Composite Structures 96 (2013) 384–395 Contents lists available at SciVerse ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct Nonlinear dynamic analysis of eccentrically stiffened imperfect functionally graded doubly curved thin shallow shells Dao Huy Bich a, 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 Article history: Available online 29 October 2012 Keywords: Functionally graded material Dynamic analysis Critical dynamic buckling load Vibration Shallow shells Stiffeners a b s t r a c t This paper presents a semi-analytical approach to investigate the nonlinear dynamic of imperfect eccentrically stiffened functionally graded shallow shells taking into account the damping subjected to mechanical loads The functionally graded shallow shells are simply supported at edges and are reinforced by transversal and longitudinal stiffeners on internal or external surface The formulation is based on the classical thin shell theory with the geometrical nonlinearity in von Karman–Donnell sense and the smeared stiffeners technique By Galerkin method, the equations of motion of eccentrically stiffened imperfect functionally graded shallow shells are derived Dynamic responses are obtained by solving the equation of motion by the Runge–Kutta method The nonlinear critical dynamic buckling loads are found according to the Budiansky–Roth criterion Results of dynamic analysis show the effect of stiffeners, damping, pre-loaded compressions, material and geometric parameters on the dynamical behavior of these structures Ó 2012 Elsevier Ltd All rights reserved Introduction Eccentrically stiffened shallow shell is a very important structure in engineering design of aircraft, missile and aerospace industries There are many researches on the static and dynamic behavior of this structure with different materials Studies on the dynamics first were carried out with eccentrically stiffened shallow shells made of homogeneous material Khalil et al [1] presented a finite strip formulation for the nonlinear analysis of stiffened plate structures subjected to transient pressure loadings using an explicit central difference/diagonal mass matrix time stepping method Shen and Dade [2] investigated dynamic analysis of stiffened plates and shells using spline gauss collocation method The free vibration of stiffened shallow shells was studied by Nayak and Bandyopadhyay [3] By using the finite element method, the stiffened shell element was obtained by the appropriate combinations of the eight-/nine-node doubly curved isoparametric thin shallow shell element with the three-node curved isoparametric beam element Sheikh and Mukhopadhyay [4] applied the spline finite strip method to investigate linear and nonlinear transient vibration analysis of plates and stiffened plates The von Karman’s large deflection plate theory has been used and the formulation was done in total Lagrange coordinate system Dynamic instability analysis of stiffened shell panels subjected to uniform in-plane harmonic edge loading and partial edge ⇑ Corresponding author E-mail address: hoainam.vu@utt.edu.vn (V.H Nam) 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.compstruct.2012.10.009 loading along the edges was studied by Patel et al [5–8] In these studies, the new formulation for the beam element requires five degrees of freedom per node as that of shell element For dynamic analysis of eccentrically stiffened laminated composite plates and shallow shells, Satish Kumar and Mukhopadhyay [9] studied the transient response analysis of laminated stiffened plates using the first order shear deformation theory Parametric study on the dynamic instability behavior of laminated composite stiffened plate was studied by Patel et al [10] The same authors [11] investigated the dynamic instability of laminated composite stiffened shell panels subjected to in-plane harmonic edge loading By using the commercial ANSYS finite element software, Less and Abramovich [12] studied the dynamic buckling of a laminated composite stringer stiffened cylindrical panel Bich et al [13] presented an analytical approach to investigate the nonlinear dynamic of imperfect reinforced laminated composite plates and shallow shells using the classical thin shell theory with the geometrical nonlinearity in von Karman–Donnell sense and the smeared stiffeners technique For functionally graded materials (FGM), many researches focused on the dynamical analysis of un-stiffened shallow shells Liew et al [14] presented the nonlinear vibration analysis of the coating FGM substrate cylindrical panel subjected to a temperature gradient arising from steady heat conduction through the panel thickness Matsunaga et al [15] investigated free vibrations and stability of FGM doubly curved shallow shells according to a 2-D higher order deformation theory Chorfi and Houmat [16] investigated nonlinear free vibrations of FGM doubly curved shallow 385 D.H Bich et al / Composite Structures 96 (2013) 384–395 shells with an elliptical plan-form Nonlinear vibrations of functionally graded doubly curved shallow shells under a concentrated force were studied by Alijani et al [17] Nonlinear dynamical analysis of imperfect functionally graded material shallow shells subjected to axial compressive load and transverse load was studied by Bich and Long [18], Dung and Nam [19] The motion, stability and compatibility equations of these structures were derived using the classical shell theory The nonlinear transient responses of cylindrical and doubly-curved shallow shells subjected to excited external forces were obtained and the dynamic critical buckling loads were evaluated based on the displacement responses using Budiansky–Roth dynamic buckling criterion Recently, some authors have studied static and dynamical behaviors of some kind of shells Najafizadeh et al [20] have studied static buckling behaviors of FGM cylindrical shell Bich et al [21] have studied the nonlinear static post-buckling of eccentrically stiffened imperfect functionally graded plates and shallow shells The nonlinear dynamical analysis of imperfect eccentrically stiffened FGM cylindrical panels based on the classical theory with the von Karman–Donnell geometrical nonlinearity are investigated by Bich et al [22] Following the idea of work [22], in this paper, dynamic governing equations taking into account the effect of damping, nonlinear vibration and dynamic critical buckling loads of eccentrically stiffened imperfect FGM doubly curved thin shallow shells are established Effects of stiffeners, material, geometric parameters and damping on the dynamic behavior of structure are considered V c zị ẳ  k 2z ỵ h ; 2h V m zị ẳ À V c ðzÞ; ð1Þ where k is the volume fraction exponent (k P 0), z is the thickness coordinate and varies from Àh/2 to h/2; the subscripts m and c refer to the metal and ceramic constituents respectively Effective properties Preff of FGM shell are determined by linear rule of mixture as Preff ẳ Prm zịV m zị þ Prc ðzÞV c ðzÞ: ð2Þ According to Eqs (1) and (2), the modulus of elasticity E and the mass density q can be expressed in the form  k 2z ỵ h ; 2h  k 2z ỵ h qzị ẳ qm V m ỵ qc V c ẳ qm ỵ qc qm ị : 2h Ezị ẳ Em V m ỵ Ec V c ẳ Em ỵ ðEc À Em Þ ð3Þ Poisson’s ratio m and linear damping coefficient e are assumed to be constants Assume that the shell is reinforced by eccentrically longitudinal and transversal homogeneous stiffeners with the elastic modulus E0 and the mass density q0 of stiffeners In order to provide the continuity between the shell and stiffeners, the full metal stiffeners are put at the metal-rich side of the shell thus E0 and q0 take the value E0 = Em, q0 = qm and conversely the full ceramic ones at the ceramic-rich side, so that E0 = Ec, q0 = qc Theoretical formulation Eccentrically stiffened FGM shallow shells (ES-FGM shallow shells) Consider a doubly curved functionally graded shallow thin shell (see Fig 1) of thickness h and in-plane edges a and b The shallow shell is assumed to have a relative small rise as compared with its span Let the (x1,x2) plane of the Cartesian coordinates overlaps the rectangular plane area of the shell Note that the middle surface of the shell generally is defined in terms of curvilinear coordinates, but for the shallow shell, so the Cartesian coordinates can replace the curvilinear coordinates on the middle surface The volume fractions of constituents are assumed to vary through the thickness according to the following power law distribution The strains at the middle surface and curvatures relating to the displacement components u, v, w based on the classical shell theory and von Karman–Donnell geometrical nonlinearity assumption are of the form [24]  2 @u @w k1 w ỵ ; @x1 @x1  2 @v @w ¼ À k2 w ỵ ; @x2 @x2 e01 ẳ v1 ẳ @2w ; @x21 e02 v2 ¼ @2w ; @x22 c012 ẳ @u @ v @w @w ỵ ỵ ; @x2 @x1 @x1 @x2 v12 ẳ 4ị @2w ; @x1 @x2 in which k1 ¼ R11 ; k2 ¼ R12 and R1, R2 are curvatures and radii of curvatures of the shell respectively Fig Configuration of an eccentrically stiffened shallow shell 386 D.H Bich et al / Composite Structures 96 (2013) 384–395 The strain components across the shell thickness at a distance z from the middle surface are given by e1 ¼ e01 À zv1 ; e2 ¼ e02 À zv2 ; c12 ẳ c012 2zv12 : 5ị From Eq (4), the strains must be relative in the deformation compatibility equation @ e01 @ e02 @ c012 ỵ ẳ @x22 @x1 @x1 @x2 @2w @x1 @x2 !2 À @2w @2w @2w @2w À k1 À k2 : @x21 @x22 @x2 @x1 ð6Þ Hooke’s stress–strain relation is applied for the shell rsh ẳ r sh Ezị e1 ỵ me2 ị; m2 Ezị ẳ e2 ỵ me1 ị; m2 ssh 12 ẳ 7ị Ezị c ; 21 ỵ mị 12 r ẳ E0 e1 ; r ẳ E0 e2 : ð8Þ Taking into account the contribution of stiffeners by the smeared stiffeners technique and omitting the twist of stiffeners because these torsion constants are smaller more than the moments of inertia [24] and integrating the stress–strain equations and their moments through the thickness of the panel, lead to the expressions for force and moment resultants of an ES-FGM shallow shells as [22]   E A1 N ẳ A11 ỵ e1 ỵ A12 e02 B11 ỵ C ịv1 B12 v2 ; s1   E A2 e2 À B12 v1 B22 ỵ C ịv2 ; N ẳ A12 e01 ỵ A22 ỵ s2     E0 A1 E A2 ; Ẫ22 ¼ ; A11 ỵ A22 ỵ D s1 D s2 A12 Ẫ12 ¼ ; Ẫ66 ¼ ; A66 D    E A1 E0 A2 A22 ỵ A212 ; D ẳ A11 ỵ s1 s2 B22 ẳ A11 B22 ỵ C ị A12 B12 ; B12 ẳ A22 B12 A12 B22 ỵ C ị; B21 ẳ A11 B12 A12 B11 ỵ C ị; B66 : A66 M ẳ B11 N1 þ BÃ21 N2 À DÃ11 v1 À DÃ12 v2 ; 15ị where E1 E1 m E1 ; A12 ẳ ; A66 ¼ ; À m2 À m2 2ð1 ỵ mị E2 E2 m E2 ; B12 ẳ ; B66 ¼ B11 ¼ B22 ¼ ; À m2 m2 21 ỵ mị E3 E3 m E3 ; D12 ¼ ; D66 ¼ D11 ¼ D22 ¼ ; m2 m2 21 ỵ mị E0 I1 B11 ỵ C ịB11 B12 B21 ; s1 D22 ẳ D22 ỵ E0 I2 B12 B12 B22 ỵ C ịB22 ; s2 D12 ẳ D12 B11 ỵ C ịB12 B12 B22 ; 11ị D21 ẳ D12 B12 B11 B22 ỵ C ịB21 ; Based on the classical shell theory and the Volmir’s assumption 2 [23] u ( w and v ( w; q1 @@t2u ! and q1 @@t2v ! 0, the nonlinear motion equations of a shallow thin shell with damping force is written in the form @N1 @N12 ỵ ẳ 0; @x1 @x2 with @N12 @N2 ỵ ẳ 0; @x1 @x2 E1 ẳ ð12Þ @ M1 @ M 12 @ M @2w @2w @2w ỵ2 ỵ ỵ N1 þ 2N12 þ N2 2 @x1 @x2 @x1 @x2 @x1 @x2 @x1 @x2 d2 h2 ỵ A2 z22 : 12 16ị D66 ẳ D66 B66 B66 : A11 ¼ A22 ¼   Ec À Em Ec Em ịkh h; E2 ẳ Em ỵ ; kỵ1 2k ỵ 1ịk ỵ 2ị  ! Em 1 ỵ h ; ỵ Ec Em ị E3 ẳ k ỵ k ỵ 4k ỵ 12 D11 ẳ D11 ỵ 10ị where Aij, Bij, Dij (i, j = 1, 2, 6) are extensional, coupling and bending stiffness of the un-stiffened shell I2 ¼ BÃ66 ¼ M 12 ¼ BÃ66 N12 À 2DÃ66 v12 ; M 12 ¼ B66 c012 À 2D66 v12 ; 14ị B11 ẳ A22 B11 ỵ C ị A12 B12 ; M ẳ B12 N1 ỵ B22 N2 D21 v1 D22 v2 ; c À 2B66 v12 ;   E0 I M ẳ B11 ỵ C ịe01 ỵ B12 e02 D11 ỵ v1 D12 v2 ; s1   E0 I2 M ¼ B12 e01 þ ðB22 þ C Þe02 À D12 v1 À D22 ỵ v2 ; s2 and 13ị Substituting Eq (13) into Eq (10) yields ð9Þ A66 012 d h1 þ A1 z21 ; 12 e01 ¼ Ẫ22 N1 À A12 N2 ỵ B11 v1 ỵ B12 v2 ; e02 ẳ A11 N2 A12 N1 ỵ B21 v1 ỵ B22 v2 ; c012 ẳ A66 ỵ 2B66 v12 ; AÃ11 ¼ st st I1 ¼ where the coupling parameters C1, C2 are negative for outside stiffeners and positive for inside ones, s1 and s2 are the spacing of the longitudinal and transversal stiffeners, A1, A2 are the cross-section areas of stiffeners, I1,I2 are the second moments of cross-section areas, z1, z2 are the eccentricities of stiffeners with respect to the middle surface of shell, and the width and thickness of longitudinal and transversal stiffeners are denoted by d1, h1 and d2, h2, respectively For later use, the strain-force resultant reverse relations are obtained from Eq (9) where and for stiffeners N 12 ¼ E A1 z E A2 z ; C2 ẳ ặ ; s1 s2 h1 ỵ h h2 ỵ h z1 ¼ ; z2 ¼ ; 2 C1 ¼ Ỉ þ k1 N1 þ k2 N þ q0 ¼ q1 @2w @w ; ỵ 2q1 e @t @t where e is damping coefficient and ð17Þ 387 D.H Bich et al / Composite Structures 96 (2013) 384–395   A1 A2 ỵ s1 s2 h=2     qc qm A1 A2 : ỵ ẳ qm ỵ h ỵ q0 kỵ1 s1 s2 Z q1 ẳ h=2 The couple of nonlinear Eqs (19) and (20) or Eqs (21) and (22) in terms of two dependent unknowns w and u are used to investigate the nonlinear vibration and dynamic stability of ES-FGM shells qzịdz ỵ q0 The rst two of Eq (16) are satisfied automatically by introducing a stress function u as 2 @ u ; @x22 N1 ¼ N2 ¼ @ u ; @x21 N12 ¼ À @ u : @x1 @x2 ð18Þ Substituting Eq (13) into the compatibility Eqs (6) and (15) into the third of Eq (17), taking into account expressions (4) and (18), yields AÃ11 Á @4u @4u À à @4u @4w þ A66 À 2AÃ12 þ AÃ22 þ BÃ21 4 2 @x1 @x1 @x2 @x2 @x1 ỵ ẳ B11 ỵ B22 @2w @x1 @x2 2B66 !2 À An imperfect ES-FGM shallow thin shell considered in this paper is assumed to be simply supported and subjected to uniformly distributed pressure of intensity q0 and axial compression of intensities r0 and p0 respectively at its cross-section Thus the boundary conditions are w ¼ 0; M ¼ 0; N1 ¼ Àr h; N12 ¼ 0; at x1 ¼ 0; a; w ¼ 0; M ¼ 0; N2 ¼ Àp0 h; N12 ¼ 0; at x2 ¼ 0; b: @4w @4w @2w @2w ỵ B12 þ k1 þ k2 2 @x1 @x2 @x2 @x2 @x1 @2w @2w ; @x21 @x22 ð23Þ ð19Þ The mentioned conditions (23) can be satisfied identically if the buckling mode shape is chosen by w ẳ f tị sin @4w @2w @w @4w q1 ỵ 2q1 e ỵ D11 ỵ D12 ỵ D21 ỵ 4DÃ66 2 @t @x1 @x1 @x2 @t 4 À Á @ w @ u @ u @4u þ DÃ22 À BÃ21 À BÃ11 þ BÃ22 À 2BÃ66 À BÃ12 @x1 @x21 @x22 @x2 @x2 À k1 Nonlinear dynamic analysis @2u @2u @2u @2w @2u @2w @2u @2w k2 ỵ2 À ¼ q0 : 2 @x1 @x2 @x1 @x2 @x1 @x22 @x2 @x1 @x2 @x1 ð20Þ mpx1 npx2 sin ; a b ð24Þ where f(t) is time dependent total amplitude and m, n are numbers of haft waves in x1 and x2 directions respectively The initial imperfection w0 is assumed to have the same form of the shell deflection w, i.e w0 ¼ f0 sin mpx1 npx2 sin ; a b ð25Þ For an initial imperfection shell: The initial imperfection of the shell considered here can be seen as a small deviation of the shell middle surface from the perfect shape, also seen as an initial deflection which is very small compared with the shell dimensions, but may be compared with the shell wall thickness Let w0 = w0(x1, x2) denote a known small imperfection, proceeding from the motion Eqs (19) and (20) of a perfect FGM shallow shell and following to the Volmir’s approach [23] for an imperfection shell, we can formulate the system of motion equations for an imperfect eccentrically stiffened functionally graded shallow shell (Imperfect ES-FGM shallow shell) as where f0 is the known initial amplitude Substituting Eqs (24) and (25) into Eq (21) and solving the resulting equation for unknown u yield Á @4u @4u À @4u @ ðw À w0 Þ Ẫ11 ỵ A66 2A12 ỵ A22 ỵ B21 2 @x1 @x1 @x2 @x2 @x41 À Á @ ðw À w0 Þ @ ðw À w0 ị ỵ B11 ỵ B22 2B66 ỵ B12 2 @x1 @x2 @x42 !2 @ ðw À w0 Þ @ ðw À w0 Þ @ w @ w @ w5 ỵ k1 ỵ k2 @x1 @x2 @x22 @x21 @x1 @x22 !2 @ w0 @ w0 @ w0 ẳ 0; 21ị ỵ4 @x1 @x2 @x21 @x22 u1 ¼ 2mpx1 2npx2 mpx1 npx2 þ u2 cos À u3 sin sin a b a b x22 x21 À r h À p0 h ; 2 u ¼ u1 cos where n k2 f m2 f ; à ; u2 ¼ 32m2 A11 32n2 k2 AÃ22 h  i À B21 m4 ỵ B11 ỵ B22 2B66 m2 n2 k2 ỵ B12 n4 k4 pa k1 n2 k2 ỵ k2 m2 u3 ẳ : A11 m4 ỵ A66 2A12 m2 n2 k2 ỵ A22 n4 k4 27ị Substituting the expressions (24)–(26) into Eq (22) and applying the Galerkin’s method to the obtained equation Z U  q1 À 2 where w is a total deflection of shell U sin mpx1 npx2 sin dx1 dx2 ¼ a b p @ u@ w @ u @ w @ u@ w ỵ2 q0 ẳ 0; @x1 @x2 @x1 @x2 @x21 @x22 @x22 @x21 a a2 h ỵ H f f02 ỵ K f f02 f r m2 ỵ p0 n2 k2 f À Á @4u @2u @2u à @ u B11 ỵ B22 2B66 B k À k 12 @x21 @x22 @x42 @x22 @x21 Z ! B2 8mnk2 B ðf À f0 Þ þ M€f þ 2Mef_ þ D þ d1 d2 ðf À f0 Þf A A 3p2 @ ðw À w0 ị @ w w0 ị @4u ỵ DÃ22 À BÃ21 2 @x1 @x2 @x2 @x1 b lead to Á @2w @w @ w w0 ị ỵ D11 ỵ 2q1 e ỵ D12 ỵ D21 ỵ 4D66 @t @x1 @t ð26Þ ð22Þ 4a4 h 4a4 d1 d2 k1 r ỵ k2 p0 ị q0 d1 d2 ẳ 0; ỵ mnp mnp6 where the coefcients are given by ð28Þ 388 D.H Bich et al / Composite Structures 96 (2013) 384–395 M¼ a4 p4 À Á q1 ; A ẳ A11 m4 ỵ A66 2A12 m2 n2 k2 ỵ A22 n4 k4 ; a2 B ẳ B21 m4 ỵ B11 ỵ B22 2B66 m2 n2 k2 ỵ B12 n4 k4 k1 n2 k2 ỵ k2 m2 ; p D ẳ D11 m4 ỵ D12 ỵ D21 þ 4DÃ66 m2 n2 k2 þ DÃ22 n4 k4 ; !   2mnk2 BÃ21 BÃ12 a2 k2 n2 k2 k1 m2 d Hẳ ỵ d ỵ 29ị d1 d2 ; 3p2 AÃ11 AÃ22 6p4 mn AÃ11 AÃ22 ! m n k4 a K¼ ỵ ;k ẳ ;d1 ẳ 1ịm 1; d2 ẳ ðÀ1Þn À 1: 16 Ẫ22 Ẫ11 b The governing Eq (28) is a basic equation for dynamic analysis of imperfect ES-FGM doubly-curved shallow thin shells in general Based on this equation, the nonlinear vibration of perfect and imperfect FGM shallow shells can be investigated and the dynamic buckling analysis of shells under various loadings can be performed Particularly for a spherical panel k1 = k2 = 1/R, for a cylindrical panel k1 = 0, k2 = 1/R, and for a plate k1 = k2 = are taken in Eqs (28) and (29) 4.1 Nonlinear vibration Consider an imperfect ES-FGM shallow shell under uniformly lateral pressure q0 = Q sin Xt and pre-loaded compressions r0, p0, Eq (28) becomes M€f þ 2Mef_ þ D þ ! B2 8mnk2 B d1 d2 f f0 ịf f f0 ị ỵ A 3p2 A p ð30Þ By using this equation, the fundamental frequencies of natural vibration of ES-FGM shell and FGM shell without stiffeners, and frequency-amplitude relation of nonlinear vibration and nonlinear response of ES-FGM shell are taken into consideration The nonlinear dynamical responses of ES-FGM shells can be obtained by solving Eq (30) by the fourth order Runge–Kutta iteration method with the time step Dt and with initial conditions to be assumed as f 0ị ẳ 0; f_ 0ị ẳ If the vibration is free and linear, and without damping, Eq (30) reduces to Mf ỵ D ỵ ! B2 f f0 ị ẳ 0: A 31ị The fundamental frequencies of natural vibration of imperfect ES-FGM shallow shells can be determined by xmn ð32Þ If the shell is perfect ES-FGM and the vibration is nonlinear forced vibration without pre-loaded compressions r0 = p0 = 0, Eq (30) can be rewritten a f ỵ 2ef_ ỵ x2 f ỵ H1 f ỵ H2 f ẳ F sinXtị; mn 33ị in which H1 ẳ 8mnk2 B d d 3p2 A M x2mn ỵH 4e p  X ẳ x2mn ỵ  8H1 3H F g ỵ g2 ; 3p g 35ị where g is the amplitude of nonlinear vibration By introducing the non-dimension frequency parameter n ¼ xXmn , Eq (35) becomes n2   8H1 3H F nẳ 1ỵ g ỵ g2 : pxmn 3p gxmn 4e ð36Þ In the case of the nonlinear forced vibration without damping, Eq (36) leads to n2 ¼ þ 8H1 3H F g þ g2 À : 3p gxmn ð37Þ If F = and without damping, the frequency–amplitude relation of the nonlinear free vibration, from Eq (37), is as n2 ẳ ỵ 8H1 3H g ỵ g2 : 3p 38ị 4.2 Nonlinear dynamic buckling Investigate the nonlinear dynamic buckling analysis of imperfect ES-FGM doubly-curved panels under lateral pressure varying as linear function of time q0 = ct (c is a loading speed) and preloaded compressions r0 = const, p0 = const, Eq (28) becomes À Á À Á Á a2 h ỵ H f f02 ỵ K f f02 f r m2 ỵ p0 n2 k2 f p 4a4 h 4a4 ỵ d1 d2 k1 r ỵ k2 p0 ị ẳ d d ct: mnp p mn ð39Þ By solving Eq (39), the dynamic critical time tcr can be obtained according to Budiansky–Roth criterion [25] This criterion is based on that, for large value of loading speed, the amplitude-time curve of obtained displacement response increases sharply depending on time and this curve obtain a maximum by passing from the slope point and at the corresponding time t = tcr the stability loss occurs Here t = tcr is called critical time and the load corresponding to this critical time is called dynamic critical buckling load qdcr = ctcr For static buckling analysis of ES-FGM shallow shell, the explicit expressions of the upper and lower buckling load were obtained in [21] Numerical results and discussions 5.1 Validation of the present formulation vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u u1 B2 t : Dỵ ẳ M A  X2 ! B2 8mnk2 B _ € d1 d2 ðf À f0 Þf Mf þ 2M ef þ D þ ðf À f0 Þ þ A 3p2 A À Á À Á Á a2 h ỵ H f f02 ỵ ỵK f f02 f r0 m2 ỵ p0 n2 k2 f 4a4 h 4a4 ỵ d1 d2 k1 r0 ỵ k2 p0 ị ẳ d d Q sin Xt: mnp p mn Seeking solution as f(t) = g sin (Xt) and applying procedure like Galerkin method to Eq (33), the frequency–amplitude relation of nonlinear forced vibration is obtained  ; H2 ¼ K M x2mn ; F¼ 4a4 d1 d2 Q : Mp6 mn ð34Þ To validate the present study, the fundamental frequency parameter of unstiffened FGM shallow shells is compared with other studies Table shows the present results in comparison with those presented by Matsunaga [15] based on the two-dimensional (2D) higher-order theory, Chorfi and Houmat [16] accorded to the first-order shear deformation theory and Alijani et al [17] used Donnell’s nonlinear shallow shell theory In p this comparison, the ffiffiffiffiffiffiffiffiffiffiffiffi ~ ¼ xmn h qc =Ec of the perfect fundamental frequency parameter x un-stiffened FGM shallow shell (a/b = 1, h/a = 0.1) with simply supported edges The material properties are Aluminum and Alumina, i.e Em = 70  109 N/m2; qm = 2702 kg/m3 and Ec = 380  109 N/m2; qc = 3800 kg/m3 respectively The Poisson’s ratio is chosen to be 389 D.H Bich et al / Composite Structures 96 (2013) 384–395 Table ~ with results reported by Matsunaga [15], Chorfi and Houmat [16] Comparison of x and Alijani et al [17] b/R2 a/R1 k Present Ref [15] Ref [16] Ref [17] 0.5 10 0.0597 0.0506 0.0456 0.0396 0.0381 0.0588 0.0492 0.0430 0.0381 0.0364 0.0577 0.0490 0.0442 0.0383 0.0366 0.0597 0.0506 0.0456 0.0396 0.0380 FGM spherical panel 0.5 0.5 0.5 10 0.0779 0.0676 0.0617 0.0520 0.0482 0.0751 0.0657 0.0601 0.0503 0.0464 0.0762 0.0664 0.0607 0.0509 0.0471 0.0779 0.0676 0.0617 0.0519 0.0482 FGM cylindrical panel 0.5 0.5 10 0.0648 0.0553 0.0501 0.0430 0.0409 0.0622 0.0535 0.0485 0.0413 0.0390 0.0629 0.0540 0.0490 0.0419 0.0395 0.0648 0.0553 0.0501 0.0430 0.0408 FGM hyperbolic paraboloidal panel À0.5 0.5 0.0597 0.5 0.0506 0.0456 0.0396 10 0.0381 0.0563 0.0479 0.0432 0.0372 0.0355 0.0580 0.0493 0.0445 0.0385 0.0368 0.0597 0.0506 0.0456 0.0396 0.0380 FGM plate 0 0.3 As can be seen, a very good agreement is obtained in the comparison with the result of Ref [17], but there are a little differences with those of Refs [15,16] because they use above mentioned other theories 5.2 Nonlinear vibration results To illustrate the proposed approach of eccentrically stiffened FGM shallow shells, the stiffened and un-stiffened FGM shallow shells are considered with in-plane edges a = b = 0.8 m; h = 0.01 m; The shells are simply supported at all its edges The combination of materials consists of Aluminum Em = 70  109 N/ m2, qm = 2702 kg/m3 and Alumina Ec = 380  109 N/m2 qc = 3800 kg/m3 The Poisson’s ratio is chosen to be 0.3 for simplicity Material of stiffeners has elastic modulus Es1 = Es1 = 380  109 N/m2, qs1 = qs2 = 3800 kg/m3 The height of stiffeners is equal to 50 mm, its width 2.5 mm, the spacing of stiffeners s1 = s2 = 0.1 m The obtained results in Table show that effects of stiffeners on the fundamental frequencies of natural vibration are considerable Obviously the natural fundamental frequencies of un-stiffened and stiffened FGM spherical panels observed to be dependent on the constituent volume fractions, they decrease when increasing the power index k, furthermore with greater value k the effect of stiffeners is observed to be stronger This is completely reasonable due to the lower value of the elasticity modulus of the metal constituent The natural frequencies of stiffened spherical panels are greater than one of un-stiffened panels Table shows high frequencies of natural vibration of spherical panels with R1 = R2 = m, k = Clearly, all modes of stiffened spherical panel are greater than ones of un-stiffened panel Especially, the difference is larger with higher modes Table shows the fundamental frequencies of doubly curved FGM shallow shells In this case, with k = 1, It seem that the natural frequencies of un-stiffened panels depend to the value jk1 + k2j, i.e they have the same value for all panels of same jk1 + k2j, the natural frequencies of un-stiffened panels increases when this value increases, when k1 + k2 = (R1 = 5, R2 = À5 or R1 = À5, R2 = 5) the nat- Table Fundamental frequencies of natural vibration (rad/s) of spherical panels R1 = R2 k Un-stiffened Stiffened 0.2 10 3291.88 2863.12 2250.15 2060.96 3446.03 3112.45 2687.97 2579.30 0.2 10 2093.17 1809.95 1436.76 1325.01 2724.96 2560.43 2376.61 2339.12 10 0.2 10 1287.67 1095.27 893.94 839.28 2412.76 2348.57 2294.78 2292.47 (plate) 0.2 10 866.22 712.41 614.33 594.95 2388.11 2365.91 2360.03 2359.30 Table Frequencies of natural vibration (rad/s) of spherical panels with R1 = R2 = m, k = x1(m = 1, n = 1) x2(m = 1, n = 2) and (m = 2, n = 1) x3(m = 2, n = 2) x4(m = 1, n = 3) and (m = 3, n = 1) x5(m = 2, n = 3) and (m = 3, n = 2) Un-stiffened Stiffened 1809.95 2437.29 3299.81 3931.47 4920.50 2560.43 6743.12 9309.30 14576.20 16063.76 ural frequencies of un-stiffened panels are equal to the natural frequencies of un-stiffened plates (see Tables and 4), but as can be seen, this phenomenon is not observed for stiffened panels For stiffened panels of the same value jk1 + k2j have different natural frequencies Except in special cases when k1 + k2 = 0, the natural frequencies of stiffened panels have the same value with stiffened respective plate The frequency–amplitude curves of nonlinear vibration of FGM spherical panels without damping are presented in Fig This figure shows that the frequency–amplitude curves of forced vibration are asymptotic with the frequency–amplitude curve of free vibration and the extreme point of stiffened panel is greater than one of un-stiffened panel In forced vibration, a value of n corresponds to the maximum five distinguished values of g Fig shows the effect of excitation force Q on the frequency– amplitude curves of nonlinear vibration of spherical panels As can be seen, when the excitation force decreases, the curves of forced vibration are closer to the curve of free vibration Figs and show the effect of volume-fraction index and radius of panels on the frequency–amplitude curve of ES-FGM spherical panel without damping Clearly, the extreme points of frequency–amplitude curve decrease when the volume-fraction index or the radius decreases Four cases of Gauss curvature of stiffened and un-stiffened panels are taken into consideration: k1k2 > and k1 + k2 > when R1 = m, R2 = 10 m, k1k2 > and k1 + k2 < when R1 = À3 m, R2 = À10 m, k1k2 < and k1 + k2 > when R1 = m, R2 = À10 m and k1k2 < and k1 + k2 < when R1 = À3 m, R2 = 10 m (Figs 6–9) It seems that the frequency–amplitude curve depends on the value k1 + k2 The minimal point of frequency–amplitude curve increases when this value decreases Especially, when k1 + k2 < 0, the frequency–amplitude curve of free vibration does not exist extreme points (see Figs and 9) 390 D.H Bich et al / Composite Structures 96 (2013) 384–395 Table Fundamental frequencies of natural vibration (rad/s) of doubly curved shallow shells jR1j jR2j k R1 > 0, R2 > R1 > 0, R2 < R1 < 0, R2 > R1 < 0, R2 < Unstiffened Stiffened Unstiffened Stiffened Unstiffened Stiffened Unstiffened Stiffened 0.2 10 2684.30 2330.04 1837.47 1686.97 3052.89 2805.76 2506.85 2436.09 1074.14 902.84 751.55 713.94 2370.00 2327.75 2300.22 2304.56 1074.14 902.84 751.55 713.94 2539.60 2505.03 2480.60 2484.22 2684.30 2330.04 1837.47 1686.97 3556.80 3361.46 3119.90 3062.03 10 0.2 10 2238.69 1938.18 1535.30 1413.86 2799.50 2614.99 2403.89 2358.66 1409.22 1203.97 975.40 911.55 2446.54 2368.80 2298.28 2291.62 1409.22 1203.97 975.40 911.55 2728.14 2666.60 2606.00 2599.46 2238.69 1938.18 1535.30 1413.86 3247.06 3102.05 2929.40 2891.28 5 0.2 10 2093.17 1809.95 1436.76 1325.01 2724.96 2560.43 2376.61 2339.12 866.22 712.41 614.33 594.95 2388.11 2365.91 2360.03 2359.30 866.22 712.41 614.33 594.95 2388.11 2365.91 2360.03 2359.30 2093.17 1809.95 1436.76 1325.01 3150.15 3020.95 2869.74 2837.71 Fig The frequency–amplitude curve of nonlinear vibration of FGM spherical panels (R = m, k = 1, Q = 105 N/m2) Fig Effect of excitation force Q on the frequency–amplitude curve of stiffened spherical panel (R = m, k = 1) Fig 10 investigate the frequency–amplitude curve of nonlinear vibration of stiffened plate and shallow shells with jR1j = jR2j, the phenomenon is similar with Figs 6–9, however when k1 + k2 = with R1 = À5, R2 = and R1 = 5, R2 = À5 the frequency–amplitude curves coincide to the frequency–amplitude curve of plate Consider the un-stiffened and stiffened perfect FGM spherical panel without damping under the uniformly harmonic load q0 (t) = Q sin (Xt), nonlinear responses are obtained solving the Eq (30) by fourth order Runge–Kutta method with the time step Dt Fig Effect of index k on the frequency–amplitude curves of stiffened spherical panels (R = m, Q = 105 N/m2) Nonlinear responses of un-stiffened FGM spherical panel with difference time steps are presented in Fig 11 As can be observed, difference of nonlinear responses of time steps Dt =  10À4 and Dt = 10À4 is very small Therefore, the next results are calculated with time step Dt = 10À4 to ensure the accuracy of this method Nonlinear responses of stiffened and un-stiffened functionally graded spherical panel with k = 1, R = are presented in Figs 12 and 13 Natural frequencies of un-stiffened and stiffened spherical panel are 1809.95 rad/s and 2560.43 rad/s, respectively (see Table 2) The excitation frequencies are much smaller (Fig 12, q0(t) = 105 sin (100t)) and much greater (Fig 13, q0(t) = 105 sin (104t)) than natural frequencies These results show that the stiffeners strongly decrease vibration amplitude of the shell when excitation frequencies are much smaller or much greater than natural frequencies When the excitation frequencies are near to natural frequencies, the interesting phenomenon is observed like the harmonic beat phenomenon of a linear vibration (Figs 14 and 15) The excitation frequencies are 2510 rad/s and 2530 rad/s which are very near to natural frequencies 2348.57 rad/s of stiffened spherical panel These results show that the amplitude of beats of stiffened panels increases rapidly when the excitation frequency approaches the natural frequencies The maximal amplitude of harmonic beat increases and the response time of beat decreases when the excitation force increases as shown in Fig 15 The deflection–velocity relation has the closed curve form as in Fig 16 Deflection f and velocity f_ are equal to at initial time and final time of beat and the contour of this relation corresponds to the period which has the greatest amplitude of beat D.H Bich et al / Composite Structures 96 (2013) 384–395 Fig Effect of radius R on the frequency–amplitude curves of stiffened spherical panels (k = 1, Q = 105 N/m2) Fig The frequency–amplitude curve of nonlinear vibration of un-stiffened shallow shells (k = 1, Q = 105 N/m2) Fig The frequency–amplitude curve of nonlinear vibration of un-stiffened shallow shells (k = 1, Q = 105 N/m2) Figs 17 and 18 show the effect of pre-loaded compressions and of known initial amplitude on the nonlinear responses of ES–FGM spherical panel In these investigations, pre-loaded compressions and known initial amplitude slightly influence on the amplitude of nonlinear vibration of panels Effect of damping on nonlinear responses is presented in Figs 19 and 20 with linear damping coefficient e = 0.3 The damping influences very small to the nonlinear response in the first vibration periods (Fig 19) however it strongly decreases amplitude at the next far periods (Fig 20) 391 Fig The frequency–amplitude curve of nonlinear vibration of stiffened shallow shells (k = 1, Q = 105 N/m2) Fig The frequency–amplitude curve of nonlinear vibration of stiffened shallow shells (k = 1, Q = 105 N/m2) Fig 10 The frequency–amplitude curve of nonlinear vibration of stiffened plate and shallow shells (k = 1, Q = 105 N/m2) 5.3 Nonlinear dynamic buckling results To evaluate the effectiveness of stiffener in the nonlinear dynamic buckling problem, we consider an imperfect ES-FGM cylindrical panel and spherical panel under lateral pressure and pre-loaded compression Materials of shells and stiffeners used in this section are the same in the previous section The effect of stiffeners to the critical buckling of perfect FGM shallow shells under only lateral pressure is investigated for two cases of cylindrical and spherical panel under uniformly lateral 392 D.H Bich et al / Composite Structures 96 (2013) 384–395 Fig 11 Nonlinear responses of un-stiffened FGM spherical panel with difference time steps (R = m, k = 1, q(t) = 105 sin 100t) Fig 12 Nonlinear q(t) = 105 sin 100t) responses Fig 13 Nonlinear q(t) = 105 sin 104t) responses of FGM spherical panel (R = m, Fig 14 Harmonic beat phenomenon of stiffened spherical panel (R = m, k = 1, q(t) = 105 sin Xt) k = 1, Fig 15 Harmonic beat phenomenon of stiffened spherical panel (R = m, k = 1, q(t) = Q sin 2530t) of FGM spherical panel (R = m, k = 1, pressure varying on time as q0 = 105t (N/m2) (see Figs 21 and 22) The critical buckling load corresponds to the buckling mode shape m = 1, n = in all cases These figures also show that there is no definite point of instability as in static analysis Rather, there is a region of instability where the slope of t vs f/h curve increases rapidly According to the Budiansky–Roth criterion, the critical time tcr can be taken as an intermediate value of this region There  ¼ as in fore one can choose the inflexion point of curve i.e ddt2f  t¼t cr Ref [26] As can be seen, the dynamic buckling loads of stiffened panels are greater than one of un-stiffened panels Fig 16 Deflection–velocity relation of stiffened spherical panel (R = m, k = 1, q(t) =  105 sin Xt) Effect of volume-fraction index k and loading speed c on critical dynamic buckling of stiffened cylindrical panels and spherical panel are showed in Table Clearly, the critical dynamic buckling of panel decrease when the volume-fraction index increases or the loading speed decreases As can be also observed in Table 5, the critical dynamic buckling load is greater than the static critical load Table shows the effect of thickness h on the critical dynamic buckling load of cylindrical and spherical panels The critical dynamic buckling loads of un-stiffened and stiffened panels increase when the thickness of panels increases In addition, Table also 393 D.H Bich et al / Composite Structures 96 (2013) 384–395 Fig 17 Effect of pre-loaded compressions on nonlinear responses of stiffened spherical panel (R = m, k = 1, q(t) = 105 sin 105t) Fig 20 Effect of damping on nonlinear responses of stiffened spherical panel (R = m, k = 1, q(t) =  103 sin 2530t) Fig 18 Effect of known initial amplitude on nonlinear responses of stiffened spherical panel (R = m, k = 1, q(t) = 105 sin 100t) Fig 21 Effect of stiffeners on dynamic buckling of cylindrical panel under lateral pressure (R = m, f0 = 0, d1 = d2 =  10À3 m, h1 = h2 =  10À2 m, s1 = s2 = 0.25 m) Fig 19 Effect of damping on nonlinear responses of stiffened spherical panel (R = m, k = 1, q(t) =  103 sin 2530t) Fig 22 Effect of stiffeners on dynamic buckling spherical panel under lateral pressure (R = 10 m, f0 = 0, d1 = d2 =  10À3 m, h1 = h2 =  10À2 m, s1 = s2 = 0.25 m) shows that the effect of stiffeners decreases when the thickness increases Table gives the results on the effect of known initial amplitude f0 on the nonlinear buckling of stiffened cylindrical and stiffened spherical panels Clearly, the known initial amplitude slightly influences on the critical dynamic buckling loads of these structures when they only subjected to lateral pressure Fig 23 shows the dynamic response of stiffened spherical panels under combination of lateral pressure varying on time q0 = 105t (N/m2) and pre-loaded compressions r0 = const, p0 = const As can be observed, the pre-loaded compressions strongly influence on Table Effect of volume fraction index k, loading speed c on the critical dynamic buckling load of ES-FGM shallow panels (Â105 N/m2) k 0.2 Cylindrical panels Static Dynamic q0 = 105t Dynamic q0 = 106t 2.3179 2.3423 2.4574 Spherical panels Static Dynamic q0 = 105t Dynamic q0 = 106t 17.7013 17.7236 17.8372 10 1.6022 1.6272 1.7441 0.9001 0.9256 1.0378 0.7461 0.7709 0.8732 12.0989 12.1219 12.2372 6.5172 6.5421 6.6619 5.2513 5.2770 5.4012 R = m, d1 = d2 =  10À3 m, h1 = h2 = 35  10À3 m, s1 = s2 = 0.25 m 394 D.H Bich et al / Composite Structures 96 (2013) 384–395 Table Effect of thickness h on the critical dynamic buckling load of ES-FGM shallow panels (Â105 N/m2) h 0.006 0.008 0.01 Cylindrical panels Un-stiffened Stiffened Difference (%) 0.2262 0.3678 62.60 0.3084 0.4635 50.29 0.4024 0.5744 42.74 1.5663 1.6855 7.61 2.1071 2.2353 6.08 2.6695 2.8099 5.26 Spherical panels Un-stiffened Stiffened Difference (%) R = 10 m, d1 = d2 =  10À3 m, h1 = h2 =  10À3 m, s1 = s2 = 0.25 m, q0 = 105t, k = Table Effect of known initial amplitude f0 on the critical dynamic buckling load of ES-FGM shallow panels (Â105 N/m2) f0 À4 10  10 À4 10 À3 R = m, d1 = d2 =  10À3 m, h1 = h2 = 35  10À3 m, s1 = s2 = 0.25 m, k = 1, q0 = 105t Cylindrical panels Spherical panels 1.6272 12.1219 1.6241 12.1071 1.6103 12.0471 1.5884 11.9739 Fig 23 Effect of pre-loaded compression on dynamic buckling stiffened spherical panel (R = 10 m, f0 = 0, d1 = d2 =  10À3 m, h1 = h2 = 3.5  10À2 m, s1 = s2 = 0.25 m) the critical dynamic buckling of stiffened spherical panel The critical dynamic buckling of panel increases when the pre-loaded compressions increase Conclusions The formulation of the governing equations of eccentrically stiffened functionally graded doubly curved imperfect shallow shells based upon the classical shell theory and the smeared stiffeners technique with von Karman–Donnell nonlinear terms is presented By using the Galerkin method, the nonlinear dynamic equation for analysis of dynamic and stability characteristics of ES-FGM shallow shells is obtained The effects of material parameters, stiffeners and initial geometrical imperfection to the static and dynamic behavior of FGM shallow shells are investigated The results of considered cases show some special phenomena as: (i) Fundamental frequencies of natural vibration of un-stiffened panels seem to depend only on the value jk1 + k2j, but this phenomenon is not observed for stiffened panels Except in (ii) (iii) (iv) (v) special cases when k1 + k2 = 0, the natural frequencies of stiffened panels have the same value with stiffened respective plate The frequency–amplitude curves of nonlinear vibration of shallow shells which have k1 + k2 = are coincide to the frequency–amplitude curve of respective plate The stiffener system strongly enhances the stability and load-carrying capacity of FGM shells The pre-loaded compressions strongly influence on the critical dynamic buckling of stiffened spherical panel The damping influences very small to the nonlinear response in the first vibration periods, however it strongly decreases amplitude at the next far periods 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] Khalil MR, Olson MD, Anderson DL Nonlinear dynamic analysis of stiffened plates Comput Struct 1988;29(6):929–41 [2] Shen PC, Dade C Dynamic analysis of stiffened plates and shells using spline gauss collocation method Comput Struct 1990;36(4):623–9 [3] Nayak AN, Bandyopadhyay JN On the free vibration of stiffened shallow shells J Sound Vib 2002;255(2):357–82 [4] Sheikh AH, Mukhopadhyay M Linear and nonlinear transient vibration analysis of stiffened plate structures Finite Elem Anal Des 2002;38(6):477–502 [5] Patel SN, Datta PK, Sheikh AH Buckling and dynamic instability analysis of stiffened shell panels 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F Nonlinear vibrations of functionally graded doubly curved shallow shells J Sound Vib 2011;330:1432–54 [18] Bich DH, Long VD Non-linear dynamical analysis of imperfect functionally graded material... vibration of stiffened shallow shells (k = 1, Q = 105 N/m2) Fig 10 The frequency–amplitude curve of nonlinear vibration of stiffened plate and shallow shells (k = 1, Q = 105 N/m2) 5.3 Nonlinear dynamic