Journal of http://jvc.sagepub.com/ Vibration and Control Nonlinear dynamic analysis of imperfect functionally graded material double curved thin shallow shells with temperature-dependent properties on elastic foundation Nguyen Dinh Duc and Tran Quoc Quan Journal of Vibration and Control published online 31 July 2013 DOI: 10.1177/1077546313494114 The online version of this article can be found at: http://jvc.sagepub.com/content/early/2013/07/31/1077546313494114 Published by: http://www.sagepublications.com Additional services and information for Journal of Vibration and Control can be found at: Email Alerts: http://jvc.sagepub.com/cgi/alerts Subscriptions: http://jvc.sagepub.com/subscriptions Reprints: http://www.sagepub.com/journalsReprints.nav Permissions: http://www.sagepub.com/journalsPermissions.nav >> OnlineFirst Version of Record - Jul 31, 2013 What is This? 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The Author(s) 2013 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1077546313494114 jvc.sagepub.com Nguyen Dinh Duc and Tran Quoc Quan Abstract This paper presents an analytical investigation on the nonlinear dynamic analysis of functionally graded double curved thin shallow shells using a simple power-law distribution (P-FGM) with temperature-dependent properties on an elastic foundation and subjected to mechanical load and temperature The formulations are based on the classical shell theory, taking into account geometrical nonlinearity, initial geometrical imperfection, temperature-dependent properties and unlike other publications, Poisson ratio is assumed to be varied smoothly along the thickness  ẳ zị The nonlinear equations are solved by the Bubnov-Galerkin and Runge-Kutta methods The obtained results show the effects of temperature, material and geometrical properties, imperfection and elastic foundation on the nonlinear vibration and nonlinear dynamical response of double curved FGM shallow shells Some results were compared with those of other authors Keywords Elastic foundation, FGM double curved thin shallow shells, imperfection, nonlinear dynamic analysis, temperaturedependent properties Introduction Functionally graded materials (FGMs), which are microscopically composites and made from a mixture of metal and ceramic constituents, have received considerable attention in recent years due to their high performance heat resistance capacity and their 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 for 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 undertaken 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 different parts of 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 et al (1999) Lam and Li Hua (1999) has taken into account the influence of boundary conditions on the frequency characteristics of a rotating truncated circular conical shell Pradhan et al (2000) studied vibration characteristics of FGM cylindrical shells under various boundary conditions Ng et al (2001) Vietnam National University, Hanoi, Vietnam Received: 22 March 2013; accepted: 19 May 2013 Corresponding author: Nguyen Dinh Duc, Vietnam National University, Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam Email: ducnd@vnu.edu.vn Downloaded from jvc.sagepub.com at University of Texas at El Paso on November 7, 2014 XML Template (2013) [26.7.2013–12:29pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130469/APPFile/SG-JVCJ130469.3d (JVC) [1–23] [PREPRINTER stage] Journal of Vibration and Control 0(0) studied the dynamic stability analysis of functionally graded cylindrical shells under periodic axial loading The group of Ng et al (2003) also published results on generalized differential quadrate for free vibration of rotating composite laminated conical shell with various boundary conditions In the same year, Yang and Shen (2003) published the nonlinear analysis of FGM plates under transverse and in-plane loads Zhao et al (2004) studied the free vibration of a two-sided simply supported laminated cylindrical panel via the mesh-free kp-Ritz method With regard to vibration of FGM plates, Vel and Batra (2004) gave a three-dimensional exact solution for the vibration of FGM rectangular plates Sofiyev and Schnack (2004) investigated the stability of functionally graded cylindrical shells under linearly increasing dynamic tensional loading and obtained the result for the stability of functionally graded truncated conical shells subjected to a periodic impulsive loading They also published the result of the stability of functionally graded ceramic–metal cylindrical shells under a periodic axial impulsive loading in 2005 Ferreira et al (2006) received natural frequencies of FGM plates by a meshless method Zhao et al (2006) used the element-free kp-Ritz method for free vibration analysis of conical shell panels Liew et al (2006a, 2006b) studied the nonlinear vibration of a coating-FGM-substrate cylindrical panel subjected to a temperature gradient and dynamic stability of rotating cylindrical shells subjected to periodic axial loads Woo et al (2006) investigated the nonlinear free vibration behavior of functionally graded plates Ravikiran Kadoli and Ganesan (2006) studied the buckling and free vibration analysis of functionally graded cylindrical shells subjected to a temperature-specified boundary condition Wu et al (2006) published their results of nonlinear static and dynamic analysis of functionally graded plates Sofiyev (2007) has considered the buckling of functionally graded truncated conical shells under dynamic axial loading Prakash et al (2007) studied the nonlinear axisymmetric dynamic buckling behavior of clamped functionally graded spherical caps Darabi et al (2008) 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 Hiroyuki Matsunaga (2008) using 2-D higher-order deformation theory Shariyat (2008a, 2008b) also obtained the dynamic thermal buckling of suddenly heated temperature- dependent FGM cylindrical shells under combined axial compression and external pressure and dynamic buckling of suddenly loaded imperfect hybrid cylindrical FGM with temperature-dependent material properties under thermo-electro-mechanical loads Allahverdizadeh et al (2008) studied nonlinear free and forced vibration analysis of thin circular functionally graded plates Sofiyev (2009) investigated the vibration and stability behavior of freely supported FGM conical shells subjected to external pressure Shen (2009) 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 Zhang and Li (2010) published the dynamic buckling of FGM truncated conical shells subjected to non-uniform normal impact load Ibrahim and Tawfik (2010) studied FGM plates subject to aerodynamic and thermal loads Fakhari and Ohadi (2011) investigated nonlinear vibration control of functionally graded plate with piezoelectric layers in thermal environment Ruan et al (2012) analyzed dynamic stability of functionally graded materials’ skew plates subjected to uniformly distributed tangential follower forces Najafov et al (2012) studied vibration and stability of axially compressed truncated conical shells with a functionally graded middle layer surrounded by elastic medium Bich et al (2012, 2013) investigated nonlinear dynamical analysis of eccentrically stiffened functionally graded cylindrical panels and shallow shells using the classical shell theory Recently, Duc (2013) investigated the nonlinear dynamic response of imperfect FGM double curved shallow shells eccentrically stiffened on an elastic foundation However, in practice, the FGM structure is usually exposed to high-temperature environments, where significant changes in material properties are unavoidable Therefore, the temperature dependence of their properties should be considered for an accurate and reliable prediction of deformation behavior of the composites Duc and Tung (2010) investigated mechanical and thermal post-buckling of FGM plates with temperature-dependent properties using first order shear deformation theory Huang and Shen (2004) studied nonlinear vibration and dynamic response of FGM plates in a thermal environment with temperaturedependent material properties – volume fraction follows a simple power law for P-FGM plate Shariyat investigated vibration and dynamic buckling control of imperfect hybrid FGM plate subjected to thermo-electromechanical conditions (2009) and dynamic buckling of suddenly load imperfect hybrid FGM cylindrical shells (2008) with temperature-dependent material properties Kim (2005) studied temperature dependent vibration analysis of functionally graded rectangular plates by the finite element method It is evident from the literature that investigations considering the temperature dependence of material properties for FGM shells are few in number It should be noted that all the publications mentioned above (Huang and Shen, 2004; Kim, 2005; Shariyat, 2009) use displacement functions and the volume fraction follows a simple power law Downloaded from jvc.sagepub.com at University of Texas at El Paso on November 7, 2014 XML Template (2013) [26.7.2013–12:29pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130469/APPFile/SG-JVCJ130469.3d (JVC) [1–23] [PREPRINTER stage] Duc and Quan This paper presents a dynamic nonlinear response of double curved FGM thin shallow shells with temperature-dependent properties subjected to mechanical load and temperature on an elastic foundation The formulations are based on the classical shell theory taking into account geometrical nonlinearity, initial geometrical with Pasternak type elastic foundation The obtained results show the effects of material, geometrical properties, elastic foundation and imperfection on the dynamical response of FGM shallow shells ðz, TÞ and Kðz, TÞ are obtained by substituting equation (2.1) into (2.2) as Double curved FGM shallow shell on elastic foundation where 2z ỵ h Vm zị ẳ 2h N , Vc zị ¼ À Vm ðzÞ ð1Þ where N is volume fraction index (0 N 1) Effective properties Preff of the FGM panel are determined by linear rule of mixture as Przị ẳ Pr Vm zị ỵ Pr Vc zị eff m c ẳ ẵEc T ị, c T ị, T ị, c T ị, Kc T ị ỵ ẵEmc ðT Þ, mc ðT Þ, mc ðT Þ, mc ðT ị, Kmc T ị   2z ỵ h N 2h 3ị Emc T ị ẳ Em T ị Ec ðT Þ, Consider an FGM double curved thin shallow shell of radii of curvature Rx ,Ry length of edges a, b and uniform thickness h A coordinate system ðx, y, zÞ is established in which ðx, yÞ plane on the middle surface of the shell and z is the thickness direction ðÀh=2 z h=2Þ, as shown in Figure For the P-FGM shell, the volume fractions of constituents are assumed to vary through the thickness according to the following power law distribution:  ẵEz, Tị, vz, Tị, z, Tị, ðz, TÞ, Kðz, Tފ ð2Þ where Pr denotes a temperature independent material property, and subscripts m and c stand for the metal and ceramic constituents, respectively Specific expressions of modulus of elasticity Eðz, TÞ, ðz, TÞ, ðz, TÞ, mc ðT Þ ¼ m ðT Þ À c ðT Þ, mc T ị ẳ m T ị c T ị, 4ị mc T ị ẳ m T ị c T ị, Kmc T ị ẳ Km T ị Kc ðT Þ The values with subscripts m and c belong to metal and ceramic respectively, and unlike other publications, the Poisson ratio is assumed to be varied smoothly along the thickness  ẳ zị It is evident from equations (2.3) and (2.4) that the upper surface of the panel (z ¼ Àh=2) is ceramic-rich, while the lower surface (z ¼ h=2) is metal-rich, and the percentage of ceramic constituent in the panel is enhanced when N increases A material property Pr, such as the elastic modulus E, Poisson ratio , the mass density , the thermal expansion coefficient and coefficient of thermal conduction K can be expressed as a nonlinear function of temperature: À Á Pr ¼ P0 PÀ1 T1 ỵ ỵ P1 T1 ỵ P2 T2 ỵ P3 T3 5ị in which T ẳ T0 ỵ Tzị and T0 ¼ 300K (room temperature); P0, P-1, P1, P2 and P3 are coefficients characteristic of the constituent materials The shell–foundation interaction is represented by the Pasternak model as qe ¼ k1 w À k2 r2 w ð6Þ where r2 ẳ @2 =@x2 ỵ @2 =@y2 , w is the deflection of the panel, k1 is Winkler foundation modulus and k2 is the shear layer foundation stiffness of the Pasternak model Figure Geometry and coordinate system of a P-functionally graded material double curved shallow shell on elastic foundation Theoretical formulation In this study, the classical shell theory is used to establish governing equations and determine the nonlinear Downloaded from jvc.sagepub.com at University of Texas at El Paso on November 7, 2014 XML Template (2013) [26.7.2013–12:29pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130469/APPFile/SG-JVCJ130469.3d [1–23] [PREPRINTER stage] (JVC) Journal of Vibration and Control 0(0) response of FGM thin shallow double curved shells (Brush and Almroth, 1975; Reddy, 2004): 1 ð7Þ where "0x u, x w=Rx ỵ w2,x =2 B C B C B "0 C B C B y C ¼ B v, y À w=Ry ỵ w2,y =2 C, @ A @ A xy u, y ỵ v, x ỵ w, x w, y kx Àwx, x B C B C Bk C B C B y C ¼ B Àwy, y C @ A @ A kxy Àw, xy ð8Þ In which u, v are the displacement components along the x, y directions, respectively Hooke law for an FGM shell is dened as x , y ị ẳ xy ¼ Z EðzÞ zj dz Àh=2 À ðzÞ Z h=2 Ezịzị j z dz I2j ẳ h=2 zị Z h=2 Ezị zj dz ẳ I1j I2j ị I3j ẳ h=2 2ẵ1 ỵ ðzÞ Š Z h=2 EðzÞðzÞ ÁTðzÞð1, zÞdz ðÈ1 , È2 Þ ¼ À Àh=2 À ðzÞ E xy 2ð1 þ Þ ð12Þ The nonlinear equilibrium equations of a perfect FGM double curved shallow shell based on the classical shell theory are (Nayfeh and Pai, 2004): @2 u @2 t @2 v ¼ 1 Mx, xx @ t Nx, x ỵ Nxy, y ẳ 1 Nxy, x ỵ Ny, y ỵ 2Mxy, xy ỵ My, yy ỵ Nx Ny þ Rx Ry ð13Þ þ Nx w, xx þ 2Nxy w, xy ỵ Ny w, yy E "x , "y ị ỵ v"y , "x ị vịT1, 1ị 1v ỵ q k1 w þ k2 r2 w ¼ 1 @2 w @2 t where ð9Þ where ÁT is the temperature rise from a stress-free initial state The force and moment resultants of the FGM shallow shell are determined by Z h=2 I1j ¼ "0x kx "x B C B C B C B C B C B C B "y C ¼ B "y C ỵ zB ky C @ A @ A @ A 2k xy xy xy where Iij ði ¼ 1, 2, 3; j ¼ 0, 1, 2ị: h=2 Ni , Mi ị ẳ i 1, zịdz i ẳ x, y, xy 10ị   c m 1 ẳ zịdz ẳ m ỵ h Nỵ1 h=2 Z h=2 ð14Þ With Volmir’s assumption (Volmir, 1972): u ( w, v ( w, in (13) the inertia 1 @@t2u ! and 1 @@t2v ! 0, the nonlinear motion equations for perfect FGM shells can be written in terms of deflection w and force resultants as Àh=2 Nx, x ỵ Nxy, y ẳ Substitution of equations (7) and (9) into equation (10) and the result into equation (10) gives the constitutive relations as À Á Nx , Ny , Mx , My ¼ ðI10 , I20 , I11 , I21 ị"0x ỵ I21 , I11 , I22 , I12 ịky ỵ ẩ1 , ẩ1 , ẩ2 , ẩ2 ịT ỵ 2I31 , I32 Þkxy Nxy , Mxy ¼ ðI30 , I31 Þ xy Mx, xx ỵ 2Mxy, xy ỵ My, yy ỵ Nx Ny þ Rx Ry ð15Þ þ Nx w, xx þ 2Nxy w, xy ỵ Ny w, yy ỵ q k1 w ỵ k2 r2 w ẳ 1 ỵ I20 , I10 , I21 , I11 ị"0y ỵ I11 , I21 , I12 , I22 ịkx Nxy, x ỵ Ny, y ¼ @2 w @2 t Calculated from equation (11) 11ị "0x ẳ D0 I10 Nx I20 Ny ỵ D1 w, xx ỵ D2 w, yy D3 ẩ1 Tị "0y ẳ D0 I10 Ny I20 Nx ỵ D1 w, yy ỵ D2 w, xx D3 ẩ1 Tị Nxy ỵ 2I31 w, xy ị 0xy ẳ I30 ð16Þ Downloaded from jvc.sagepub.com at University of Texas at El Paso on November 7, 2014 XML Template (2013) [26.7.2013–12:29pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130469/APPFile/SG-JVCJ130469.3d [1–23] [PREPRINTER stage] (JVC) Duc and Quan where From the constitutive relations (16) in conjunction with equation (20): D0 ¼ , D1 ¼ I10 I11 À I20 I21 I210 À I220 D2 ¼ I10 I21 À I20 I11 , D3 ẳ I10 I20 17ị Once again substituting equation (16) into the expression of Mij in (11), then Mij into the equation (15) leads to Nx, x þ Nxy, y ¼ Nxy, x þ Ny, y ẳ P1 r4 f ỵ P2 r4 w ỵ þNx w, xx þ 2Nxy w, xy þ Ny w, yy ỵ P3 ẳ where P1 ẳ D0 D2 , P2 ẳ D0 I11 D1 ỵ I21 D2 ị I12 19ị f(x,y) is stress function dened by Nx ẳ f, yy , Ny ¼ f, xx , Nxy ¼ Àf, xy ð20Þ For imperfect FGM shells, equation (18) is modied into form as   P1 r4 f ỵ P2 r4 w ỵ f, yy w, xx ỵ w, xx 2f, xy w, xy ỵ w, xy ỵ 24ị where Nx Ny ỵ ỵ q k1 w Rx Ry  Setting equation (23) into equation (22) gives the compatibility equation of an imperfect FGM double curved shell as w, xy À w, xx w, yy ỵ 2w, xy w, xy B C B w, xx wÃ, yy À w, yy wÃ, xx C r4 f ỵ P3 r4 w P4 B C ẳ 0: @ A w, yy w, xx À À Rx Ry 18ị @2 w ỵ k2 r2 w ẳ 1 @ t  "0x ¼ D0 ðI10 f, yy I20 f, xx ỵ D1 w, xx ỵ D2 w, yy D3 ẩ1 Tị "0y ẳ D0 I10 f, xx I20 f, yy ỵ D1 w, yy þ D2 w, xx À D3 È1 ÁTÞ ðÀf, xy ỵ 2I31 w, xy ị 0xy ẳ I30 23ị D2 , I10 P4 ẳ D0 I10 25ị Equations (21) and (24) are nonlinear equations in terms of variables w and f and are used to investigate the dynamic response of thick double curved shallow FGM shells on elastic foundations subjected to mechanical, thermal and thermo-mechanical loads Depending on the in-plane restraint at the edges, three cases of boundary conditions, labeled as Cases 1, and may be considered (Reddy, 2004; Duc and Tung, 2010): Case 1: Four edges of the shallow shell are simply supported and freely movable (FM) The associated boundary conditions are w ¼ Nxy ¼ Mx ¼ 0, Nx ¼ Nx0 at x ¼ 0, a   ỵ f, xx w, yy ỵ w, yy 21ị f, yy f, xx @2 w ỵ þ q À k1 w þ k2 r2 w ¼ 1 @ t Rx Ry à in which w ðx, yÞ is a known function representing initial small imperfection of the shell Following Volmir’s approach, the geometrical compatibility equation for an imperfect double curved shallow shell is written as 26ị w ẳ Nxy ẳ My ẳ 0, Ny ẳ Ny0 at y ¼ 0, b: Case 2: Four edges of the shallow shell are simply supported and immovable (IM) In this case, boundary conditions are w ¼ u ¼ Mx ¼ 0, Nx ¼ Nx0 at x ¼ 0, a w ¼ v ¼ My ¼ 0, Ny ¼ Ny0 at y ẳ 0, b 27ị Case 3: All edges are simply supported Two edges x ¼ 0, a are freely movable, whereas the remaining two edges y ¼ 0, b are immovable For this case, the boundary conditions are dened as "0x, yy ỵ "0y, xx À xy, xy ¼ w, xy À w, xx w, yy ỵ 2w, xy w, xy w, xx w, yy À w, yy wÃ, xx À w, yy w, xx : Rx Ry 22ị w ẳ Nxy ẳ Mx ¼ 0, Nx ¼ Nx0 at x ¼ 0, a w ¼ v ¼ My ¼ 0, Ny ¼ Ny0 at y ¼ 0, b Downloaded from jvc.sagepub.com at University of Texas at El Paso on November 7, 2014 ð28Þ XML Template (2013) [26.7.2013–12:29pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130469/APPFile/SG-JVCJ130469.3d (JVC) [1–23] [PREPRINTER stage] Journal of Vibration and Control 0(0) where Nx0 , Ny0 are in-plane compressive loads at movable edges (i.e Case and the first of Case 3) or are fictitious compressive edge loads at immovable edges (i.e Case and the second of Case 3) In the present study, the edges of curved shallow shells are assumed to be simply supported and immovable (Case 2) Taking into account temperature-dependent material properties, the mentioned conditions (27) can be satisfied if the deflectionw, wà and the stress function f are written in the form similar to (Duc and Tung, 2010): mx ny sin w ẳ tị sin a b mx ny w ¼ 0 sin sin a b 2mx 2ny f ẳ A1 cos ỵ A2 cos a b mx ny þ A3 sin sin a b mx ny 1 þ A4 cos cos þ Nx0 y2 þ Ny0 x2 a b 2 A simply supported FGM double curved shallow shell on elastic foundations with all immovable edges is considered The FGM shell is subjected to uniform external pressure q and simultaneously exposed to temperature environments or subjected to temperature gradient which varies through the thickness of the shell The in-plane condition on immovability at all edges, i.e u ¼ at x ¼ 0, a and v ¼ at y ¼ 0, b, is fulfilled in an average sense as (Shen, 2004; Kim, 2005; Duc and Tung, 2010) Z bZ @u dx dy ¼ 0, @x Z aZ 0 b @v dydx ¼ @y ð31Þ From equations (7) and (16) the following expressions, in which equation (20) and imperfections have been included, can be obtained: @u ¼ D0 ðI10 f, yy À I20 f, xx ỵ D1 w, xx ỵ D2 w, yy D3 È1 Þ @x w À w2,x À w, x w, x ỵ Rx @v ẳ D0 I10 f, xx I20 fyy ỵ D1 w, yy ỵ D2 w, xx À D3 È1 Þ @y w w2,y w, y w, y ỵ Ry 29ị m ẳ m=a, n ẳ n=b, m, n ẳ 1, 2, , are natural numbers representing the number of half waves in the x and y directions respectively;  is the deflection amplitude; 0 ¼ const, varying between and 1, represents the size of the imperfections We should note that the choice of f in (29) is different from the form used in Duc and Tung, 2010 and Duc, 2013 The coefficients Ai ði ¼ Ä 3Þ are determined by substitution of equation (29) into equation (24) as P4 2n P4 2m  ỵ 20 ị, A2 ẳ  ỵ 20 ị, 32m 322n   P4 n 2m A3 ¼ À þ  À P3 , A4 ¼ 0: Á2 Rx Ry 2m ỵ 2n a 32ị Substitution of equation (29) into equation (32) and then the result into equation (31) gives fictitious edge compressive loads as &   4 I10 I20 2 I  ỵ I  ị ỵ Nx0 ẳ ẩ1 ỵ 21 n 11 m mn2 mn2 Rx Ry " #   P4 n 2m n o ỵ ỵ4  P3 mb2 Rx Ry 2 ỵ 2 A1 ẳ 30ị m n ỵ I10 2m ỵ I20 2n ị ỵ 20 ị, 33ị Table Material properties of the constituent materials of the considered functionally graded material shells Material Property P0 PÀ1 P1 P2 P3 Si3N4 (Ceramic) E (Pa)  (kg/m3) ðKÀ1 Þ k ðW=mKÞ  E (Pa)  (kg/m3) ðKÀ1 Þ k ðW=mKÞ  348.43e9 2370 5.8723eÀ6 13.723 0.24 201.04e9 8166 12.330eÀ6 15.379 0.3177 0 0 0 0 0 À3.70eÀ4 9.095eÀ4 0 3.079eÀ4 8.086eÀ4 0 2.160eÀ7 0 0 À6.534eÀ7 0 0 À8.946eÀ11 0 0 0 0 SUS304 (Metal) Downloaded from jvc.sagepub.com at University of Texas at El Paso on November 7, 2014 XML Template (2013) [26.7.2013–12:29pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130469/APPFile/SG-JVCJ130469.3d (JVC) [1–23] [PREPRINTER stage] Duc and Quan &   4 I10 I20 2 Ny0 ¼ È1 þ ðI  þ I  Þ À þ 11 n 21 m mn2 mn2 Ry Rx " # '   P4 n 2m m ỵ ỵ4 À P3  Á2 2 na R R x y m ỵ n ỵ I20 2m ỵ I10 2n ị ỵ 20 ị: Specic expressions of parameter È1 in two cases of thermal loading will be determined Subsequently, substitution of equation (29) into equation (21) and applying the Galerkin procedure for the resulting equation yields   ab n 2m ẵP3 ỵ P1 P4 ị ỵ ỵ P2 P1 P3 ị 2m þ 2n Rx Ry  2 À Á P4 n 2m ỵ k2 2m ỵ 2n k1  Rx Ry 2m ỵ 2n " #   8m n P4 n 2m ỵ ỵ P3  ỵ 0 ị Rx Ry 2m ỵ 2n   ! P4 m 2n ỵ ỵ P1 P4 m n  þ 20 Þ 6m n Ry Rx Á P4 ab m ỵ 4n  ỵ 0 ị þ 20 Þ À 64 Á ab À Nx0 2m þ Ny0 2n ð þ 0 Þ À   Nx0 Ny0 4q ab @2  ỵ ẳ 1 ỵ ỵ m n Rx m n @2 t Ry ð34Þ Where m, n are odd numbers This is a basic equation governing the nonlinear dynamic response for thin imperfect FGM double curved shallow shells under mechanical, thermal and thermo-mechanical loading conditions In what follows, some thermal loading conditions will be considered Introducing Nx0 , Ny0 at (33) into equation (34) gives ỵ m1 tị ỵ m2 2 tị ỵ m3 3 tị tị ỵ m4 tị0 ỵ m5  tị0 ỵ ỵ m 0 ẳ m q ỵ m m6 tị20 35ị and specific expressions of coefficients mi ði ¼ Ä 8Þ are given in Appendix A and ðtÞ -deflection of  middle point of the plate tị ẳ wxẳa=2 yẳb=2 For linear free vibration for FGM plate equation (35) gets the form: :: tị ỵ m1 tị ẳ ð36Þ The fundamental frequency of natural vibration of the FGM plate can be determined: !L ẳ p m1 37ị Equation (35) – for obtaining the nonlinear dynamic response the initial conditions are assumed as : 0ị ẳ 0 , 0ị ¼ The applied loads are varying as function of time The nonlinear dynamic response of the FGM shell acted on by the harmonic uniformly excited transverse load qtị ẳ Q0 sin t are obtained by solving equation (35) combined with the initial conditions and with the use of the Runge-Kutta method Numerical results and discussion Here, several numerical examples will be presented for perfect and imperfect simply supported midplanesymmetric of the FGM shell The typical values of the coefficients of the materials mentioned in equation (5) are listed in Table (Reddy and Chin, 1998) Table Frequency of natural vibration (rad/s) of spherical shallow shells with Rx ¼ Ry ¼ ðmÞ and N ¼ T-ID T-D ðm, nÞ Present Bich et al (2012) Present (1,1) (1,2) and (2,1) (2,2) (1,3) and (3,1) (2,3) and (3,2) 3.4582e3 7.3694e3 11.1643e3 15.9583e3 18.3469e3 2.5604e3 6.7431e3 9.3093e3 14.5762e3 16.0638e3 2.4229e3 6.0439e3 9.5415e3 14.3293e3 16.1738e3 T-D: temperature dependent; T-ID: temperature independent Table A comparison the fundamental natural ffi pffiffiffiffiffiffiffiffiffiffiamong frequency É ¼ !L h c =Ec of the square Al/Al2O3 functionally graded material plates pffiffiffiffiffiffiffiffiffiffiffi É ¼ !L h c =Ec a=h N Matsunaga (2008) Shariyat (2009) Present 0.5 0.5 0.2121 0.1819 0.1640 0.05777 0.04917 0.04426 0.2083 0.1762 0.1594 0.05682 0.04876 0.04369 0.2287 0.1575 0.1380 0.4865 0.0386 0.0297 10 Downloaded from jvc.sagepub.com at University of Texas at El Paso on November 7, 2014 XML Template (2013) [26.7.2013–12:29pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130469/APPFile/SG-JVCJ130469.3d (JVC) [1–23] [PREPRINTER stage] Journal of Vibration and Control 0(0) À Á À Á Ã1=2 Table Natural frequency parameter !L a2 =h 0 À 2 E0 for Si3N4/SUS304 double curved shallow shells in thermal environments T-ID (m,n) Tm ¼ 300 K Tm ¼ 300 K Tc ¼ 400 K Tc ¼ 600 K (1,1) (1,2) (2,2) (1,3) (2,3) (1,1) (1,2) (2,2) (1,3) (2,3) and (2,1) and (3,1) and (3,2) and (2,1) and (3,1) and (3,2) T-D Shen (2004) Present Present Shen (2004) 7.514 17.694 26.717 32.242 39.908 7.305 17.486 26.506 31.970 39.692 7.0683 18.9712 29.8519 34.7710 40.6462 7.0683 18.9712 30.8519 36.7710 44.6462 6.7888 16.1335 25.4717 30.6960 40.0313 6.7574 15.5673 28.4734 33.0257 43.9825 7.474 17.607 26.590 32.088 39.721 7.171 17.213 26.109 31.557 39.114 T-D: temperature dependent; T-ID: temperature independent Figure Effect of b=a on nonlinear dynamic response of the functionally graded material shell (temperature-independent T ẳ 300 Kị) b=h ẳ 30, m ẳ n ¼ 1, N ¼ 1, Rx ¼ Ry ¼ 0:6 mị, Q0 ẳ 75000 N=m2 ị,  ẳ 3000 rad=sị, 0 ẳ 0: k1 ẳ k2 ẳ : b=a ¼ 1; À À À : b=a ¼ 2Þ The metal-rich surface temperature Tm is maintained at a stress-free initial value while ceramic-rich surface temperature Tc is elevated and nonlinear steady temperature conduction is governed by one-dimensional Fourier equation ! d dT Kzị ẳ 0, Tz ẳ h=2ị ¼ Tc , dz dz Tðz ¼ h=2Þ ¼ Tm Downloaded from jvc.sagepub.com at University of Texas at El Paso on November 7, 2014 ð38Þ XML Template (2013) [26.7.2013–12:29pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130469/APPFile/SG-JVCJ130469.3d (JVC) [1–23] [PREPRINTER stage] Duc and Quan Figure Effect of b=a on nonlinear dynamic response of the functionally graded material shell (temperature-dependent, T ẳ T zị, Tc ẳ 500 Kị, Tm ẳ 300 Kị) b=h ẳ 30, m ¼ n ¼ 1, N ¼ 1, Rx ¼ Ry ¼ 0:6 ðmÞ, Q0 ¼ 75000 ðN=m2 Þ,  ¼ 3000 rad=sị, 0 ẳ 0: k1 ẳ k2 ẳ 0: ðÀÀ : b=a ¼ 1; À À À : b=a ¼ 2Þ Figure Effect of b=h on nonlinear dynamic response of the functionally graded material shell (temperature-independent, ÁT ¼ 300 Kị) b=a ẳ 2, N ẳ 1, m ẳ n ẳ 1, Rx ẳ Ry ẳ 0:6 mị, Q0 ¼ 75000 ðN=m2 Þ,  ¼ 3000 ðrad=sÞ, 0 ¼ 0: k1 ¼ k2 ¼ Downloaded from jvc.sagepub.com at University of Texas at El Paso on November 7, 2014 XML Template (2013) [26.7.2013–12:29pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130469/APPFile/SG-JVCJ130469.3d 10 (JVC) [1–23] [PREPRINTER stage] Journal of Vibration and Control 0(0) Figure Effect of b=h on nonlinear dynamic response of the functionally graded material shell (temperature-dependent, T ẳ T zị, Tc ẳ 500 Kị, Tm ẳ 300 Kị) b=a ẳ 2, N ẳ 1, m ¼ n ¼ 1, Rx ¼ Ry ¼ 0:6 mị, Q0 ẳ 75000 N=m2 ị,  ẳ 3000 rad=sị, 0 ¼ 0: k1 ¼ k2 ¼ ðÀÀÀ : b=h ẳ 20; : b=h ẳ 30ị Figure Effect of imperfection 0 on nonlinear dynamic response of functionally graded material shell (temperature-independent, T ẳ 300 Kị) b=a ¼ 1, b=h ¼ 30, N ¼ 1, m ¼ n ¼ 1, Rx ¼ Ry ¼ 0:6 ðmÞ, k1 ¼ k2 ¼ Downloaded from jvc.sagepub.com at University of Texas at El Paso on November 7, 2014 XML Template (2013) [26.7.2013–12:29pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130469/APPFile/SG-JVCJ130469.3d (JVC) [1–23] [PREPRINTER stage] Duc and Quan 11 Figure Effect of imperfection 0 on nonlinear dynamic response of functionally graded material shell (temperature-dependent, ÁT ¼ Tzị), b=a ẳ 1, b=h ẳ 30, N ẳ 1, m ¼ n ¼ 1, Rx ¼ Ry ¼ 0:6 mị, k1 ẳ k2 ẳ 0: : 0 ẳ : 0 ¼ 0:001; À À : 0 ẳ 0:003ị Figure Deflection-velocity relation (d=dt ) with temperature-independent, T ẳ 300 Kị of the functionally graded material shallow spherical shell Downloaded from jvc.sagepub.com at University of Texas at El Paso on November 7, 2014 XML Template (2013) [26.7.2013–12:29pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130469/APPFile/SG-JVCJ130469.3d 12 (JVC) [1–23] [PREPRINTER stage] Journal of Vibration and Control 0(0) Figure Deflection-velocity (d=dtÀ) with temperature-dependent, T ẳ T zị, Tc ẳ 500 Kị, Tm ¼ 300 ðKÞ of the functionally graded material shallow spherical shell Figure 10 Effect of  on nonlinear dynamic response of the functionally graded material shell (temperature-independent, ÁT ¼ 300 Kị b=a ẳ 1, b=h ẳ 30, N ẳ 1, m ¼ n ¼ 1, Rx ¼ Ry ¼ 0:6 mị, Q0 ẳ 75000 N=m2 ị, 0 ẳ 0: k1 ¼ k2 ¼ Downloaded from jvc.sagepub.com at University of Texas at El Paso on November 7, 2014 XML Template (2013) [26.7.2013–12:29pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130469/APPFile/SG-JVCJ130469.3d (JVC) [1–23] [PREPRINTER stage] Duc and Quan 13 Figure 11 Effect of  on nonlinear dynamic of the functionally graded material shell (T À D, ÁT ¼ Tzị, Tc ẳ 500 Kị, Tm ẳ 300 Kị) b=a ¼ 1, b=h ¼ 30, N ¼ 1, m ¼ n ẳ 1, Rx ẳ Ry ẳ 0:6 mị, Q0 ¼ 75000 ðN=m2 Þ, 0 ¼ 0: k1 ¼ k2 ¼ ðÀÀÀ :  ¼ 2500ðrad=sÞ; À À À :  ẳ 3000rad=sịị Figure 12 Effect of amplitude Q0 on dynamic response of the functionally graded material shell (temperature-independent, T ẳ 300 Kị) b=a ẳ 1, b=h ẳ 30, N ¼ 1, m ¼ n ¼ 1, Rx ¼ Ry ẳ 0:6 mị,  ẳ 3000 rad=sị, 0 ẳ 0: k1 ¼ k2 ¼ Downloaded from jvc.sagepub.com at University of Texas at El Paso on November 7, 2014 XML Template (2013) [26.7.2013–12:29pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130469/APPFile/SG-JVCJ130469.3d 14 (JVC) [1–23] [PREPRINTER stage] Journal of Vibration and Control 0(0) Figure 13 Effect of amplitude Q0 on dynamic response of the functionally graded material shell (T D, T ẳ T zị, Tc ẳ 500 Kị, Tm ẳ 300 Kị) b=a ẳ 1, b=h ¼ 30, N À¼ 1, m Á¼ n ¼ 1, Rx ẳ Ry ẳ 0:6 mị,  ẳ 3000 rad=sị, 0 ẳ 0: k1 ẳ k2 ẳ : Q0 ¼ 75000 N=m2 ; À À À : Q0 ẳ 95000 N=m2 ị Figure 14 Effect of elastic foundations k1 , k2 on nonlinear dynamic response of the functionally graded material shell (temperatureindependent, T ẳ 300 Kị) b=a ¼ 1, b=h ¼ 30, N ¼ 1, m ¼ n ẳ 1, Rx ẳ Ry ẳ 0:6 mị, Q0 ¼ 75000 ðN=m2 Þ,  ¼ 3000 ðrad=sÞ, 0 ¼ Downloaded from jvc.sagepub.com at University of Texas at El Paso on November 7, 2014 XML Template (2013) [26.7.2013–12:29pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130469/APPFile/SG-JVCJ130469.3d (JVC) [1–23] [PREPRINTER stage] Duc and Quan 15 Figure 15 Effect of elastic foundations k1 , k2 (temperature-dependent, ÁT ¼ T zị, Tc ẳ 500 Kị, Tm ẳ 300 Kị) on nonlinear dynamic response of the functionally graded material shell b=a ¼ 1, b=h ¼ 30, N ¼ 1, m ¼ n ¼ 1, Rx ¼ Ry ¼ 0:6 ðmÞ, Q0 ẳ 75000 N=m2 ị,  ẳ 3000 rad=sị, 0 ¼ ðÀÀ: k1 ¼ 20, k2 ¼ 100; À : k1 ẳ 50, k2 ẳ 200ị Figure 16 Effect of temperature (temperature-independent) on nonlinear dynamic response of the shell b=a ¼ 1, b=h ¼ 30, N ¼ 1, m ¼ n ¼ 1, Rx ¼ Ry ¼ 0:6 mị, Q0 ẳ 75000 N=m2 ị,  ẳ 3000 rad=sị, 0 ẳ Downloaded from jvc.sagepub.com at University of Texas at El Paso on November 7, 2014 XML Template (2013) [26.7.2013–12:29pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130469/APPFile/SG-JVCJ130469.3d 16 (JVC) [1–23] [PREPRINTER stage] Journal of Vibration and Control 0(0) Figure 17 Effect of temperature (temperature-dependent) on nonlinear dynamic response of the shell b=a ¼ 1, b=h ¼ 30, N ¼ 1, m ¼ n ¼ 1, Rx ẳ Ry ẳ 0:6 mị, Q0 ẳ 75000 N=m2 ị,  ẳ 3000 rad=sị, 0 ẳ : Tc ẳ 400 Kị, Tm ẳ 300 Kị; : Tc ẳ 500 Kị, Tm ẳ 300 Kị Figure 18 The frequency-amplitude relation of nonlinear free vibration !nl - (temperature-dependent, T ẳ 300 Kị) Downloaded from jvc.sagepub.com at University of Texas at El Paso on November 7, 2014 XML Template (2013) [26.7.2013–12:29pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130469/APPFile/SG-JVCJ130469.3d (JVC) [1–23] [PREPRINTER stage] Duc and Quan 17 Figure 19 The frequency-amplitude relation of nonlinear free vibration !nl - (temperature-independent, T ẳ T zị, Tc ¼ 500 ðKÞ, Tm ¼ 300 ðKÞ) Figure 20 Effect of Poisson ratio on dynamic response of the functionally graded material spherical shallow shell b=a ¼ 1, b=h ¼ 30, N ¼ 1, m ¼ n ¼ 1, Rx ¼ Ry ẳ 0:6 mị, Q0 ẳ 75000 N=m2 ị,  ¼ 3000 ðrad=sÞ, 0 ¼ 0, Tc ¼ 500 ðKÞ, Tm ẳ 300 Kị :  ẳ zị; :  ẳ constị Downloaded from jvc.sagepub.com at University of Texas at El Paso on November 7, 2014 XML Template (2013) [26.7.2013–12:29pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130469/APPFile/SG-JVCJ130469.3d (JVC) [1–23] [PREPRINTER stage] 18 Journal of Vibration and Control 0(0) Figure 21 Nonlinear dynamic responses of functionally graded material spherical shallow shells (without the temperature gradient, 1: Bich (2012); 2: Present) Using KðzÞ in equation (3), the solution of equation (38) may be found in terms of polynomial series, and the first eight terms of this series gives the following approximation (Shen, 2004; Kim, 2005; Duc and Tung, 2010): P5 Kmc =Kc ịj jẳ0 jNỵ1 Hẳ P5 jẳ0 r mc ỵEmc c ỵ Ecjỵ1ịNỵ2 Emc mc ỵ jỵ2ịNỵ2 Kmc =Kc ịj jNỵ1 43ị P5 rN Kmc =Kc ịj jẳ0 jNỵ1 Kmc =Kc ịj jẳ0 jNỵ1 Tzị ẳ Tm ỵ T T P5 39ị where r ẳ 2z ỵ hị=2h and ÁT ¼ Tc À Tm is defined as the temperature change between two surfaces of the FGM shallow shell Introduction of equation (39) into equation (12) gives the thermal parameter as ẩ1 ẳ L HịhT The equation of nonlinear free vibration of perfect FGM shells can be obtained from (35) :: tị ỵ m1 tị ỵ m2 2 tị ỵ m3 3 tị ẳ !NL Emc mc P ỵ Ec mc ỵ Emc c ị N ỵ 1ịmc mc   Z1 d þ N À c À mc À c mc  P ẳ mc Ec mc ỵ c Emc ị Emc mc ỵ Emc mc c 41ị 42ị 44ị Seeking solution as tị ẳ cos!tị and applying a procedure such as Galerkin’s method to equation (44), the frequency-amplitude relation of nonlinear free vibration is obtained ð40Þ where Lẳ Ec c jNỵ2  1 3m3 2 ẳ !L ỵ  4!L 45ị where !NL is the nonlinear vibration frequency and  is the amplitude of nonlinear vibration Table illustrates the numerical calculation of the fundamental frequency of natural vibration (37) applied for the spherical shallow shells in the temperature dependent case (T-D) and the temperature independent one (T-ID) We can conclude that the temperature has a significant effect on the frequency Downloaded from jvc.sagepub.com at University of Texas at El Paso on November 7, 2014 XML Template (2013) [26.7.2013–12:29pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130469/APPFile/SG-JVCJ130469.3d (JVC) [1–23] [PREPRINTER stage] Duc and Quan 19 Figure 22 Nonlinear response of functionally graded material cylindrical panel b ¼ a ¼ 1:5 mị, h ẳ 0:008 mị, N ẳ 5, Rx ẳ 1, Ry ẳ mị, k1 ẳ k2 ẳ 0, ÁT ¼ Bich (2012); Present of the FGM shells In the case of T-ID, we have made a comparison with Bich et al (2012) and we have found good agreement between the two calculations pffiffiffiffiffiffiffiffiffiffiffiffi Table presents the calculation of É ¼ !L h c =Ec for Al/Al2O3 FGM (i.e.1=Rx ¼ 1=Ry ¼ 0) in the case of T-D without the plastic foundation (k1 ¼ k2 ¼ 0, b=a ¼ 1, b=h ¼ 30, N ¼ 1, Tc ẳ 500 Kị, Tm ẳ 300 Kị) We can see that there is great agreement between our results and those reported in Matsunaga (2008) and Shariyat (2009) illustrates the calculation of ÀTableÁ À4 Á Ã1=2 !L a2 =h 0 À 2 E0 for Si3N4/SUS304 FGM plate in a thermal environment for two cases (T-ID and T-D) with b=a ¼ 1, b=h ¼ 30, 1=Rx ¼ 1=Ry ¼ 0, k1 ¼ k2 ¼ 0, N ¼ This finding agrees greatly with the one captured by Shen (2004) for FGM plates Figures and describe the effect of a geometric parameter b=a on the nonlinear dynamic response of the double curved FGM shallow shells in the case of T-ID (Figure 2) and T-D (Figure 3) Obviously, the larger the ratio b=a, the stronger the effect on the nonlinear dynamic response of the shell Figures and illustrate the effect of the geometric parameter b=h (where h is the height of the shell) on the nonlinear dynamic response of the double curved FGM shallow shells for the T-ID and T-D cases, respectively It is clear that the larger the ratio b=h, the stronger the effect on the nonlinear dynamic response of the FGM shallow shells Figures and illustrate the effect of the imperfect coefficient 0 on the nonlinear dynamic response of FGM shallow shells under the conditions: b=a ¼ 1, b=h ¼ 30, N ¼ 1, k1 ¼ k2 ¼ Also, the imperfect coefficient has a significant effect on the dynamic response of the FGM shell Figures and describe the deflection-velocity relation of the spherical FGM shallow shell in the T-ID case (Figure 8) and in the T-D case (Figure 9) under the conditions: b=a ¼ 1, b=h ¼ 30, N ¼ 1, Rx ẳ Ry ẳ 0:6 mị, Q0 ẳ 75000 N=m2 ị,  ẳ 3000 rad=sị, 0 ẳ 0, k1 ¼ k2 ¼ Obviously, the temperature has a significant effect on the deflection-velocity relation of the shell Figures 10 and 11 illustrate the effect of the frequency  on the nonlinear dynamic response of the double curved FGM shallow shell in the T-ID case (Figure 10) and in the T-D case (Figure 11), respectively At a frequency , its effect on the dynamic response of the shell in the T-D case is stronger and faster than its effect in the T-ID case Figures 12 and 13 show the effect of the dynamical load (i.e an amplitude Q0 ) on the nonlinear dynamic response of the double curved FGM shallow shell As a result, the larger the amplitude of the dynamic loads are Downloaded from jvc.sagepub.com at University of Texas at El Paso on November 7, 2014 XML Template (2013) [26.7.2013–12:29pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130469/APPFile/SG-JVCJ130469.3d (JVC) [1–23] [PREPRINTER stage] 20 Journal of Vibration and Control 0(0) Figure 23 Nonlinear dynamic response of the functionally graded material spherical shallow shell b ¼ a ¼ mị, h ẳ 0:01 mị, Rx ẳ Ry ẳ ðmÞ and the higher the temperature is, the stronger its effects on dynamic of the FGM shell Figures 14 and 15 illustrate the effect of the elastic coefficient k1 , k2 on the nonlinear dynamic response of the double curved FGM shallow shells under the conditions: b=a ¼ 1, b=h ¼ 30, N ¼ 1, Q0 ¼ 75000 ðN=m2 Þ,  ¼ 3000 ðrad=sÞ, 0 ¼ 0: We conclude that these coefficients have a strong effect on the nonlinear dynamic response of the FGM shells Compared to the case corresponding to the coefficient k1 , the Pasternak type elastic foundation with the coefficient k2 has a stronger effect Figures 16 and 17 show the effect of the temperature on the nonlinear dynamic response of the double curved FGM shallow shell Obviously, the temperature has a strong effect on the dynamic response of the shell Figures 18 and 19 illustrate the frequency-amplitude relation of nonlinear free vibration !nl - in the case of T-ID (Figure 18) and T-D (Figure 19) The higher the temperature, the stronger the amplitude of nonlinear free vibration of the FGM shell Unlike other publications, the Poisson ratio is assumed to be varied smoothly along the thickness  ẳ zịin this framework The analytical calculation and transformation are, therefore, more complex However, Figure 20 shows us that there is no significant change in the dynamic response of the FGM spherical shallow shell in case of  ¼ ðzÞ and  ¼ const In the case of 1=Rx ¼ 1=Ry , the shell is a spherical shallow shell Figure 21 shows the nonlinear dynamic response of the FGM spherical shallow shell (without the temperature gradient ÁT ¼ 0) has a good agreement with the recent report by Bich et al (2012) In particular, Rx ¼ or 1=Rx ¼ makes the FGM shells in the form of the cylindrical shallow panel Figure 22 shows that our findings are in good agreement with the calculation in Bich (2012) applied for the same system Figure 23 compares the nonlinear dynamic response of FGM spherical shallow shells in the presence of the temperature gradient (i.e ÁT ¼ 300K) and in the absence of temperature gradient (i.e ÁT ¼ 0) reported in recent work Duc (2013) This result shows us the significant effect of the temperature on the dynamic response between the bending and time Conclusion This paper presents an analytical investigation on the nonlinear dynamic response of double curved P-FGM thin shallow shells with temperature-dependent properties on an elastic foundation and subjected to mechanical load and temperature The formulations are based on the classical shell theory taking into account geometrical nonlinearity, initial geometrical imperfection with Pasternak type elastic foundation The nonlinear Downloaded from jvc.sagepub.com at University of Texas at El Paso on November 7, 2014 XML Template (2013) [26.7.2013–12:29pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130469/APPFile/SG-JVCJ130469.3d (JVC) [1–23] [PREPRINTER stage] Duc and Quan 21 equations are solved by Bubnov-Galerkin and the Runge-Kutta methods The obtained results show the effects of material and geometrical properties, imperfection, elastic foundation and temperature on the dynamical response of P-FGM double curved thin shallow shells Therefore, when we change these factors, we can control the dynamic response and vibration of the FGM shell actively Some results were compared with those of other authors Funding This work was supported by the grant in mechanics of the National Foundation for Science and Technology Development of Vietnam – NAFOSTED The authors are grateful for this support References Allahverdizadeh A, Naei MH and Nikkhah Bahrami M (2008) Nonlinear free and forced vibration analysis of thin circular functionally graded plates Journal of Sound and Vibration 310: 966–984 Bich DH, Dung DV and Nam VH (2012) Nonlinear dynamical analysis of eccentrically stiffened functionally graded cylindrical panels Journal of Composite Structures 94: 2465–2473 Bich DH, Dung DV and Nam 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À Á2  2 P4 n 2m ỵ k2 2m ỵ 2n k1 P3 ỵ P1 P4 ị n ỵ m ỵ P2 P1 P3 ị 2m þ 2n À À Á2 1 Rx Ry Rx Ry 2m ỵ 2n & 2 ẩ1 m ỵ n ị 16 ỵ I21 2n ỵ I11 2m Þ 1 mn 1 mn2 Rx " #     ' I10 I20 P4 n 2m n ỵ ỵ ỵ4 P3 mn2 Rx Rx Ry mb2 Rx Rx Ry 2m ỵ 2n ( " # )     P4 n 2m m 16 4 I10 I20 2 ỵ I11 n ỵ I21 m ị ỵ ỵ4 P3 na2 Ry mn2 1 mn2 Ry mn2 Ry Ry Rx Rx Ry 2m ỵ 2n m1 ¼ Downloaded from jvc.sagepub.com at University of Texas at El Paso on November 7, 2014 XML Template (2013) [26.7.2013–12:29pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130469/APPFile/SG-JVCJ130469.3d (JVC) [1–23] [PREPRINTER stage] Duc and Quan 23 " #     ! À32mn2 P4 n 2m P4 m 2n P m2 ẳ ỵ ỵ P   P À Á2 m n ab1 6m n Ry Rx 3a2 b2 1 Rx Ry 2m ỵ 2n ! I10 2m ỵ I20 2n ị I20 2m ỵ I10 2n ị ỵ mn2 1 Rx Ry " # &     ' m2 2 4 I10 I20 P4 n 2m n 2 ỵ I  ỵ I  ị þ þ þ À P À Á2 21 n 11 m mn2 Rx Rx Ry mb2 Rx a 1 mn2 Rx Rx Ry 2m ỵ 2n ( " # )     n2 2 4 I I P   m 10 20 n I11 2n ỵ I21 2m ị ỵ þ m À P3 þ þ4 À Á2 mn2 Ry Ry Rx na2 Ry b 1 mn2 Ry Rx Ry 2m ỵ 2n P4 m3 ẳ m ỵ 4n ỵ I10 4m ỵ 2I20 2m 2n ỵ I10 4n ị 81 161 " #     ! À32mn2 P4 n 2m P4 m 2n ỵ ỵ m4 ẳ P3 À À P1 P4 m n À Á2 ab1 6m n Ry Rx 3a2 b2 1 Rx Ry 2m ỵ 2n " # &     ' m2 2 4 I10 I20 P4 2n 2m n 2 I  ỵ I  ị ỵ þ þ þ À P À Á2 21 n 11 m mn2 Rx Rx Ry mb2 Rx a 1 mn2 Rx Rx Ry 2m ỵ 2n ( " # )     n2 2 4 I10 I20 P4 n 2m m 2 I11 n ỵ I21 m ị ỵ ỵ ỵ ỵ4 P3 mn2 Ry Ry Rx na2 Ry b 1 mn2 Ry Rx Ry 2m ỵ 2n ! I10 2m ỵ I20 2n ị I20 2m ỵ I10 2n ị ỵ mn2 1 Rx Ry 3P4 m5 ẳ  ỵ 4n ỵ I10 4m ỵ 2I20 2m 2n ỵ I10 4n Þ 81 161 m Á Á P4 À m6 ẳ m ỵ 4n ỵ I10 4m þ 2I20 2n 2m þ I10 4n 41 81 Á ẩ1 m7 ẳ  ỵ 2n 1 m 16 m8 ¼ 1 mn2   4È1 1 m9 ẳ ỵ m n Rx Ry Downloaded from jvc.sagepub.com at University of Texas at El Paso on November 7, 2014 ... dynamic analysis of imperfect functionally graded material double curved thin shallow shells with temperature-dependent properties on elastic foundation Journal of Vibration and Control 0(0) 1–23... effects of temperature, material and geometrical properties, imperfection and elastic foundation on the nonlinear vibration and nonlinear dynamical response of double curved FGM shallow shells. .. compared with those of other authors Keywords Elastic foundation, FGM double curved thin shallow shells, imperfection, nonlinear dynamic analysis, temperaturedependent properties Introduction Functionally