Composite Structures 102 (2013) 306–314 Contents lists available at SciVerse ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct Corrigendum Corrigendum to ‘‘Nonlinear dynamic response of imperfect eccentrically stiffened FGM double curved shallow shells on elastic foundation’’ [Compos Struct 99 (2013) 88–96] Nguyen Dinh Duc ⇑ University of Engineering and Technology, Vietnam National University, 144 Xuan Thuy, Cau Giay, Hanoi, Viet Nam a r t i c l e i n f o Article history: Available online 10 April 2013 Keywords: Nonlinear dynamic Eccentrically stiffened FGM double curved shallow shells Imperfection Elastic foundation a b s t r a c t This paper presents an analytical investigation on the nonlinear dynamic response of eccentrically stiffened functionally graded double curved shallow shells resting on elastic foundations and being subjected to axial compressive load and transverse load The formulations are based on the classical shell theory taking into account geometrical nonlinearity, initial geometrical imperfection and the Lekhnitsky smeared stiffeners technique with Pasternak type elastic foundation The non-linear equations are solved by the Runge–Kutta and Bubnov–Galerkin methods Obtained results show effects of material and geometrical properties, elastic foundation and imperfection on the dynamical response of reinforced FGM shallow shells Some numerical results are given and compared with ones of other authors Ó 2013 Elsevier Ltd All rights reserved Introduction Functionally Graded Materials (FGMs), which are microscopically composites and made from mixture of metal and ceramic constituents, have received considerable attention in recent years due to their high performance heat resistance capacity and excellent characteristics in comparison with conventional composites By continuously and gradually varying the volume fraction of constituent materials through a specific direction, FGMs are capable of withstanding ultrahigh temperature environments and extremely large thermal gradients Therefore, these novel materials are chosen to use in temperature shielding structure components of aircraft, aerospace vehicles, nuclear plants and engineering structures in various industries As a result, in recent years important studies have been researched about the stability and vibration of FGM plates and shells The research on FGM shells and plates under dynamic load is attractive to many researchers in the world Firstly we have to mention the research group of Reddy et al The vibration of functionally graded cylindrical shells has been investigated by Loy, Lam and Reddy [1] Lam and Hua has taken into account the influence of boundary conditions on the frequency characteristics of a rotating truncated circular conical shell [2] In [3] Pradhan et al studied vibration characteristics of FGM cylindrical shells under various boundary conditions Ng et al studied the DOI of original article: http://dx.doi.org/10.1016/j.compstruct.2012.11.017 ⇑ Tel.: +84 37547989; fax: + 84 37547724 E-mail address: ducnd@vnu.edu.vn 0263-8223/$ - see front matter Ó 2013 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.compstruct.2013.03.009 dynamic stability analysis of functionally graded cylindrical shells under periodic axial loading [4] The group of Ng and Lam also published results on generalized differential quadrate for free vibration of rotating composite laminated conical shell with various boundary conditions in 2003 [5] In the same year, Yang and Shen [6] published the nonlinear analysis of FGM plates under transverse and in-plane loads In 2004, Zhao et al studied the free vibration of two-side simply-supported laminated cylindrical panel via the mesh-free kpRitz method [7] About vibration of FGM plates Vel and Batra [8] gave three dimensional exact solution for the vibration of FGM rectangular plates Also in this year, Sofiyev and Schnack investigated the stability of functionally graded cylindrical shells under linearly increasing dynamic tensional loading in [9] and obtained the result for the stability of functionally graded truncated conical shells subjected to a periodic impulsive loading [10], and they published the result of the stability of functionally graded ceramic– metal cylindrical shells under a periodic axial impulsive loading in 2005 [11] Ferreira et al received natural frequencies of FGM plates by meshless method [12], 2006 In [13], Zhao et al used the element-free kp-Ritz method for free vibration analysis of conical shell panels Liew et al studied the nonlinear vibration of a coating-FGM-substrate cylindrical panel subjected to a temperature gradient [14] and dynamic stability of rotating cylindrical shells subjected to periodic axial loads [15] Woo et al investigated the non linear free vibration behavior of functionally graded plates [16] Kadoli and Ganesan studied the buckling and free vibration analysis of functionally graded cylindrical shells subjected to a temperature-specified boundary condition [17] Also in this year, 307 N.D Duc / Composite Structures 102 (2013) 306–314 Wu et al published their results of nonlinear static and dynamic analysis of functionally graded plates [18] Sofiyev has considered the buckling of functionally graded truncated conical shells under dynamic axial loading [19] Prakash et al studied the nonlinear axisymmetric dynamic buckling behavior of clamped functionally graded spherical caps [20] In [21], Darabi et al obtained the nonlinear analysis of dynamic stability for functionally graded cylindrical shells under periodic axial loading Natural frequencies and buckling stresses of FGM plates were analyzed by Matsunaga using 2-D higher-order deformation theory [22] In 2008, Shariyat also obtained the dynamic thermal buckling of suddenly heated temperature-dependent FGM cylindrical shells under combined axial compression [23] and external pressure and dynamic buckling of suddenly loaded imperfect hybrid FGM cylindrical with temperature-dependent material properties under thermo-electromechanical loads [24] Allahverdizadeh et al studied nonlinear free and forced vibration analysis of thin circular functionally graded plates [25] Sofiyev investigated the vibration and stability behavior of freely supported FGM conical shells subjected to external pressure [26], 2009 Shen published a valuable book ‘‘Functionally Graded materials, Non linear Analysis of plates and shells’’, in which the results about nonlinear vibration of shear deformable FGM plates are presented [27] Last years, Zhang and Li published the dynamic buckling of FGM truncated conical shells subjected to non-uniform normal impact load [28], Bich and Long studied the non-linear dynamical analysis of functionally graded material shallow shells subjected to some dynamic loads [29], Dung and Nam investigated the nonlinear dynamic analysis of imperfect FGM shallow shells with simply supported and clamped boundary conditions [30] Bich et al has also considered the nonlinear vibration of functionally graded shallow spherical shells [31] In fact, the FGM plates and shells, as other composite structures, ussually reinforced by stiffening member to provide the benefit of added load-carrying static and dynamic capability with a relatively small additional weight penalty Thus study on static and dynamic problems of reinforced FGM plates and shells with geometrical nonlinearity are of significant practical interest However, up to date, the investigation on static and dynamic of eccentrically stiffened FGM structures has received comparatively little attention Recently, Bich et al studied nonlinear dynamical analysis of eccentrically stiffened functionally graded cylindrical panels [32] This paper presents an dynamic nonlinear response of double curved shallow eccentrically stiffened shells FGM resting on elastic foundations and being subjected to axial compressive load and transverse load The formulations are based on the classical shell theory taking into account geometrical nonlinearity, initial geometrical imperfection and the Lekhnitsky smeared stiffeners technique with Pasternak type elastic foundation The nonlinear transients response of doubly curved shallow shells subjected to excited external forces obtained the dynamic critical buckling loads are evaluated based on the displacement response using the criterion suggested by Budiansky–Roth Obtained results show effects of material, geometrical properties, eccentrically stiffened, elastic foundation and imperfection on the dynamical response of FGM shallow shells Eccentrically stiffened double curved FGM shallow shell on elastic foundations Consider a ceramic–metal stiffened FGM double curved shallow shell of radii of curvature Rx, Ry length of edges a, b and uniform thickness h resting on an elastic foundation A coordinate system (x, y, z) is established in which (x, y) plane on the middle surface of the panel and z is thickness direction (Àh/ z h/2) as shown in Fig z b a h y x Ry Rx Fig Geometry and coordinate system of an eccentrically stiffened double curved shallow FGM shell on elastic foundation The volume fractions of constituents are assumed to vary through the thickness according to the following power law distribution V m zị ẳ N 2z ỵ h ; 2h V c zị ẳ V m ðzÞ ð1Þ where N is volume fraction index (0 N < 1) Effective properties Preff of FGM panel are determined by linear rule of mixture as Preff zị ẳ Prm V m zị ỵ Prc V c ðzÞ ð2Þ where Pr denotes a material property, and subscripts m and c stand for the metal and ceramic constituents, respectively Specific expressions of modulus of elasticity E(z) and q(z) are obtained by substituting Eq (1) into Eq (2) as ẵEzị; qzị ẳ Em ; qm ị ỵ Ecm ; qcm ị N 2z ỵ h 2h 3ị where Ecm ¼ Ec À Em ; qcm ¼ qc À qm ; mzị ẳ const ẳ m 4ị It is evident from Eqs (3), (4) that the upper surface of the panel (z = Àh/2) is ceramic-rich, while the lower surface (z = h/2) is metal-rich, and the percentage of ceramic constituent in the panel is enhanced when N increases The panel–foundation interaction is represented by Pasternak model as qe ¼ k1 w À k2 r2 w 2 ð5Þ 2 where r = @ /@x + @ /@ y ,w is the deflection of the panel, k1 is Winkler foundation modulus and k2 is the shear layer foundation stiffness of Pasternak model Theoretical formulation In this study, the classical shell theory and the Lekhnitsky smeared stiffeners technique are used to obtain governing equations and determine the nonlinear dynamical response of FGM curved panels The strain across the shell thickness at a distance z from the mid-surface are 1 e0x ex kx B e C B e0 C B C @ y A ¼ @ y A À z@ ky A cxy 2kxy cxy ð6Þ where e0x ; e0x and c0xy are normal and shear strain at the middle surface of the shell, and kx, ky,kxy are the curvatures The nonlinear strain–displacement relationship based upon the von Karman theory for moderately large deflection and small strain are: 308 N.D Duc / Composite Structures 102 (2013) 306–314 1 1 e0x u;x w=Rx ỵ w2;x =2 wx;x kx B e0 C B C B C B C @ y A ¼ @ v ;y À w=Ry þ w2;y =2 A; @ ky A ¼ @ wy;y A w;xy kxy c0xy u;y ỵ v ;x ỵ w;x w;y ð7Þ In which u, v are the displacement components along the x, y directions, respectively The force and moment resultants of the FGM panel are determined by ðNi ; Mi ị ẳ Z x2 h z z2 x1 O h=2 ri 1; zị dz i ẳ x; y; xy ð8Þ Àh=2 The constitutive stress–strain equations by Hooke law for the shell material are omitted here for brevity The shell reinforced by eccentrically longitudinal and transversal stiffeners is shown in Fig The shallow shell is assumed to have a relative small rise as compared with its span The contribution of stiffeners can be accounted for using the Lekhnitsky smeared stiffeners technique Then intergrading the stress–strain equations and their moments through the thickness of the shell, the expressions for force and moment resultants of an eccentrically stiffened FGM shallow shell are obtained [32]: E1 EA1 E m E2 E2 m ỵ ex ỵ e0y ỵ C kx ky 2 1Àm s1 1Àm 1Àm À m2 E1 m E1 EA2 E m E2 Ny ẳ e ỵ ỵ ey 2 kx ỵ C ky m2 x À m2 s2 1Àm À m2 E2 E2 m E3 EI1 E3 m kx Mx ẳ ỵ C e0x ỵ e þ ky À m2 À m2 y À m2 s1 À m2 E2 m E2 E3 m E3 EI2 My ¼ e ỵ ỵ C e0y kx ỵ ky À m2 x À m2 À m2 À m2 s2 N xy ẳ E1 c0xy 2E2 kxy 21 ỵ mị M xy ¼ E2 c0xy À 2E3 kxy 21 ỵ mị 9ị s1 s1 s1 s1 s2 a Fig Configuration of an eccentrically stiffened shallow shells Mx;xx þ 2Mxy;xy þ My;yy þ @2w @t ð11Þ where Z h A A q A A qẳ qzị dz ỵ q0 þ ¼ qm þ cm h þ q0 þ s1 s2 Nþ1 s1 s2 À2h ð12Þ 2 in which q0 is the mass density of stiffeners; q @@t2u ! and q @@t2v ! into consideration because of u ( w,v ( w the Eq (11) can be rewritten as: Mx;xx ỵ 2Mxy;xy ỵ My;yy þ Nx Ny þ þ Nx w;xx þ 2Nxy w;xy þ Ny w;yy þ q Rx Ry À k1 w þ k2 r2 w ¼ q Ecm h E1 ẳ Em ỵ Nỵ1 @2w @t 13ị Calculating from Eq (9), obtained: E2 ẳ 10ị In above relations (9) and (10), the quantities A1, A2 are the cross section areas of the stiffeners and I1, I2, z1, z2 are the second moments of cross section areas and eccentricities of the stiffeners with respect to the middle surface of the shell respectively, E is elasticity modulus in the axial direction of the corresponding stiffener witch is assumed identical for both types of stiffeners (Fig 2) In order to provide continuity between the shell and stiffeners, suppose that stiffeners are made of full metal (E = Em) if putting them at the metal-rich side of the shell, and conversely full ceramic stiffeners (E = Ec) at the ceramic-rich side of the shell [32] The nonlinear dynamic equations of a FGM shallow shells based on the classical shell theory are [33] N xy;x ỵ Ny;y Nx Ny ỵ ỵ Nx w;xx ỵ 2Nxy w;xy ỵ Ny w;yy ỵ q Rx Ry k1 w ỵ k2 r2 w ẳ q where @2u @t @2v ¼q @t s2 N x;x ỵ N xy;y ẳ q s2 b z1 Nx ẳ Ecm Nh 2N ỵ 1ịN ỵ 2ị ! Em 1 E3 ¼ À þ h þ Ecm N þ N þ 4N ỵ 12 EA1 z1 EA2 z2 C1 ẳ ; C2 ¼ s1 s2 s2 x y e ẳ A22 Nx A12 Ny ỵ B11 kx þ B12 ky e ¼ A11 Ny À A12 Nx þ B21 kx þ B22 ky c0xy ¼ A66 Nxy þ 2A66 kxy ð14Þ where EA1 E1 EA2 E1 ỵ ẳ ỵ ; A 22 D s1 À m2 D s2 À m2 E1 m 21 ỵ mị ; A66 ẳ A12 ¼ E1 D À m2 2 EA1 E1 EA2 E1 E1 m Dẳ ỵ þ À 2 s1 1Àm s2 1Àm 1Àm E2 E2 m E2 E2 m B11 ẳ A22 C ỵ ; B A A12 12 22 ẳ A11 C ỵ 2 1Àm 1Àm 1Àm À m2 E2 m E2 E2 m E2 B12 ¼ A22 A12 ỵ C ; B21 ẳ A11 A12 ỵ C1 2 2 1m 1m 1m 1m E2 B66 ẳ E1 15ị A11 ẳ Substituting once again Eq (14) into the expression of Mij in (9) leads to: M x ẳ B11 Nỵx B21 Ny D11 kx D12 ky M x ẳ B12 Nỵx B22 NÀy D21 kx À D22 ky M xy ¼ B66 NÀxy 2D66 kxy ð16Þ 309 N.D Duc / Composite Structures 102 (2013) 306–314 The approximate solutions of w, w⁄ and f satisfying boundary conditions (23) are assumed to be [27–31] where D11 D22 D12 D21 D66 EI1 E3 E2 E2 m B11 ẳ ỵ C þ B21 s1 À m À m2 À m2 EI2 E3 E2 E2 m ẳ ỵ C2 ỵ B12 B22 s2 À m À m2 À m2 E3 m E2 E2 m ¼ À C1 þ B22 B12 À À m2 À m2 À m2 E3 m E2 E2 m B21 ẳ C2 ỵ B11 m2 À m2 À m2 E3 E2 ¼ À B66 21 ỵ mị 21 ỵ mị 17ị Ny ẳ f;xx ; Nxy ẳ f;xy 24aị w ẳ W0 sin km x sin dn y 24bị f ẳ gtịẵsin km x sin dn y À hðxÞ À xðyÞ ð24cÞ where km = mp/a, dn = np/b and W is the maximum deflection; W0 is a constant; h(x) and x(y) chosen such that: gh00 xị ẳ p0 h gx00 yị ẳ r0 h Then Mij into Eq (13) and f(x, y) is stress function defined by Nx ¼ f;yy ; w ¼ WðtÞ sin km x sin dn y ð18Þ Subsequently, substitution of Eqs (24a and b) into Eqs (22) and (24c) into Eq (19) and applying the Galerkin procedure for the resulting equation yield leads to:  g A11 m4 þ ðA66 À 2A12 Þm2 n2 k2 þ A22 n4 k4 ỵ ỵ f;xx w;yy ỵ ỵ D12 ỵ D21 þ þ f;yy w;xx À 2f ;xy w;xy f;yy f;xx @2w ỵ ỵ q k1 w ỵ k2 r2 w ẳ q Rx Ry @t 19ị g a4 e0x ẳ A22 f;yy A12 f;xx ỵ B11 w;xx þ B12 w;yy e0y ¼ A11 f;xx À A12 f;yy þ B21 w;xx þ B22 w;yy c0xy ¼ ÀA66 f;xy þ 2A66 w;xy Þ D11 m4 þ ðD12 þ D21 þ 4D66 Þn2 m2 k2 þ D22 n4 k4 a4 k2 W p2 a2 m2 ỵ k2 n2 ị ẳ q W p2 h a2 m2 r ỵ n2 p0 k2 Þ À k1 À k2 A11 f;xxxx ỵ A66 2A12 ịf;xxyy ỵ A22 f;yyyy ỵ B21 w;xxxx ð22Þ @2W @t2 ð27Þ where m,n are odd numbers, and k ¼ ab Eliminating g from two obtained equations leads to non-linear second-order ordinary differential equation for f(t): " ð21Þ Ã 32 k2 p2 hW W gmnp2 ỵ m r ỵ n2 p0 k2 ị a ! a2 p2 m2 n2 k2 16h r p0 16q ỵ ỵ ỵ k1 W À 2g mnp2 Rx Ry mnp2 a Ry Rx ð20Þ Setting Eq (21) into Eq (20) gives the compatibility equation of an imperfect eccentrically stiffened shallow FGM shells as ỵ B11 ỵ B22 2B66 ịw;xxyy ỵ B12 w;yyyy ¼ w2;xy À w;xx w;yy w;yy À wÃ;yy w;xx w;xx w2 ;xy ỵ w;xx w;yy Rx Ry p4 ỵ e0x;yy þ e0y;xx À c0xy;xy ¼ w2;xy À w;xx w;yy À w2 ;xy ỵ w;xx w;yy From the constitutive relations (18) in conjunction with Eq (14) one can write ð26Þ Ã B21 m4 ỵ B11 ỵ B22 B66 ịn2 m2 k2 ỵ B12 n4 k4 W W0ị in which w (x, y) is a known function representing initial small imperfection of the eccentrically stiffened shallow shells The geometrical compatibility equation for an imperfect shallow shells is written w;yy À wÃ;yy w;xx À wÃ;xx À : Rx Ry n2 k2 m2 W W ị ỵ Rx Ry p4  ⁄ À p à !  à þ W B21 m4 þ ðB11 þ B22 À 2B66 ịm2 n2 k2 ỵ B12 n4 k4 16 mnk2 ỵ W W 20 ẳ p B21 f;xxxx ỵ B12 f;yyyy ỵ B11 ỵ B22 À 2B66 Þf;xxyy À D11 w;xxxx À D22 w;yyyy D12 ỵ D21 ỵ 4D66 ịw;xxyy ỵ D11 w;xxxx 4D66 ÞwÃ;xxyy a2 À For an imperfect FGM curved panel, Eq (13) are modified into form D22 wÃ;yyyy ð25Þ ! a2 m2 ỵ k2 n2 ị p4 P2 a4 P1 ỵ 2 p2 m2 a2 !2 Ry þ m n k P2 p m n k 15 þ À P3 À þ P1 a Ry Rx P a4 Ry Rx " ! # m2 n2 k2 16mnk2 16mnp2 k2 P2 ỵ W W 20 ị ỵ a Ry Rx 3P a P1 ! 2 2 2 32mn p k P 32mnk m n k ỵ W ỵ WW W ị þ P1 3a P1 3a Ry Rx 512m2 n2 k 16h r p0 16q @2W ỵ WW W 20 ị þ ¼q 2 9a P1 mnp Rx Ry mnp @t ỵ W W ị4 p 2 p2 2 n k2 Rx ! P2 P1 # ð28Þ Eqs (19) and (22) are nonlinear equations in terms of variables w and f and used to investigate the nonlinear dynamic and nonlinear stability of thick imperfect stiffened FGM double curved panels on elastic foundations subjected to mechanical, thermal and thermo mechanical loads where Nonlinear dynamic analysis The obtained Eq (28) is a governing equation for dynamic imperfect stiffened FGM doubly-curved shallow shells in general _ The initial conditions are assumed as Wð0Þ ¼ W ; Wð0Þ ¼ The nonlinear Eq (28) can be solved by the Newmark’s numerical integration method or Runge–Kutta method In the present study, suppose that the stiffened FGM shallow shell is simply supported at its all edges and subjected to a transverse load q(t), compressive edge loads r0(t) and p0(t) The boundary conditions are w ¼ N xy ¼ M x ¼ 0; Nx ¼ Àr h at x ¼ 0; a w ¼ N xy ¼ M y ¼ 0; Ny ¼ p0 h at y ẳ 0; b: 23ị where a and b are the lengths of in-plane edges of the shallow shell P1 ẳ A11 m4 ỵ A66 2A12 ịm2 n2 k2 ỵ A22 n4 k4 P2 ẳ B21 m4 ỵ B11 ỵ B22 2B66 ịm2 n2 k2 ỵ B12 n4 k4 2 ð29Þ 4 P3 ẳ D11 m ỵ D12 ỵ D21 ỵ 4D66 ịm n k ỵ D22 n k 4.1 Nonlinear vibration of eccentrically stiffened FGM shallow shell Consider an imperfect stiffened FGM shallow shell acted on by uniformly distributed excited transverse q(t) = Qsin Xt, i.e p0 = r0 = 0, from (28) we have 310 N.D Duc / Composite Structures 102 (2013) 306314 Q W ỵ Q W W ị ỵ Q W W 20 Q W ỵ Q WðW @2W À W Þ À Q W W W 20 ỵ Q sin Xt ẳ q @t 30ị where Q4 ẳ Q5 ẳ p2 a2 m2 ỵ k2 n2 ị þ 32mnp2 k2 P2 3a4 P1 p4 P2 a4 P1 À p2 m2 a2 Ry ! p4 P2 p2 m2 P2 2 m ỵ k n ị þ a2 a2 a4 P1 a2 RP ! 2 4 p m P2 p m À P3 ỵ W W ị a RP a R P1 " # 16m3 nk2 16mnp2 k2 P 2 À þ W À W0 þ ÀW a2 3P1 R a4 P1 p2 h W ! n2 k2 P2 Rx P1 ! !2 p2 m2 n2 k2 P2 p4 m2 n2 k2 Q2 ẳ ỵ ỵ P3 ỵ ỵ P1 a Ry Rx P a Ry Rx ! m2 n2 k2 16mnk2 16mnp2 k2 P Q3 ẳ ỵ a Ry Rx 3P1 a4 P1 Q ẳ k1 ỵ k2 4.2.1 Imperfect eccentrically stiffened FGM cylindrical panel acted on by axial compressive load Eq (28) in this case Rx ? 1, Ry = R, p0 = q = 0;r0 – can be rewritten as: ỵ m2 r k1 k2 32mnp2 k2 P2 32mnk2 m2 ỵ WW À W Þ 3a P1 3a2 R P1 512m2 n2 k @ W À W W À W 20 ¼q 9a4 P1 @t 31ị ! 32mnk2 m2 n2 k2 ỵ 3a2 Ry Rx P 512m2 n2 k 9a4 P1 16Q Q7 ¼ mnp2 W p2 h a2 m2 r ẳ W k1 ỵ k2 q 32ị 33ị where denoting H1 ẳ x2L ẳ H2 ẳ H3 ¼ q Q4 À Q3 À Q5 q xNL m2 r0 ẳ k1 ỵ k2 34ị q 35ị where xNL is the nonlinear vibration frequency and s is the amplitude of nonlinear vibration 4.2 Nonlinear dynamic buckling analysis of imperfect eccentrically stiffened FGM shallow shell The aim of considered problems is to search the critical dynamic buckling loads They can be evaluated based on the displacement responses obtained from the motion Eq (28) This criterion suggested by Budiansky and Roth is employed here as it is widely accepted This criterion is based on that, for large values of loading speed the amplitude-time curve of obtained displacement response increases sharply depending on time and this curve obtained a maximum by passing from the slope point, and at the time t = tcr a stability loss occurs, and here t = tcr is called critical time and the load corresponding to this critical time is called dynamic critical buckling load p m P2 À p p4 P a4 P1 m4 P3 ỵ a4 R P1 ỵ ! 512m2 n2 k 9a4 P1 ð37Þ p2 a2 m2 ỵ k2 n2 ị ỵ " p4 P2 a4 P1 À p2 m2 P2 a2 RP À p2 m P a2 RP 1 16m3 nk2 16mnp2 k2 P2 À a 3P R a4 P1 R P1 # 32mnp2 k2 P2 32mnk2 m2 512m2 n2 k ỵ ỵ W2 9a4 P1 3a4 P1 3a2 R P1 ỵ Q6 1 8H2 3H3 2 ẳ xL ỵ s ỵ s 3px2L 4x2L m2 ỵ k2 n2 ị ỵ Taking of W – 0, i.e considering the shell after the loss of stability we obtain a2 Seeking solution as W(t) = scos xt and applying procedure like Garlerkin method to Eq (33), the frequency-amplitude relation of nonlinear free vibration is obtained a2 ỵ W3 p2 h Q ỵ Q Þ p2 a2 RP a2 RP " 16m3 nk2 16mnp2 k2 P2 À W2 À a 3P1 R a4 P1 # 32mnp2 k2 P2 32mnk2 m2 ỵ 3a4 P1 3a2 R P1 The equation of nonlinear free vibration of a perfect FGM shallow panel can be obtained from: ỵ H1 W þ H2 W þ H W ¼ W p m P2 From Eq (30) the fundamental frequencies of natural vibration of the imperfect stiffened FGM shell can be determined by the relation: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðQ ỵ Q ị 36ị The static critical load can be determined by the equation to be € ¼ 0; W ¼ reduced from Eq (36) by putting W Q6 ¼ xL ¼ p2 p4 a4 m4 P3 ỵ W 38ị From Eq (38) the upper buckling load can be determined by W=0 rupper ¼ a2 m2 h p k1 ỵ k2 p2 a2 m2 þ k2 n2 Þ þ p4 P2 p2 m2 P2 p2 m2 P2 p4 a4 P À a2 RP a2 RP ỵ a4 P3 ỵ m4 ! R P1 ð39Þ dr0 And the lower buckling load is found using the condition dW ¼ 0, it follows: rlower ẳ a2 k1 ỵ k2 p2 hm p2 m2 P2 p4 " a2 RP ỵ a4 p2 a2 m2 ỵ k2 n2 ị ỵ P3 þ m4 R2 P À p4 P p2 m P a4 P À a2 RP1 9a4 P 1024m2 n2 k #2 16m3 nk2 16mnp2 k2 P 32mnp2 k2 P 32mnk2 m2 ỵ a2 3P R a4 P1 3a4 P1 3a2 R P1 " 512m2 n2 k 1 16m3 nk2 16mnp2 k2 P þþ4 À 9a4 P1 a2 3P R a4 P1 ## 32mnp2 k2 P 32mnk2 m2 ỵ ð40Þ 3a4 P1 3a2 R P1 311 N.D Duc / Composite Structures 102 (2013) 306–314 4.2.2 Imperfect eccentrically stiffened shallow FGM cylindrical panel subjected to transverse load Eq (28) in this case Rx ? 1,Ry = R, p0 = r0 = can be rewritten as: " W Àk1 À k2 p2 a2 m2 ỵ k2 n2 ị p m P2 p4 P2 a4 P1 ỵ p2 n2 k2 P2 ! a2 RP # ỵ W W Þ 4 xL(rad/s) N 1 p m À P3 À RP a a RP 1 " # 16m3 nk2 16mnp2 k2 P 2 ỵ W À W0 a2 3RP a4 P1 Table The dependence of the fundamental frequencies of nature vibration of spherical FGM double curved shallow shell on volume ratio N 32mnp2 k2 P2 32m3 nk2 ỵ WW W Þ 3a4 P1 3Ra2 P1 2 16q @2W 2 512m n k À W W W0 ỵ ẳq 9a P1 mnp @t Reinforced Unreinforced 56.130  105 39.034  105 31.982  105 24.047  105 55.667  105 38.515 105 31.441 105 23.477 105 ỵ W ð41Þ The static critical load can be determined by the equation to be re€ ¼ 0; W ¼ and using condition duced from Eq (41) by putting W dq ¼ dW 4.2.3 Imperfect eccentrically stiffened FGM shallow spherical panel under transverse load Eq (28) in this case Rx = Ry = R, p0 = r0 = can be rewritten as: " W Àk1 À k2 p2 2 p4 P2 ! p2 m2 ỵ n2 k2 P2 # m ỵ k n ị ỵ a P1 a2 R P1 ! !2 2 2 2 p m ỵ n k P p m ỵ n k 15 ỵ W W ị4 P3 À P1 a R P1 a4 R ! " # m2 ỵ n2 k2 16mnk2 16mnp2 k2 P ỵ W W 20 a2 R 3P1 a4 P1 ! 2 2 2 32mnk m ỵ n k 32mnp k P ỵ W ỵ WW W Þ P1 3a4 P1 3a2 R 512m2 n2 k 16q @2W À W W À W 20 ỵ ẳq 42ị 9a P1 mnp @t a2 The static critical load can be determined by the equation to be € ¼ 0; W ¼ and using condireduced from Eq (42) by putting W dq tion dW ¼ Numerical results and discussion The eccentrically stiffened FGM shells considered here are shallow shell with in-plane edges: Table The dependence of the fundamental frequencies of nature vibration of spherical FGM double curved shallow shell on elastic foundations K1, K2 xL (rad/s) Reinforced Unreinforced K1 = 200, K2 = K1 = 200, K2 = 10 K1 = 200, K2 = 20 K1 = 200, K2 = 30 K1 = 0, K2 = 10 K1 = 100, K2 = 10 K1 = 150, K2 = 10 K1 = 200, K2 = 10 33.574  10 39.034  10 44.079  10 48.535  10 26.734  105 31.534  10 35.585  10 39.034  10 32.865  105 38.515  105 43.273  105 46.371  105 25.646  105 30.078  105 35.033  105 38.515  105 Table Comparison of - with result reported by Bich et al [32], Alijani et al [34], Chorfi and Houmat [35] and Matsunaga [36] (a/Rx, b/Ry) N Present Ref [32] Ref [34] Ref [35] Ref [36] 0.5 10 0.0562 0.0502 0.0449 0.0385 0.0304 0.0597 0.0506 0.0456 0.0396 0.0381 0.0597 0.0506 0.0456 0.0396 0.0380 0.0577 0.0490 0.0442 0.0383 0.0366 0.0588 0.0492 0.0403 0.0381 0.0364 FGM cylindrical panel (0, 0.5) 0.0624 0.5 0.0528 0.0494 0.0407 10 0.0379 0.0648 0.0553 0.0501 0.0430 0.0409 0.0648 0.0553 0.0501 0.0430 0.0408 0.0629 0.0540 0.0490 0.0419 0.0395 0.0622 0.0535 0.0485 0.0413 0.0390 FGM plate (0, 0) a ¼ b ¼ m; h ¼ 0:01 m; Ec ¼ 380  109 N=m2 ; qm ¼ 2702 kg=m ; qc ¼ 3800 kg=m3 ; s1 ¼ s2 ¼ 0:4; z1 ẳ z2 ẳ 0:0225 mị; 43ị 5.7419 x 10 Reinforced, Rx =Ry =3(m), N=5 m ¼ 0:3 Table presents the dependence of the fundamental frequencies of nature vibration of spherical FGM shallow shell on volume ratio N in which m ¼ n ¼ 1; a ¼ b ẳ mị, h ẳ 0:01 mị; K ¼ 200; K ¼ 10, R¼ x Ry ¼ mị; W ẳ 1e From the results of Table 1, it can be seen that the increase of volume ration N will lead to the decrease of frequencies of nature vibration of spherical FGM shallow shell Table presents the frequencies of nature vibration of spherical double curved FGM shallow shell depending on elastic foundations These results show that the increase of the coefficients of elastic foundations will lead to the increase of the frequencies of nature vibration Moreover, the Pasternak type elastic foundation has the greater influence on the frequencies of nature vibration of FGM shell than Winkler model does Based on (28) the nonlinear vibration of imperfect eccentrically stiffened shells under various loading cases can be performed Reinforced, Rx =R(y)=3(m), N=0 5.7418 Unreinforced, Rx =Ry =3(m), N=5 Unreinforced, Rx =Ry =3(m), N=0 5.7417 ωNL (rad/s) Em ¼ 70  109 N=m2 ; 5.7416 5.7415 5.7414 5.7413 0.05 τ 0.1 Fig Frequency-amplitude relation 0.15 312 N.D Duc / Composite Structures 102 (2013) 306–314 Fig Effect of eccentrically stiffeners on nonlinear dynamic response of the shallow spherical FGM shell Fig Influence of elastic foundations on nonlinear dynamic response of the eccentrically stiffened shallow spherical FGM shell Fig Deflection-velocity relation of the eccentrically stiffened shallow spherical FGM shell Fig Effect of volume metal-ceramic on nonlinear response of the eccentrically stiffened shallow spherical FGM shell Particularly for spherical panel we put R1x ¼ R1y in (28), for cylindrical shell R1x ¼ and for a plate R1x ¼ R1y ¼ Table presents the comparison on the fundamental frequency qffiffiffiffi parameter - ¼ xL h qEcc (In the Tables 1–3, xL is calculated from Eq (32)) given by the present analysis with the result of Alijani et al [34] based on the Donnell’s nonlinear shallow shell theory, Chorfi and Haumat [35] based on the first–order shear deformation theory and Matsunaga [36] based on the two-dimensional (2D) higher order theory for the perfect unreinforced FGM cylindrical panel The results in Table were obtained with m = n = 1, a = b = (m), h = 0.02 (m), K1 = 0, K2 = 0; W⁄ = and with the chosen material properties in (43) As in Table 3, we can observe a very good agreement in this comparison study Fig shows the relation frequency-amplitude of nonlinear free vibration of reinforced and unreinforced spherical shallow FGM shell on elastic foundation (calculated from Eq (35)) with m ¼ n ¼ 1; a ẳ b ẳ mị, h ẳ 0:01 mị; K ẳ 200, K ẳ 10; Rẳ x Ry ẳ mị; W ẳ 1e As expected the nonlinear vibration frequencies of reinforced spherical shallow FGM shells are greater than ones of unreinforced shells The nonlinear Eq (28) is solved by Runge–Kutta method The below figures, except Fig 6, are calculated basing on k1 = 100; k2 = 10 Fig shows the effect of eccentrically stiffeners on nonlinear respond of the FGM shallow shell on elastic foundation The FGM N.D Duc / Composite Structures 102 (2013) 306–314 313 Fig Effect of dynamic loads on nonlinear response Fig 10 Influence of initial imperfection on nonlinear dynamic response of the eccentrically stiffened spherical panel Fig Effect of Rx on nonlinear dynamic response shell considered here is spherical panel Rx = Ry = m Clearly, the stiffeners played positive role in reducing amplitude of maximum deflection Relation of maximum deflection and velocity for spherical shallow shell is expressed in Fig Fig shows influence of elastic foundations on nonlinear dynamic response of spherical panel Obviously, elastic foundations played negative role on dynamic response of the shell: the larger k1 and k2 coefficients are, the larger amplitude of deflections is Fig shows effect of volume metal-ceramic on nonlinear dynamic response of the eccentrically stiffened shallow spherical FGM shell Figs and show effect of dynamic loads and Rx on nonlinear dynamic response of the eccentrically stiffened shallow spherical FGM shell Fig 11 Nonlinear dynamic response of eccentrically stiffened spherical and cylindrical FGM panel Fig 10 shows influence of initial imperfection on nonlinear dynamic response of the eccentrically stiffened spherical panel The increase in imperfection will lead to the increase of the amplitude of maximum deflection Fig 11 shows nonlinear dynamic response of shallow eccentrically stiffened spherical and eccentrically stiffened cylindrical FGM panels For eccentrically stiffened cylindrical FGM panel, in this case, the obtained results is identical to the result of Bich in [32] Concluding remarks This paper presents an analytical investigation on the nonlinear dynamic response of eccentrically stiffened functionally graded double curved shallow shells resting on elastic foundations and 314 N.D Duc / Composite Structures 102 (2013) 306–314 being subjected to axial compressive load and transverse load The formulations are based on the classical shell theory taking into account geometrical nonlinearity, initial geometrical imperfection and the Lekhnitsky smeared stiffeners technique with Pasternak type elastic foundation The nonlinear equations 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Effect of dynamic loads on nonlinear response Fig 10 Influence of initial imperfection on nonlinear dynamic response of the eccentrically stiffened spherical panel Fig Effect of Rx on nonlinear dynamic. .. elastic foundations on nonlinear dynamic response of the eccentrically stiffened shallow spherical FGM shell Fig Deflection-velocity relation of the eccentrically stiffened shallow spherical FGM shell