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Accepted Manuscript An Analytical Approach: Nonlinear Vibration of Imperfect Stiffened Fgm Sandwich Toroidal Shell Segments Containing Fluid Under External ThermoMechanical Loads Dao Huy Bich, Dinh Gia Ninh PII: DOI: Reference: S0263-8223(16)32131-6 http://dx.doi.org/10.1016/j.compstruct.2016.11.065 COST 8026 To appear in: Composite Structures Received Date: Revised Date: Accepted Date: 11 October 2016 20 November 2016 23 November 2016 Please cite this article as: Huy Bich, D., Gia Ninh, D., An Analytical Approach: Nonlinear Vibration of Imperfect Stiffened Fgm Sandwich Toroidal Shell Segments Containing Fluid Under External Thermo-Mechanical Loads, Composite Structures (2016), doi: http://dx.doi.org/10.1016/j.compstruct.2016.11.065 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain AN ANALYTICAL APPROACH: NONLINEAR VIBRATION OF IMPERFECT STIFFENED FGM SANDWICH TOROIDAL SHELL SEGMENTS CONTAINING FLUID UNDER EXTERNAL THERMO-MECHANICAL LOADS Dao Huy Bicha, Dinh Gia Ninhb* a Professor, D Sci, Department of Mathematics, Mechanics and Informatics, Vietnam National University, Hanoi, Vietnam Address: No 144 Xuan Thuy Street, Cau Giay District, Hanoi b Lecturer, Ph D, School of Mechanical Engineering, Hanoi University of Science and Technology, Hanoi, Vietnam *Corresponding author Tel: +84 988 287 789 Email address: ninhdinhgia@gmail.com and ninh.dinhgia@hust.edu.vn Highlights • • • • A new analytical approach to the nonlinear dynamical buckling of imperfect stiffened three-layered toroidal shell segment containing fluid under external thermal environment is studied The lowest natural frequencies corresponding to particular modes of both convex and concave are found in the considered case The fluid remarkably influenced on nonlinear vibration response of FGM sandwich toroidal shell segment Definitely, it makes the amplitudes of vibration of shell and frequencies decreased considerably The change of external temperature makes the deflection as well as the amplitudes of shell rocketed Abstract: An analytical study to the nonlinear vibration of imperfect stiffened FGM sandwich toroidal shell segment containing fluid in external thermal environment is approached in this present The toroidal shell segments consist of two types convex shell and concave shell which are reinforced by ring and stringer stiffeners system Material properties of shell are assumed to be continuously graded in the thickness direction Based on the classical thin shell theory with geometrical nonlinearity in von Karman-Donnell sense, Stein and McElman assumption, and the smeared stiffeners technique theoretical formulations are established In addition, the dynamical pressure of fluid is taken into account The fluid is assumed to be non-viscous and ideal incompressible The nonlinear vibration analyses of full-filled fluid toroidal shell segment are solved by using numerical method fourth-order Runge-Kutta Furthermore, effects of geometrical and material parameters, imperfection, fluid and change of temperature field on the nonlinear vibration responses of shells are shown in obtained results It is hoped that the obtained results will be used as benchmark solutions for an analytical approach of fluid-structures vibration in further research Key words: Toroidal shell segment; thermal vibration; fluid-structures; imperfection; fullfilled fluid Introduction The sandwich structures have become pay attention in structural applications The smooth and continuous changes in material properties makes sandwich FGMs to avoid interface problems and unexpected thermal stress concentrations On the other hand, the sandwich structures also have the mentionable properties, especially thermal and sound insulation Sofiyev et al [1, 2] investigated the parametric instability of simply-supported sandwich cylindrical shell with a FGM core under static and time dependent periodic axial compressive loads; the influences of shear stresses and rotary inertia on the vibration of FG coated sandwich cylindrical shells resting on Pasternak elastic foundation The free vibration of sandwich plates with FGM face sheets in various thermal environments to improve highorder sandwich plate theory using Hamilton's principle was studied by Khalili and Mohammadi [3] A new approach was used to reduce the equations of motion and then solved them for both un-symmetric and symmetric sandwich plates Xia and Shen [4] anlyzed the small and large-amplitude vibration of compressively and thermally post-buckled sandwich plates with FGM face sheets in thermal environment using a higher-order shear deformation plate theory The formulations were based on a general von-Karman-type equation that included a thermal effect and the equations of motion were solved by an improved perturbation technique Wang and Shen [5] studied the nonlinear dynamic response of sandwich plates with FGM face sheets assumed to be graded in the thickness direction according to the Mori–Tanaka scheme resting on elastic foundations in thermal environments Sburlati [6] gave an analytical solution in the framework of the elasticity theory to indicate the elastic bending response of axisymmetric circular sandwich panels with functionally graded material cores and homogeneous face-sheets The elastic solution was obtained using a Plevako representation, which reduced the problem to the search of potential functions satisfying linear fourth-order partial differential equations Furthermore, Taibi et al [7] proposed the deformation behavior of shear deformable FG sandwich plates resting on Pasternak foundation under thermo-mechanical loads Ninh and Bich [8] investigated the nonlinear torsional buckling and post buckling of eccentrically stiffened ceramic FGM metal layer cylindrical shell under thermo-mechanical load A layerwise shear deformation theory proposed by Ferreira for FGM sandwich shells and laminated composite shells using a differential quadrature finite element method (DQFEM) was analyzed by Liu et al [9] The combination of the DQFEM with Ferreira’s layerwise theory allows a very accurate prediction of the field variables On the other hand, the fluid-structures problems have been attracted the special interest of many authors in the world Chen et al [10] researched the vibration of fluid-filled orthotropic FGM cylindrical shells based on the three-dimensional fundamental equations of anisotropic elasticity The frequency equation was deduced for an FGM cylindrical shell filled with a compressible, non-viscous fluid medium and the concept “added mass effect” appeared to show the effect of fluid to structures An analytical approach to research the elastic dynamic responses of FG plates to underwater shock with applications in deep sea exploration and naval and coastal engineering was proposed by Liang et al [11] Taylor‫׳‬s one dimensional fluid solid interaction (FSI) model was extended to fit a three dimensional model appropriate to FG plates The extended FSI model and Laplace transform were integrated into the state space method, with the transient solution in the time domain being obtained by using the numerical inversion of the Laplace transform Sheng and Wang [12] investigated into the vibration of FGM cylindrical shells with flowing fluid in an elastic medium under mechanical and thermal loads using a modal expansion method Based on the first-order shear deformation theory (FSDT) and the fluid velocity potential, the dynamic equation of functionally graded cylindrical shells with flowing fluid was derived Amabili et al [13, 14] presented circular cylindrical shells conveying incompressible flow using Donnell's nonlinear theory retaining in-plane displacements and the Sanders–Koiter non-linear theory; geometrically nonlinear vibrations of thin infinitely long rectangular plates under axial flow and concentrated harmonic excitation using von Karman non-linear plate theory and geometric imperfections by employing Lagrangian approach The fluid was modelled by potential flow theory but the effect of steady viscous forces was investigated; moreover the flow perturbation potential was derived by applying the Galerkin technique A thin-walled beam made of functionally graded material (FGM) which was used as rotating blades in turbomachinery under aerothermoelastic loadingwas presented Fazelzadeh and Hosseini [15] Based on first-order shear deformation theory, the governing equations included the effects of the presetting angle, the secondary warping, temperature gradient through the wall thickness of the beam and also the rotational speed In addition, quasi-steady aerodynamic pressure loadings were determined using first-order piston theory, and steady beam surface temperature was obtained from gas dynamics theory Eghtesad et al [16] researched the Smoothed Particle Hydrodynamics (SPH) method to investigate elastic–plastic deformation of AL and ceramic–metal FGM plates under the impact of water in a fluid–solid interface A new scheme called corrected smooth particle method (CSPM) was applied to both fluid and solid particles to improve the free surface behavior Khorshidi and Bakhsheshy [17] investigated the vibration analysis of a functionally graded (FG) rectangular plate partially in contact with a bounded fluid The fluid velocity potential satisfying fluid boundary conditions was derived, and wet dynamic modal functions of the plate were expanded in terms of finite Fourier series for compatibility requirement along the contacting surface between the plate and the fluid The vibration of FGM structures also is studied by many researchers Loy et al [18] investigated vibration of FGM cylindrical shells based on Love’s shell theory and the eigenvalue governing equation was obtained using Rayleigh-Ritz method The nonlinear dynamic buckling of FGM cylindrical shells under time-dependent axial load and radial load using an energy method and Budiansky-Roth criterion were presented by Huang and Han [19, 20] Sofiyev [21] researched the dynamic buckling of truncated conical shells with FGM coatings subjected to a time dependent axial load in the large deformation The method of solution utilizes superposition principle and Galerkin procedure The nonlinear vibration of FGM cylindrical shell improved Donnell equations using Galerkin procedure was studied by Bich and Nguyen [22] The dynamic instability of simply supported, functionally graded (FG) truncated conical shells under static and time dependent periodic axial loads using Galerkin method was analyzed by Sofiyev and Kuruoglu [23] The domains of principal instability were determined by using Bolotin’s method Shariyat [24] presented dynamic buckling of a pre-stressed, suddenly heated imperfect FGM cylindrical shell and dynamic buckling of a mechanically loaded imperfect FGM cylindrical shell in thermal environment, with temperature-dependent properties Free vibration of simply supported FGM sandwich spherical and cylindrical shell geometries with a three-dimensional exact shell model and different two-dimensional computational models was investigated by Fantuzzi et al [25] Ferreira et al [26-27] analyzed nonlinear vibration of microstructure-dependent FGM piezoelectric material beams in pre/post buckling regimes based on Timoshenko beam theory with various inplane and out-of-plane boundary conditions in thermo-mechanical loads; the geometric nonlinear analysis of FGM plates and shells using the Marguerre shell element, modified to incorporate the graded properties across the thickness The toroidal shell segment structures containing fluid have been applied in mechanical engineering, aerospace engineering, bio-mechanical engineering such as fusion reactor vessels, underwater toroidal pressure hull and nuclear reactor In the past, the initial postbuckling behavior of toroidal shell segments under several loading conditions using the basic of Koiter’s general theory was studied by Hutchinson [28] Stein and McElman [29] considered the buckling of homogenous and isotropic toroidal shell segments Recently, Bich et al [30-34] have investigated the stability buckling of functionally graded toroidal shell segment under thermomechanical load based on the classical thin shell theory and the smeared stiffeners technique Bich et al [35] have just studied the nonlinear dynamical investigation of eccentrically stiffened FGM toroidal shell segments surrounded by elastic foundation in thermal environment To the best of the authors’ knowledge, there are no any publications for nonlinear vibration of imperfect FGM sandwich toroidal shell segments containing fluid with external thermal environment Based on the classical shell theory with the nonlinear strain5 displacement relation of large deflection, the Galerkin method, Volmir’s assumption and the numerical method using fourth-order Runge-Kutta are implemented for nonlinear vibration responses of fluid-shells The fluid is assumed to be non-viscous and ideal incompressible The temperature impacts on from external environment to shells In addition, effects of geometrical and material parameters, imperfection and fluid on the nonlinear vibration responses of shells are shown in figures and discussion Formulation of the problem 2.1 FGM sandwich The sandwich toroidal shell segment of thickness h, length L, which is formed by rotation of a plane circular arc of radius R about an axis in the plane of the curve is shown in Figure The coordinate system (x, y, z) is located on the middle surface of the shell, x and y is the axial and circumferential directions, respectively and z is the normal to the shell surface The thickness of the shell is defined in a coordinate system (y, z) in Fig In this paper, FGM core and ceramic core structures are investigated For FGM core, the inner layer (z = h/2) and the outer layer (z = -h/2) are isotropic homogenous with ceramic and metal, respectively Suppose that the material composition of the shell varies smoothly along the thickness in such a way that the inner surface is ceramic, the outer surface is metal and the core is FGM For ceramic core, the inner layer (z = h/2) and the outer layer (z = -h/2) are metal when supposing that the material composition of the shell varies smoothly along the thickness from ceramic core to outside metal The thickness of the shell is h, ceramic-rich and metal rich are h c, h m, respectively for FGM core and FGM coatings are hz1 and hz2 as in Fig.2 The subscripts m and c are refered to the metal and ceramic constituents respectively Denote Vm and Vc as volume - fractions of metal and ceramic phases respectively, where Vm + Vc = According to the mentioned law, the volume fraction is expressed as For FGM core:  Vc ( z ) = 0,   k   z + h / − hm   , Vc ( z ) =   h − hc − hm    Vc ( z ) = 1,   − h h  ≤ z ≤ − − hm  2  h  h  −  − hm  ≤ z ≤  − hc , k ≥ 2  2  h h   − hc  ≤ z ≤ 2  (1) For ceramic core: k   z −h/2  ,  Vc ( z ) =    − hz1   Vc ( z ) = 1,  k   z + h/2  , Vc ( z ) =   hz   h h   − hz1  ≤ z ≤ , k ≥ 2  h  h  −  − hz  ≤ z ≤  − hz1 , 2  2  h h  − ≤ z ≤ − − hz  2  (2) According to the mentioned law, the Young modulus, the mass density and the thermal expression coefficient of FGM core shell are expressed of the form respectively E ( z ) = EmVm ( z ) + EcVc ( z ) = E m + ( Ec − E m )Vc ( z ) , ρ ( z ) = ρ mVm ( z ) + ρcVc ( z ) = ρ m + ( ρ c − ρ m )Vc ( z ) , α ( z ) = α mVm ( z ) + α cVc ( z ) = α m + (α c − α m )Vc ( z ) the Poisson’s ratio ν is assumed to be constant Fig.1 The coordinate system of ES-FGM core toroidal shell segment containing fluid a b Fig.2 The material properties of FGM sandwich:a) FGM core; b) ceramic core 2.2 Constitutive relations and governing equations 2.2.1 Constitutive relations A sufficiently shallow toroidal shell segment in the region of the equator of the torus is analyzed in Ref [28-35] The eccentrically stiffened FGM sandwich toroidal shell segment is reinforced by string and ring stiffeners In order to provide continuity within the shell and stiffeners and easier manufacture, the homogeneous stiffeners can be used Since pure ceramic ones are brittleness metal stiffener are used and put at metal side of the shell With the indicated law in (1) and (2), the outer surface is metal, so the external metal stiffeners are put at outer side of the shell Fig describes the geometry and coordinate system of stiffened FGM core shell (a) (b) Fig Geometry and coordinate system of a stiffened FGM sandwich toroidal shell segment containing fluid (a) stringer stiffeners; (b) ring stiffeners The von Karman type nonlinear kinematic relation for the strain component across the shell thickness at a distance z from the middle surface are of the form [36]:  ε x   ε x0   χ x     0    ε y  =  ε y  − z χ y  , γ  γ   2χ   xy   z   xy  (3) 0 where ε x0 and ε y are normal strains, γ xy is the shear strain at the middle surface of the shell and χ x and χ y are the curvatures and χ xy is the twist According to the classical shell theory the strains at the middle surface and curvatures are related to the displacement components u, v, w in the x, y, z coordinate directions as [36]:   ∂u w  ∂w  ∂w ∂wo − +   +   ∂x R  ∂x  ∂x ∂x   ε x0       ∂v w  ∂w  ∂w ∂wo − +   + εy  =  ; ∂y a  ∂y  ∂y ∂y γ     xy   ∂u ∂v ∂w ∂w ∂w ∂wo ∂w ∂wo   ∂y + ∂x + ∂x ∂y + ∂x ∂y + ∂y ∂x     ∂ 2w     χ x   ∂x2    ∂ w  χy  =   ,  χ   ∂y   xy   ∂ w   ∂x∂y    (4) where wo(x, y) is a known function representing initial imperfection of the shell From Eqs (4) the strains must be satisfied in the deformation compatibility equation 2 ∂ ε x0 ∂ ε y ∂ γ xy ∂ w ∂ w  ∂ w ∂ wo   ∂ w ∂ wo  − + − = − − + + + ∂y ∂x ∂x∂y R∂y a∂x  ∂x∂y ∂x∂y   ∂x ∂x  ∂ w ∂ wo  + ∂y  ∂y   (5)  Hooke’s law for toroidal shell segment is defined as E ( z) E ( z )α ( z ) (ε x + νε y ) + ∆T , ∆T = T − T0 , −ν −ν E ( z) E ( z )α ( z ) = (ε y + νε x ) + ∆T , −ν −ν E ( z) = γ xy , 2(1 +ν ) σ xsh = σ ysh σ xysh (6) and for metal stiffeners st σ xst = E m ε x + E mα m ∆T ; σ y = Emε y + Emα m∆T By intergrating the stress – strain equations and their moments through the thickness of the shell and using the smeared stiffeners technique; the expressions for force and moment resultants of a FGM core toroidal shell segment can be obtained as [33, 36]:  E A N x =  A11 + m ε x0 + A12ε y0 − ( B11 + C1 ) χ x − B12 χ y + Φ a + Φ*a , s1    E A  N y = A12ε x0 +  A22 + m ε y0 − B12 χ x − ( B22 + C2 ) χ y + Φ a + Φ*a* , s2   N xy = A66γ xy0 − B66 χ xy ,  E I  M x = ( B11 + C1 )ε x0 + B12ε 0y −  D11 + m  χ x − D12 χ y + Φ m + Φ *m , s1    E I  M y = B12ε x0 + ( B22 + C2 )ε 0y − D12 χ x −  D22 + m  χ y + Φ m + Φ*m* , s2   (7) (8) M xy = B66γ xy0 − D66 χ xy , where Aij , B ij , D ij (i, j = 1, 2, 6) are extensional, coupling and bending stiffenesses of the shell without stiffeners Figure Effect of L/R on nonlinear vibration response of imperfect FGM core convex shell without fluid Figure 10 Effect of L/R on nonlinear vibration response of full-filled fluid imperfect FGM core concave shell 23 Figure 11 Effect of L/R on nonlinear vibration response of imperfect FGM core concave shell without fluid Effect of ceramic layer Figure 12 Effect of ceramic layer on nonlinear vibration response of imperfect FGM core convex shell full-filled fluid and without fluid 24 Figure 13 Effect of ceramic layer on nonlinear vibration response of imperfect FGM core concave shell full-filled fluid and without fluid Fig 12 and Fig 13 describe effect of ceramic layer on nonlinear vibration response of imperfect FGM core toroidal shell segments It is observed that the amplitudes of FGM shell are higher than those of FGM sandwich shell with the same geometries It means that the mechanical characteristics of FGM sandwich structures are stronger than those of FGM structures, ie FGM sandwich structures work better Effect of volume fraction k The effect of volume-fraction k on nonlinear vibration of stiffened imperfect FGM core convex and concave is depicted in Fig 14 and Fig 15, respectively When the value of volume-fraction increases, the amplitudes of nonlinear vibration of FGM core shell is also go up This is suitable for the real characteristic of material and regulation because the higher the value of k is, the richer metal shell is and it has less stiffness than a richer ceramic one 25 Figure 14 Effect of volume fraction k on nonlinear vibration response of full-filled fluid imperfect FGM core convex shell Figure 15 Effect of volume fraction k on nonlinear vibration response of full-filled fluid imperfect FGM core concave shell Effect of imperfection Fig 16 shows effect of imperfection on nonlinear vibration response of FGM core concave shell As can be seen, the amplitude-time curves increase with increase of initial imperfection 26 and the amplitudes of imperfect full-filled fluid FGM core shell are greater than those of perfect one Under dynamical load perfect structures work better Figure 16 Effect of imperfection on nonlinear vibration response of full-filled fluid FGM core concave shell Effect of pre-axial compression loads Figure 17 Effect of pre-axial compression loads on nonlinear vibration response of fullfilled fluid imperfect FGM core convex shell 27 Figure 18 Effect of pre-axial compression loads on nonlinear vibration response of fullfilled fluid imperfect FGM core concave shell Fig 17 and Fig 18 indicate the effect of pre-axial compression load on nonlinear vibration response of full-filled fluid FGM core convex and concave shell, respectively For convex shell, under inverse pressure of fluid and pre-axial compression load, the amplitudes of nonlinear vibration response of convex shell will be much more negative and increase when the values of axial compression load increase For concave shell, the phenomenon is also similar, however due to the feature of concave structures, the amplitudes of nonlinear vibration of concave shell will not be different definitely as those of nonlinear vibration of convex shell Effect of temperature The effect of temperature on nonlinear vibration of full-filled fluid imperfect FGM core and ceramic core is presented from Fig 19 to Fig 24 Based on the figures, some conclusion should be given as follows: - - - The change of outside temperature significantly influences on nonlinear vibration response of FGM core shell as well as ceramic core Particularly, the amplitudes of nonlinear vibration response have soared many times when the change of temperature increases Under external pre-heated shell, the temperature field makes the shell deflected inward (positive deflection) before it is impacted on more by lateral load Thus, the amplitudes of shell will be positive The effect of fluid contradicts the impact of temperature and contributes the decrease of the amplitudes Furthermore, the fluid makes the frequencies plummeted (i.e the number of responses is much less than when shell is full-filled fluid) 28 - In addition, the amplitudes of nonlinear vibration of convex shells are about 10 times higher than those of concave shells Figure 19 Effect of temperature on nonlinear vibration response of full-filled fluid imperfect FGM core convex shell Figure 20 Effect of temperature on nonlinear vibration response of full-filled fluid imperfect FGM core concave shell 29 Figure 21 Effect of temperature and fluid on nonlinear vibration response of imperfect FGM core convex shell Figure 22 Effect of temperature and fluid on nonlinear vibration response of imperfect FGM core concave shell 30 Figure 23 Effect of temperature and fluid on nonlinear vibration response of imperfect ceramic core convex shell Figure 24 Effect of temperature and fluid on nonlinear vibration response of imperfect ceramic core concave shell Comparison on nonlinear vibration response of FGM core and ceramic core shell 31 Figure 25 Nonlinear vibration response of imperfect FGM core and ceramic core convex shell Figure 26 Nonlinear vibration response of imperfect FGM core and ceramic core concave shell With the geometrical parameters of problem: h = 0.01m; hc = hm = 0.2h; hx1 = hx2 = 0.2h and based on Fig 25 and Fig 26, the amplitudes of nonlinear vibration of FGM core shells are lower than those of ceramic core shells in the same thickness of layers It means that the load bearing capacity of ceramic core shells will be better than one of FGM core shells 32 Beating phenomenon Figure 27 Nonlinear response of imperfect FGM core convex shell with full-filled fluid and without fluid Figure 28 Nonlinear response of imperfect FGM core concave shell with full-filled fluid and without fluid 33 The beating phenomenon will happen when the frequency of the excitation is near to the natural frequency of the shell Fig 27 illustrates the nonlinear response of imperfect FGM core convex shell with full-filled fluid and without fluid For full-filled fluid convex shell, the natural frequency is ω1 = 219.69 rad/s under excitation q = 1000sin(225t) For shell without fluid, the natural frequency is ω1 = 1723.24 rad/s under excitation q = 1000sin(1700t) Fig 28 indicates the nonlinear response of imperfect FGM core concave shell with full-filled fluid and without fluid The natural frequency is ω1 = 160.16 rad/s under excitation q = 1000sin(155t) for full-filled fluid shell while the natural frequency is ω1 = 1256.22 rad/s under excitation q = 1000sin(1300t) for shell without fluid As can be seen that the number of beats of no-fluid shell is greater many times than one of full-filled fluid shell for both type of shells, but the number of beats and vibration amplitude of convex shell are smaller than ones of concave shell Under dynamical loads the bearing load capacity of convex shells is better Conclusions The nonlinear vibration of imperfect eccentrically stiffened full-filled fluid imperfect toroidal shell segments under mechanical loads with external temperature is investigated Based on the classical shell theory by using the Galerkin method, Volmir’s assumption and the numerical method using fourth-order Runge-Kutta, the nonlinear vibration responses of both convex and concave shells are analyzed in details Effects of geometrical and material parameters, imperfection, fluid and the change of temperature on the nonlinear vibration response of shells are illustrated in obtained results Some pivotal concluding remarks are obtained as follows: - - - A good agreement has been obtained in comparison with the previous publication The lowest natural frequencies corresponding to particular modes of both convex and concave are found in the considered case Geometrical ratios, imperfection, ceramic layer remarkably impact on the nonlinear vibration of both convex and concave shell In addition, the amplitudes of nonlinear vibration response of convex shell are always lower than those of concave shell with the same parameters It means that the carrying capacity of convex shells is better than those of concave shells The fluid remarkably influenced on nonlinear vibration response of FGM sandwich toroidal shell segment Definitely, it makes the amplitudes of vibration of shell and frequencies decreased considerably The change of external temperature makes the deflection as well as the amplitudes of shell rocketed When the frequencies of external force are near to the natural frequencies of the shell, the beating phenomenon is observed and investigated Acknowledgements 34 This work was supported by The scientific research Project of Hanoi University of Science and Technology under Grant number T2016-PC-056 The authors are grateful for this support References [1] A H Sofiyev, N Kuruoglu Dynamic instability of three-layered cylindrical shells containing an FGM interlayer Thin-Walled Structures, 93 (2015) 10-21 [2] A H Sofiyev, D Hui, A M Najafov, S Turkaslan, N Dorofeyskaya, G Q Yuan Influences of shear stresses and rotary inertia on the vibration of functionally graded coated sandwich cylindrical shells resting on the Pasternak elastic foundation Journal of Sandwich Structures and Materials, (2015), doi: 10.1177/1099636215594560 [3] S M R Khalili, Y Mohammadi Free vibration analysis of sandwich plates with functionally graded face sheets and temperature-dependent material properties: A new approach European Journal of Mechanics-A/Solids, 35 (2012) 61-74 [4] X Xia, H S Shen 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shell rocketed Abstract: An analytical study to the nonlinear vibration of imperfect stiffened FGM sandwich toroidal shell segment containing fluid in external thermal environment