape The review of the literature signifies that there are very little researches on the nonlinear dynamic analysis of FGM stiffened shells surrounded by an elastic foundation by analytical approach In this paper, the dynamic behavior of eccentrically stiffened FGM (ES-FGM) cylindrical circular shells reinforced by eccentrically ring and stringer stiffener system on internal and (or) external surface of shell under external pressure loads is investigated The nonlinear dynamic equations are derived by using the classical shell theory with the nonlinear strainedisplacement relation of large deflection, the smeared stiffeners technique and Galerkin method The present novelty is that an approximate three-term solution of deflection including the pre-buckling shape, the linear buckling shape and the nonlinear buckling shape are more correctly chosen and the frequencyeamplitude relation of nonlinear vibration is obtained in explicit form In addition, the nonlinear dynamic responses are found by using fourth order RungeeKutta method and the dynamic buckling loads of stiffened FGM shells are investigated according to BudianskyeRoth criterion The results show that the stiffener, volume-fractions index and geometrical parameters strongly influence to the dynamic behavior of shells Formulation 2.1 FGM power law properties Functionally graded material in this paper, is assumed to be made from a mixture of ceramic and metal in two cases: inside 43 ceramic surface, outside metal surface and outside ceramic surface, inside metal surface The volume-fractions is assumed to be given by a power law  Vin ẳ Vin zị ẳ  2z ỵ h k ; Vou ẳ Vou zị ẳ Vin zị; 2h (1) where h is the thickness of shell; k ! is the volume-fraction index; z is the thickness coordinate and varies from Àh/2 to h/2; the subscripts in and ou refer to the inside and outside material constituents, respectively For case of inside ceramic surface and outside metal surface Vin ¼ Vc and Vou ¼ Vm, for the case of outside ceramic surface and inside metal surface Vin ¼ Vm and Vou ¼ Vc In which, Vc is volumefraction of ceramic and Vm is volume-fraction of metal Effective properties Preff of FGM shell are determined by linear rule of mixture as Preff ẳ Prou zịVou zị ỵ Prin ðzÞVin ðzÞ: (2) According to the mentioned law, the Young’s modulus and the mass density of shell can be expressed in the form 2z ỵ hk ; 2h 2z ỵ hk ẳ rou ỵ rin rou ị ; 2h Ezị ẳ Eou Vou ỵ Ein Vin ẳ Eou ỵ Ein Eou ị rzị ẳ rou Vou ỵ rin Vin (3) For case of inside ceramic surface and outside metal surface Ein ¼ Ec, rin ¼ rc and Eou ¼ Em, rou ¼ rm, for the case of outside ceramic surface and inside metal surface Ein ¼ Em, rin ¼ rm and Eou ¼ Ec, rou ¼ rc Ec, rc, Em, rm are the Young’s modulus and the mass density of ceramic and metal, respectively 2.2 Constitutive relations and governing equations Consider a functionally graded cylindrical thin shell surrounded by an elastic foundation with length L, mean radius R and reinforced by closely spaced (Najafizadeh et al., 2009; Brush and Almroth, 1975; Reddy and Starnes, 1993) pure-metal ring and stringer stiffener systems (see Fig 1) The stiffener is located at outside surface for outside metal surface case and at inside surface for inside metal surface case The origin of the coordinate O locates on the middle surface and at the left end of the shell, x,y ¼ Rq and z axes are in the axial, circumferential, and inward radial directions respectively According to the von Karman nonlinear strainedisplacement relations (Brush and Almroth, 1975), the strain components at the middle surface of perfect circular cylindrical shells are the form   vu vw ỵ ; vx vx   vv w vw ỵ 0y ¼ ; vy R vy ε0x ¼ g0xy vu vv vw vw ỵ ỵ ; ẳ vy vx vx vy cx ¼ (4) v2 w v2 w v2 w c c ; ; ¼ ; ¼ y xy vxvy vx2 vy2 where ε0x and ε0y are normal strains, g0xy is the shear strain at the middle surface of shell, cx, cy, cxy are the change of curvatures and twist of shell, and u ¼ u(x,y), v ¼ v(x,y), w ¼ w(x,y) are displacements along x, y and z axes respectively 44 D.V Dung, V.H Nam / European Journal of Mechanics A/Solids 46 (2014) 42e53 Fig Configuration of an eccentrically stiffened cylindrical shell surrounded by an elastic medium The strains across the shell thickness at a distance z from the mid-surface are represented by εx ¼ ε0x À zcx ; εy ¼ ε0y À zcy ; gxy ¼ g0xy À 2zcxy : (5)   Em Is cx À D12 cy ; Mx ẳ B11 ỵ Cs ị0x ỵ B12 0y D11 ỵ ss   Em Ir cy ; My ẳ B12 0x ỵ B22 ỵ Cr ị0y D12 cx D22 ỵ sr (10) Mxy ¼ B66 g0xy À 2D66 cxy ; The deformation compatibility equation is derived from Eq (4) v2 0x vy2 ỵ v2 ε0y vx2 À v2 g0xy vxvy 1v w ẳ ỵ R vx2 !2 v w vxvy À 2 v wv w : vx2 vy2 (6) where Aij, Bij, Dij (i,j ¼ 1,2,6) are extensional, coupling and bending stiffness of the un-stiffened FGM cylindrical shell, Nx, Ny are inplane normal force intensities, Nxy is in-plane shearing force intensity, Mx, My are bending moment intensities and Mxy is twisting moment intensity The stressestrain relations for FGM shells are Ezị x ỵ ny ; 1n Ezị ẳ y ỵ nx ; n2 Ezị g ; ẳ 21 ỵ nị xy ssh x ¼ ssh y ssh xy (7) sh where the Poisson’s ratio n is assumed to be constant, ssh x ; sy are normal stress in x, y direction of un-stiffened shell, respectively, ssh xy is shearing stress in of un-stiffened shell The stressestrain relation is applied for homogenous stiffeners sst s ¼ Es εx ; sst r ¼ Er y ;  Em As x ỵ A12 0y B11 ỵ Cs ịcx B12 cy ; ss   Em Ar Ny ẳ A12 0x ỵ A22 ỵ y B12 cx B22 ỵ Cr ịcy ; sr  (8) A11 ỵ Nxy ẳ A66 g0xy À 2B66 cxy ; E1 ; À n2 A12 ¼ E1 n ; À n2 A66 ¼ E1 ; 21 ỵ nị B11 ẳ B22 ẳ E2 ; À n2 B12 ¼ E2 n ; n2 B66 ẳ E2 ; 21 ỵ nị D11 ¼ D22 ¼ E3 ; À n2 D12 ¼ E3 n ; À n2 D66 ¼ E3 ; 21 ỵ nị (11) with  Ein Eou Ein Eou ịkh2 h; E2 ẳ ; kỵ1 2k ỵ 1ịk ỵ 2ị !  Eou 1 ỵ h3 ; E3 ẳ ỵ Ein Eou ị k ỵ k ỵ 4k ỵ 12  st where sst s ; sr are normal stress of stringer and ring stiffeners, respectively Es, Er are Young’s modulus of stringer and ring stiffeners, respectively In this paper, the stringer and ring are assumed to be metal stiffeners, so Es ¼ Er h Em Taking into account the contribution of stiffeners by the smeared stiffeners technique and omitting the twist of stiffeners because the torsion constants are smaller more than the moment of inertia (Brush and Almroth, 1975) and integrating the stressestrain equations and their moments through the thickness of shell, the expressions for force and moment resultants of an ES-FGM cylindrical shell are of the form Nx ẳ A11 ẳ A22 ẳ (9) E1 ẳ Eou ỵ ds h3s dr h3r ỵ As z2s ; Ir ẳ þ Ar z2r ; 12 12 Em As zs Em Ar zr Cs ẳ ặ ; Cr ẳ ặ ; ss sr Is ẳ zs ẳ hs ỵ h ; zr ẳ (12) hr ỵ h ; where the coupling parameters Cs and Cr are negative for outside stiffeners and positive for inside ones The spacing of the longitudinal and transversal stiffeners is denoted by ss and sr, respectively The width and thickness of the stringer and ring stiffeners are denoted by ds, hs and dr, hr, respectively The quantities As, Ar are the cross-section areas of stiffeners and Is, Ir, zs, zr are the second moments of cross section areas and the eccentricities of stiffeners with respect to the middle surface of shell respectively D.V Dung, V.H Nam / European Journal of Mechanics A/Solids 46 (2014) 42e53 = h From the constitutive relations (9), one can obtain inversely 0x ẳ A*22 Nx A*12 Ny ỵ 0y ẳ A*11 Ny A*12 Nx ỵ g0xy ẳ A*66 þ 2B*66 cxy ; B*11 cx B*21 cx þ þ r1 ¼ B*12 cy ; B*22 cy ;  D  Em As ; ss D ;  ¼ A*22 ẳ  D A22 ỵ  Em Ar ; sr rin rou  kỵ1 h ỵ rm As Ar ỵ rm : ss sr (18) Nx ẳ v2 ; vy2 Ny ¼ v2 ; vx2 Nxy ¼ À v2 : vxvy (19) A*66 ¼ B*11 ẳ A*22 B11 ỵ Cs ị A*12 B12 ; Substituting Eq (13) into the compatibility Eq (6) and Eq (15) into the third of Eq (17), taking into account Eqs (4) and (19), yields (14) A*11 B*22 ¼ A*11 B22 ỵ Cr ị A*12 B12 ; B*12 ẳ A*22 B12 A*12 B22 ỵ Cr ị; B*21 ẳ A*11 B12 A*12 B11 ỵ Cs ị; B*66 ẳ rou ỵ As Ar ỵ rm ss sr Considering the first two of Eq (17), a stress function may be defined as ; A66    E A E A D ẳ A11 ỵ m s A22 ỵ m r A212 ; ss sr A*12 ẳ A12 A11 ỵ rzịdz ỵ rm h= (13) in which A*11 ¼ Z2 45 B66 : A66  v4 v4  * v4 v4 w þ A66 À 2A*12 þ A*22 þ B*21 4 2 vx vx vy vy vx  v4 w  v w v2 w * ỵ B*11 ỵ B*22 2B*66 ỵ B ỵ 12 vx2 vy2 vy4 R vx2 !2 v2 w v2 w v2 w5 À4 À ¼ 0; vxvy vx vy2 (20) Substituting Eq (13) into Eq (10) leads to Mx ẳ B*11 Nx ỵ B*21 Ny D*11 cx My ẳ B*12 Nx ỵ B*22 Ny D*21 cx Mxy ¼ B*66 Nxy À 2D*66 cxy ; À D*12 cy ; À D*22 cy ; r1 (15) v4 v2 v2 v2 w v2 v2 w v2 v2 w ỵ vxvy vxvy vx2 vy2 vy4 R vx2 vy2 vx2 ! v2 w v2 w q0 ỵ k1 w k2 ỵ ẳ 0: vx2 vy B*12 in which D*11 ẳ D11 ỵ D*22  v4 w v2 w vw v4 w  ỵ D*11 ỵ D*12 þ D*21 þ 4D*66 þ 2r1 ε vt vt vx vx2 vy2   4 v w v v4 ỵ D*22 B*21 B*11 ỵ B*22 2B*66 vy vx vx2 vy2 Em Is B11 ỵ Cs ịB*11 B12 B*21 ; ss Em Ir ẳ D22 ỵ B12 B*12 B22 ỵ Cr ịB*22 ; sr D*12 ẳ D12 B11 ỵ Cs ịB*12 B12 B*22 ; (21) (16) D*21 ẳ D12 B12 B*11 B22 ỵ Cr ÞB*21 ; Eqs (20) and (21) are a nonlinear equation system in terms of two dependent unknowns w and They are used to investigate the dynamic characteristics of ES-FGM circular cylindrical shells D*66 ¼ D66 À B66 B*66 : Dynamic Galerkin method approach The nonlinear equations of motion of a thin circular cylindrical shell based on the classical shell theory and the assumption (Darabi et al., 2008; Sofiyev and Schnack, 2004; Volmir, 1972) u