DSpace at VNU: Nonlinear thermal vibration of eccentrically stiffened Ceramic-FGM-Metal layer toroidal shell segments surrounded by elastic foundation

Thin-Walled Structures 104 (2016) 198–210 Contents lists available at ScienceDirect Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws Full length article Nonlinear thermal vibration of eccentrically stiffened Ceramic-FGM-Metal layer toroidal shell segments surrounded by elastic foundation Dinh Gia Ninh a,n, Dao Huy Bich b a b School of Mechanical Engineering, Hanoi University of Science and Technology, Hanoi, Vietnam Vietnam National University, Hanoi, Vietnam art ic l e i nf o a b s t r a c t Article history: Received 11 February 2016 Received in revised form 15 March 2016 Accepted 15 March 2016 The eccentrically stiffened Ceramic-FGM-Metal layer toroidal shell segments which applied for heatresistant, lightweight structures in aerospace, mechanical, and medical industry and so forth are the new structures Thus, the nonlinear vibration of eccentrically stiffened (ES) Ceramic-FGM-Metal (C-FGM-M) layer toroidal shell segments surrounded by an elastic medium in thermal environment is investigated in this paper Based on the classical shell theory with the geometrical nonlinear in von Karman-Donnell sense, Stein and McElman assumption and the smeared stiffeners technique, the governing equations of motion of ES-C-FGM-M layer toroidal shell segments are derived The dynamical characteristics of shells as natural frequencies, nonlinear frequency-amplitude relation, and nonlinear dynamic responses are considered Furthermore, the effects of characteristics of geometrical ratios, ceramic layer, elastic foundation, pre-loaded axial compression and temperature on the dynamical behavior of shells are studied & 2016 Elsevier Ltd All rights reserved Keywords: Toroidal shell segments FGM core FGM sandwich shell Non-linear vibration Thermal environment Stiffeners Introduction Toroidal shell has been applied in practical fields as rocket fuel tanks, fusion reactor vessels, diver's oxygen tanks, satellite support structures, and underwater toroidal pressure hull There are many studies with vibration and buckling problems in this structure such as Jiang and Redekop [1], Buchanan and Liu [2], Wang et al [3,4] and Tizzi [5] One of the special structures of toroidal shell is toroidal shell segment Stein and McElman [6] carried out the homogenous and isotropic toroidal shell segments about the buckling problem Moreover, the initial post-buckling behavior of toroidal shell segments subjected to several loading conditions based on the basic of Koiter's general theory was performed by Hutchinson [7] Parnell [8] gave a simple technique for the analysis of shells of revolution applied to toroidal shell segments Recently, there have had some new publications about toroidal shell segment structure Bich et al [9–11] has studied the buckling and nonlinear buckling of functionally graded toroidal shell segment under lateral pressure based on the classical thin shell theory, the smeared stiffeners technique and the adjacent equilibrium criterion Furthermore, the nonlinear buckling and post-buckling of ESFGM toroidal shell segments under torsional load based on the analytical approach are investigated by Ninh et al [12,13] n Corresponding author E-mail addresses: ninhdinhgia@gmail.com, ninh.dinhgia@hust.edu.vn (D.G Ninh) http://dx.doi.org/10.1016/j.tws.2016.03.018 0263-8231/& 2016 Elsevier Ltd All rights reserved Today, sandwich FGM structures have received mentionable attention in structural applications The smooth and continuous change in material properties enables sandwich FGMs to avoid interface problems and unexpected thermal stress concentrations Furthermore, the sandwich structures also have the remarkable properties, especially thermal and sound insulation Sofiyev and Kuruoglu [14] investigated the parametric instability of simplysupported sandwich cylindrical shell with a FGM core under static and time dependent periodic axial compressive loads The governing equations of sandwich cylindrical shell with an FGM core were derived to reduce the second order differential equation with the time-dependent periodic coefficient or Mathieu-type equation by using the Garlerkin's method and the equation was solved by Bolotin's method Moreover, the dynamic instability of threelayered cylindrical shells containing a FG interlayer under static and time dependent periodic axial compressive loads was studied by Sofiyev and Kuruoglu [15] The expressions for boundaries of unstable regions of three-layered cylindrical shell with an FG interlayer were found The bending response of the sandwich panel with FG skins using Fourier conduction equation and obtaining temperature distribution under thermal mechanical load based on higher order sandwich plate theory was studied by Sadighi et al [16] Taibi et al [17] analyzed the deformation behavior of shear deformable FG sandwich plates resting on Pasternak foundation under thermo-mechanical loads The influences of shear stresses and rotary inertia on the vibration of FG coated sandwich cylindrical shells resting on Pasternak elastic foundation based on the D.G Ninh, D.H Bich / Thin-Walled Structures 104 (2016) 198–210 modification of Donnell type equations of motion were examined by Sofiyev et al [18] The basic equations were reduced to an algebraic equation of the sixth order and numerically solving this algebraic equation gave the dimensionless fundamental frequency Xia and Shen [19] dealt with the small and large-amplitude vibration of compressively and thermally post-buckled sandwich plates with FGM face sheets in thermal environment The formulations were based on a higher-order shear deformation plate theory and a general von-Karman-type equation that includes a thermal effect and the equations of motion were solved by an improved perturbation technique The refined hierarchical kinematics quasi-3D Ritz models for free vibration analysis of doubly curved FGM shells and sandwich shells with FGM core were investigated by Fazzolari and Carrera [20] Sburlati [21] presented an analytical solution in the framework of the elasticity theory to describe 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 The effect of continuously grading fiber orientation face sheets on free vibration of sandwich panels with functionally graded using generalized power-law distribution was investigated by Aragh and Yas [22] Woodward and Kashtalyan [23] analyzed a three-dimensional elasticity for a sandwich panel with stiffness of the core graded in the thickness direction, on the basic of the developed 3D elasticity solution subjected to distributed and concentrated loadings The vibration problem about the shell structures have been attracted a large number of studies The free vibration analysis of function analysis of FGM cylindrical shell with holes was studied with the variational equation and the unified displacement modeshape function of the shells with various boundary conditions by Cao and Wang [24] Sofiyev and Kuruoglu [25] carried out the vibration and stability of FGM orthotropic cylindrical shells under external pressures using the shear deformation shell theory The basic equations of shear deformable FG shell were derived using Donnell shell theory and solved using the Galerkin method Bich et al [26] investigated the characteristics of free vibration and nonlinear responses, using the governing equations of motion of eccentrically stiffened functionally graded cylindrical panels with geometrically imperfections based on the classical shell theory with the geometrical nonlinearity in von Karman-Donnell sense and smeared stiffeners technique Moreover, Bich and Nguyen [27] presented the study of the nonlinear vibration of a functionally graded cylindrical shell subjected to axial and transverse mechanical loads based on improved Donnell equations The nonlinear forced vibration of infinitely long FGM cylindrical shell using the Lagrangian theory and multiple scale method was presented by Du et al [28] The energy approach was applied to derive the reduced low-dimensional nonlinear ordinary differential equations of motion Sheng et al [29] investigated the nonlinear vibrations of FGM cylindrical shell based on Hamilton's principle, von-Karman nonlinear theory and first-order shear deformation theory The vibration of FGM cylindrical shells under various boundary conditions with the strain displacement relations form Love's shell theory were studied by Pradhan et al [30] Based on the Rayleigh method, the governing equations were derived and the natural frequencies were investigated depending on the constituent volume fractions and boundary condition The strainsdisplacement relations from Love's shell theory and energy functional with the Rayleigh–Ritz method to solve the governing equation were used Strozzi and Pellicano [31] analyzed the nonlinear vibrations of FGM circular cylindrical shells using the Sanders-Koiter theory The displacement fields were expanded by means of double mixed series based on Chebyshev orthogonal 199 polynomials for the longitudinal variable and harmonic functions for the circumferential variable Sofiyev [32] studied the dynamic behavior of FGM structures such as dynamic response of an FGM cylindrical shell subjected to combined action of the axial tension, internal compressive load and ring-shaped compressive pressure with constant velocity based on the von Karman-Donnell type nonlinear kinematics using the superposition and Galerkin methods The nonlinear vibration of simply supported FGM cylindrical shells with embedded piezoelectric layers using a semi analytical approach was addressed by Jafari et al [33] The governing differential equations of motion of the FG cylindrical shell were derived using the Lagrange equations under the assumption of the Donnell's nonlinear shallow-shell theory Firooz and Seyed [34] investigated the nonlinear free vibration of prestressed circular cylindrical shells placed on Pasternak foundation using the nonlinear Sanders-Koiter shell theory to derive strain-displacement The governing equations in linear state were solved by the Rayleigh–Ritz procedure Based on strain-displacement relations from the Love's shell theory and the eigenvalue governing equation using Rayleigh–Ritz method, Loy et al [35] gave the study at vibration filed of functionally cylindrical shells The dynamic behavior of FGM truncated conical shells subjected to asymmetric internal ring-shaped moving loads using Hamilton's principle based on the first order shear deformation theory was studied by Malekzadeh and Daraie [36] The vibration of FGM cylindrical shells on elastic foundations using wave propagation to solve dynamical equations were analyzed by Abdul et al [37] The shell was assumed to be simply supported with movable edges and the equations of motion were reduced using Galerkin method to a system of infinite nonlinear ordinary differential equations with quadratic and cubic nonlinearities Haddadpour et al [38] performed free vibration analysis of functionally graded cylindrical shells using the equations of motion based on Love's shell theory and the von Karman-Donnell type of kinematic nonlinearity for the thermal effects investigated by specifying arbitrary high temperature on the outer surface and ambient temperature on the inner surface of the cylindrical Based on the first-order shear deformation theory of shells, the influences of centrifugal and Coriolis forces in combination with the other geometrical and material parameters on the free vibration behavior of rotating functionally graded truncated conical shells subjected to different boundary conditions were investigated by Malekzadeh and Heydarpour [39] Noda [40], Praveen et al [41] firstly realized the heat-resistant FGM structures and studied material properties dependent on temperature in thermo-elastic analyses Sheng and Wang [42] researched the nonlinear response of functionally graded cylindrical shells under mechanical and thermal loads using Karman nonlinear theory The coupled nonlinear partial differential equations are discretized based on a series expansion of linear modes and a multiterm Galerkin's method Furthermore, Shen [43] took into account the nonlinear vibration of shear deformable FGM cylindrical shells of finite length embedded in a large outer elastic medium and in thermal environments The motion equations were based on a higher order shear deformation shell theory that included shell-foundation interaction The transient thermoelastic analysis of functionally graded cylindrical shells under moving boundary pressure and heat flux was presented by Malekzadeh and Heydarpour [44] The hyperbolic heat conduction equations were used to include the influence of finite heat wave speed The resulting system of differential equations was solved using Newmark's time integration scheme in the temporal domain Kiani et al [45] investigated the thermoelastic dynamic behavior of an FGM doubly curved panel under the action of thermal and mechanical loads based on the first-order shear deformation theory of modified Sanders assumptions applying the Laplace 200 D.G Ninh, D.H Bich / Thin-Walled Structures 104 (2016) 198–210 transformation Furthermore, Malekzadeh et al [46,47] studied the influences of thermal environment on the free vibration characteristics of functionally graded shells and panels based on the first-order shear deformation theory By taking into account both the temperature dependence of material properties, which were assumed to be graded in the thickness direction, and the initial thermal stresses, the equations of motion and the related boundary conditions were derived using Hamilton's principle To the best of the authors' knowledge, there has not been any study to the nonlinear thermal vibration of eccentrically stiffened Ceramic-FGM-Metal layer toroidal shell segments surrounded by an elastic foundation In the present paper, the nonlinear vibration of eccentrically stiffened FGM sandwich toroidal shell segments on elastic medium in thermal environment are investigated Based on the classical shell theory with the nonlinear strain-displacement relation of large deflection, the Galerkin method, Stein and McElman assumption, Volmir's assumption and the numerical method using fourth-order Runge-Kutta are performed for dynamic analysis of shells to give expression of natural frequencies and nonlinear dynamic responses Governing equations 2.1 Ceramic-FGM-Metal layer (C-FGM-M) The coordinate system (x1, x2, z) is located on the middle surface of the shell, x1 and x2 is the axial and circumferential directions, respectively and z is the normal to the two axes Consider 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 as shown in Fig in a coordinate system (x1, x2, z) consists of ceramic, FGM and metal described in Fig The thickness of the shell is defined in a coordinate system (x2, z) in Fig The inner layer (z ¼ h/2) and the outer layer (z ¼ Àh/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 The thickness of the shell, ceramic-rich and metal rich are h, hc, hm, respectively Thus, the thickness of FGM core is h À hc À hm The subscripts m and c are referred to the metal and ceramic Fig The coordinate system of ES-C-FGM-M toroidal shell segment D.G Ninh, D.H Bich / Thin-Walled Structures 104 (2016) 198–210 201 Fig The material characteristic of C-FGM-M constituents respectively Denote Vm and Vc as volume - fractions of metal and ceramic phases respectively, where Vm ỵVc ẳ1 According to the mentioned law, the volume fraction is expressed as ⎧ ⎪Vc (z ) = 0, ⎪ ⎪ ⎪ ⎛ ⎞k ⎨Vc (z ) = ⎜ z + h/2 − hm ⎟ , ⎪ ⎝ h − hc − hm ⎠ ⎪ ⎪ Vc (z ) = 1, ⎪ ⎩ ⎞ ⎛h h − ≤ z ≤ − ⎜ − hm ⎟ ⎠ ⎝ 2 the middle surface are of the form [48]: ⎛h ⎞ ⎛h ⎞ −⎜ − h m ⎟ ≤ z ≤ ⎜ − h c ⎟ , k ≥ ⎝2 ⎠ ⎝2 ⎠ ⎞ ⎛h h ⎜ − hc ⎟ ≤ z ≤ ⎠ ⎝2 Fig Geometry and coordinate system of a stiffened C-FGM-M toroidal shell segment on elastic foundation (a) stringer stiffeners; (b) ring stiffeners ε1 = ε10 − zχ1 ; ε2 = ε20 − zχ2 ; γ12 = γ12 − 2zχ12 , (2) E(z ) = EmVm(z ) + EcVc (z ) = Em + (Ec − Em)Vc (z ), where ε10 and ε20 are normal strains, γ12 is the shear strain at the middle surface of the shell and χ1 and χ2 are the curvatures and χ12 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 x1, x2, z coordinate directions as [48]: α(z ) = αmVm + αcVc = αm + (αc − αm)Vc (z ), ε10 = 2 ⎛ ∂w ⎞ ⎛ ∂w ⎞ ∂u w ∂v w − + ⎜ − + ⎜ ⎟ ; ε20 = ⎟ ; ⎝ ∂x1 ⎠ ⎝ ∂x2 ⎠ ∂x1 R ∂x2 a γ12 = ∂u ∂v ∂w ∂w ∂ 2w ∂ 2w ∂ 2w ; χ1 = ; χ2 = ; χ12 = + + 2 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1∂x2 ∂x1 ∂x2 (1) According to the mentioned law, the Young modulus and thermal expansion coefficient of C-FGM-M shell are expressed of the form the Poisson's ratio ν is assumed to be constant 2.2 Constitutive relations and governing equations Hooke's law for toroidal shell segment is defined as For the middle surface of a toroidal shell segment, from the Fig we have: r = a − R(1 − sin φ), where a is the equator radius and φ is the angle between the axis of revolution and the normal to the shell surface For a sufficiently shallow toroidal shell in the region of the equator of the torus, the angle φ is approximately equal to π/2, thus sin φ ≈ 1; cos φ ≈ and r ¼a [6] The form of governing equation is simplified by putting: dx1 = Rdφ, dx2 = adθ The radius of arc R is positive with convex toroidal shell segment and negative with concave toroidal shell segment The shell is surrounded by an elastic foundation with Winkler foundation modulus K1 (N/m3) and the shear layer foundation stiffness of Pasternak model K2 (N/m) Suppose the eccentrically stiffened C-FGM-M (ES-C-FGM-M) 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 Because pure ceramic ones are brittleness we used metal stiffener and put them at metal side of the shell With the law indicated in (1) the outer surface is metal, so the external metal stiffeners are put at outer side of the shell Fig depicts the geometry and coordinate system of stiffened C-FGM-M shell on elastic foundation The von Karman type nonlinear kinematic relation for the strain component across the shell thickness at a distance z from (3) E(z ) E(z ) α(z ) (ε1 + νε2) − ΔT , ΔT = T − T0, 1−ν − ν2 E(z ) E(z ) α(z ) = (ε2 + νε1) − ΔT , 1−ν − ν2 E(z ) γ = 2(1 + ν ) 12 σ1sh = σ2sh sh σ12 (4) and for metal stiffeners σ1st = Emε1 − EmαmΔT; σ2st = Emε2 − EmαmΔT where T0 is initial value of temperature at which the shell is thermal stress free By integrating 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 C-FGM-M toroidal shell segment can be obtained as [13,48]: ⎛ E A ⎞ N1 = ⎜ A11 + m ⎟ε10 + A12 ε20 − (B11 + C1)χ1 − B12χ2 − Φa − Φa*, s1 ⎠ ⎝ ⎛ E A ⎞ N2 = A12 ε10 + ⎜ A22 + m ⎟ε20 − B12χ1 − (B22 + C2)χ2 − Φa − Φa**, s2 ⎠ ⎝ − 2B66χ12 , N12 = A66 γ12 (5) 202 D.G Ninh, D.H Bich / Thin-Walled Structures 104 (2016) 198–210 ⎛ E I⎞ M1 = (B11 + C1)ε10 + B12ε20 − ⎜ D11 + m ⎟χ1 − D12χ2 − Φm − Φm*, s1 ⎠ ⎝ to the middle surface of the shell respectively The nonlinear equilibrium equations of a toroidal shell segment under a lateral pressure q, an axial compression p and surrounded by an elastic foundation based on the classical shell theory are given by [48]: ⎛ E I ⎞ M2 = B12ε10 + (B22 + C2)ε20 − D12χ1 − ⎜ D22 + m ⎟χ2 − Φm − Φm**, s2 ⎠ ⎝ − 2D66χ12 , M12 = B66γ12 (6) where Aij , Bij , Dij (i, j¼1, 2, 6) are extensional, coupling and bending stiffnesses of the shell without stiffeners E1 A11 = A22 = B11 = B22 = − ν2 E2 − ν2 D11 = D22 = , , E3 − ν2 B12 = , E1ν A12 = , − ν2 E2ν − ν2 D12 = , E3ν − ν2 E1 , 2(1 + ν ) A66 = B66 = , E3 , 2(1 + ν ) ∂x12 (7) where h /2 ∫−h /2 E (z )dz = E mh + Ecmh c + E1 = ∂N12 ∂N2 ∂ 2v + = ρ1 , ∂x1 ∂x2 ∂t ∂ 2M1 E2 , 2(1 + ν ) D66 = ∂N1 ∂N12 ∂ 2u + = ρ1 , ∂x1 ∂x2 ∂t Ecm(h − h c − h m) , k+1 ∂ 2M12 ∂ 2M2 N ∂ 2w ∂ 2w ∂ 2w ∂ 2w + + N1 + 2N12 + N2 − ph + ∂x1∂x2 ∂x1∂x2 R ∂x2 ∂x1 ∂x2 ∂x1 ⎞ ⎛ ∂ 2w N2 w w w ∂ ∂ ∂ ⎟⎟ = ρ + + q − K1w + K2⎜⎜ + , + 2ρ1ε a ∂t ∂x22 ⎠ ∂t ⎝ ∂x1 (11) +2 where K1 (N/m3) is linear stiffness of foundation, K2 (N/m) is the shear modulus of the sub-grade, ε is damping coefficient and ρ1 = ρm h + ρcm hc + ⎞ E h h E h2 Ecm ⎛ h ⎜ − h c ⎟(h−h c −h m) E2 = ∫ E (z )zdz = cm c − cm c + ⎠ −h /2 2 k + 1⎝ Ecm − (h−h c −h m)2 , (k + 1)(k + 2) ρcm (h − hc − hm) k+1 ⎛A A ⎞ + ρm ⎜ + ⎟ s2 ⎠ ⎝ s1 (12) h /2 h /2 ∫−h /2 E (z )z 2dz = E3 = By substituting Eq (3) into (5) and (6) and then into Eq (11), the term of displacement components are expressed as follows: ⎞2 ⎛h ⎞ 2Ecm Ecm ⎛ h ⎜ − h c ⎟ (h−h c −h m) − ⎜ − h c ⎟ (h−h c −h m)2 ⎠ ⎠ k + 1⎝ (k + 1)(k + 2) ⎝ E h3 2Ecm (h−h c −h m)3 + c c (k + 1)(k + 2)(k + 3) ⎞⎤ ⎞ Echh c ⎛ h Em ⎡ 3hh m ⎛ h ⎟ ⎜ ⎜ + − hc + − h m⎟⎥ ⎢ hm + ⎠⎦ ⎠ ⎝2 ⎣ ⎝2 ⎤ ⎞ ⎞⎛ h ⎛h E ⎡ + m ⎢ (h−h c −h m)3 − 3⎜ − h m⎟⎜ − h c ⎟(h − h c − h m)⎥ ⎠ ⎠⎝ ⎝2 ⎦ ⎣ Y11(u) + Y12(v) + Y13(w ) + P1(w ) = ρ1 ∂ 2u , ∂t Y21(u) + Y22(v) + Y23(w ) + P2(w ) = ρ1 ∂ 2v , ∂t + Y31(u) + Y32(v) + Y33(w ) + P3(w ) + Q 3(u, w ) + R3(v, w ) − ph in which Ecm ¼Ec ÀEm +q− E Az E A z C1 = − m 1 , C2 = − m 2 , s1 s2 ∂ 2w ∂w 1 (Φa + Φa*) − (Φa + Φa**) = ρ1 + 2ρ1ε , R a ∂t ∂t ∂ 2w ∂x12 (13) where the linear operators Yij( )(i, j = 1, 2, 3) and the nonlinear A1 = h1d1, A2 = h2d2, I1 = (8) d1h13 d h3 + A1z12, I2 = 2 + A2 z22 12 12 and Φa = 1−ν Φa** = * = Φm d2 s2 d1 s1 h /2 ∫−h /2 E (z )α (z )ΔTdz; −h /2 ∫−h /2 −h −h /2 ∫−h /2 −h Φa* = d1 −h /2 E mα mΔTdz; ∫ s1 −h /2 − h1 E mα mΔTdz; Φ m = ** = E mα mΔTzdz; Φ m 1−ν d2 s2 h /2 ∫−h /2 E (z )α (z )ΔTzdz; −h /2 ∫−h /2 −h E mα mΔTzdz Nonlinear analysis (9) If ΔT ¼ const Φa = P ΔT ; 1−ν In the present paper, the simply supported boundary conditions are considered w = 0, v = 0, M1 = E α (h − h c − h m) P = Emα mh + Ecαchc + Emα m(h − h c ) + m cm k+1 E α (h − h c − h m) E α (h − h c − h m) + cm m + cm cm k+1 2k + d h d h 1 2 Φa* = Emα mΔT ; Φa** = Emα mΔT s1 s2 at x1 = and x1 = L (14) The approximate solutions of the system of Eq (13) satisfying the conditions (14) can be expressed as: mπx1 nx mπx1 nx sin ; v = Vmn(t )sin cos ; L 2a L 2a mπx1 nx w = Wmn(t )sin sin , L 2a u = Umn(t )cos in which αcm = αc − αm operators Pi( )(i = 1, 2, 3), Q3 and R3 are demonstrated in Appendix A Eq (13) are the nonlinear governing equations used to investigate the nonlinear dynamical responses of ES-C-FGM-M toroidal shell segments surrounded by elastic foundation in thermal environment (10) The spacings of the stringer and ring stiffeners are denoted by s1 and s2 respectively 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 (15) where Umn, Vmn, Wmn are the time depending amplitudes of vibration, m and n are numbers of half wave in axial direction and wave in circumferential direction, respectively Substituting Eq (15) into Eq (13) and then applying the Galerkin method leads to: D.G Ninh, D.H Bich / Thin-Walled Structures 104 (2016) 198–210 g1 2 y11Umn + y12Vmn + y13Wmn + n1Wmn = ρ1 d Umn dt ωmn = , 203 ρ1 (21) y21Umn + y22Vmn + y23Wmn + n2Wmn y31Umn + y32Vmn + y33Wmn + p + n 6VmnWmn + d Wmn dt hπ 2m2 L2 dt Solving Eq (20) leads to exact solution but implicit expression while Eq (21) performs approximate frequencies but explicit expression and simpler , Wmn + n3Wmn + n4Wmn + n5UmnWmn ⎤ 4δ1δ ⎡ 1 ⎢ q − (Φa + Φa*) − (Φa + Φa**)⎥ ⎦ R a mnπ ⎣ = ρ1 = ρ1 d Vmn + 2ρ1ε dWmn , dt 3.2 Frequency amplitude curve (16) where yij; ni are given in Appendix B Otherwise, the Volmir's assumption [49] can be used in the dynamic analysis Taking the inertia forces ρ1(d2U /dt 2) → and ρ1(d2V /dt 2) → into consideration because of u < < w, v < < w , Eq (16) can be rewritten as follows: Consider nonlinear vibration of a toroidal shell segment under a uniformly distributed transverse load q = Q sin Ωt including thermal effects Assuming pre-loaded compression p, Eq (18) takes the form ρ1 y11Umn + y12 Vmn + y13 Wmn + n1Wmn = 0, d2Wmn dWmn + 2ρ1ε + g1Wmn − g2Wmn + g3Wmn dt dt ⎤ 4δ1δ2 ⎡ 1 = Q sin Ωt − (Φa + Φa*) − (Φa + Φa**)⎥ 2⎢ ⎣ ⎦ R a mnπ y21Umn + y22 Vmn + y23 Wmn + n2Wmn = 0, y31Umn + y32 Vmn + y33 Wmn + p hπ 2m2 L2 =ρ1 d 2Wmn dt + 2ρ1ϵ Eq (22) can be rewritten as 2 Wmn + n3Wmn + n4Wmn + n5UmnWmn + n 6VmnWmn + d Wmn dt ⎤ 4δ1δ ⎡ 1 ⎢ q − (Φa + Φa*) − (Φa + Φa**)⎥⎦ R a mnπ ⎣ dWmn dt (17) d2Wmn dWmn 2 + ωmn (Wmn − HWmn + KWmn ) − F sin Ωt + G = 0, dt g1 ρ1 bration of the toroidal shell segment and H ¼g2/g1, K ¼g3/g1, ⎤ 4δ1δ2 ⎡ 4δ1δ2 ⎢ (Φ + Φ*) + (Φa + Φa**)⎥ Q, G = a a F= mnπ 2ρ1 ⎣ R mnπ 2ρ1 a ⎦ For seeking amplitude-frequency relation of nonlinear vibration we substitute (24) into Eq (23) to give (18) 2 Y = A(ωmn − Ω2)sin Ωt + 2εAΩ cos Ωt − ωmn HA2 sin2Ωt + Kωmn A sin3Ωt − F sin Ωt + G = where 2 y31(y12 y23 − y22 y13 ) ∫0 n (y21y13 − y11y23 ) + , y11y22 − y12 y21 π /2Ω Y sin Ωtdt = 0, the frequency-amplitude relation of nonlinear vibration is obtained (19) Ω2 − ⎛ 4ε 3K ⎞ F 4G ⎜1 − Ω = ωmn HA + A⎟− + ⎝ π 3π ⎠ A Aπ By denoting α = Ω 3.1 Natural frequencies α2 − Taking linear parts of the set of Eq (16) and putting p¼ 0, q¼ and ε ¼0 the natural frequencies of the shell can be directly calculated by solving determinant y11 + ρ1ω2 y12 y21 y22 + ρ1ω y23 y31 y32 y33 + ρ1ω2 =0 (20) Solving Eq (20) leads to three angular frequencies of the toroidal shell segment in the axial, circumferential and radial directions, and the smallest one is being considered In other hand, the fundamental frequencies of the shell can be approximately determined by explicit expression in Eq (18) 2 /ωmn (26) Eq (26) is rewritten as 4ε 3K F 4G HA + A − + α=1− 2 3π πωmn Aωmn Aπωmn (27) For the nonlinear vibration of the shell without damping ( ε = 0), this relation has of the form α2 = − y13 (25) Integrating over a quarter of vibration period y (y y − y11y23 ) hπ m −p , − 32 21 13 y11y22 − y12 y21 y11y22 − y12 y21 L2 y (y n2 − y22 n1) y (y n1 − y11n2) n5(y12 y23 − y22 y13 ) + + 32 12 g2 = n3 + 31 12 y11y22 − y12 y21 y11y22 − y12 y21 y11y22 − y12 y21 (y n2 − y22 n1) n (y21n1 − y11n2) − g3 = − n4 − n5 12 y11y22 − y12 y21 y11y22 − y12 y21 (23) is the fundamental frequency of linear vi- Wmn = A sin Ωt , dWmn + 2ρ1ε + g1Wmn − g2Wmn + g3Wmn dt dt ⎤ 4δ1δ2 ⎡ 1 = ⎢ q − (Φa + Φa*) − (Φa + Φa**)⎥⎦ R a mnπ ⎣ g1 = − y33 − + 2ε where ωmn = Solving the first and the second obtained equations with respect to Umn and Vmn and then substituting the results into the third equation yields ρ1 (22) 3K F 4G HA + A − + 2 3π Aωmn Aπωmn (28) If F¼ 0, G ¼0 i.e no force excitation and thermal effect acting on the shell, the frequency-amplitude relation of the free nonlinear vibration without damping is obtained 3K ⎞ 2 ⎛ ⎜1 − ωNL = ωmn HA + A⎟ ⎝ 3π ⎠ where ωNL is the nonlinear vibration frequency (29) 204 D.G Ninh, D.H Bich / Thin-Walled Structures 104 (2016) 198–210 3.3 Nonlinear vibration responses Consider an eccentrically stiffened functionally graded toroidal shell segment acted on by a uniformly distributed transverse load q(t ) = Q sin Ωt and a pre-loaded axial compression p, the set of motion Eq (16) has of the form y11Umn + y12 Vmn + y13 Wmn + n1Wmn = ρ1 d 2Umn y21Umn + y22 Vmn + y23 Wmn + n2Wmn = ρ1 y31Umn + y32 Vmn + y33 Wmn + p + n 6VmnWmn + = ρ1 d 2Wmn dt hπ 2m2 L2 dt αm = 23 × 10−6 0C −1; , dt 2 Wmn + n3Wmn + n4Wmn + n5UmnWmn dWmn dt (30) And the motion Eq (18) by the use of Volmir's assumption becomes ρ1 d2Wmn dWmn + 2ρ1ε + g1Wmn − g2Wmn + g3Wmn dt dt ⎤ 4δ1δ2 ⎡ 1 = Q sin Ωt − (Φa + Φa*) − (Φa + Φa**)⎥ 2⎢ ⎣ ⎦ R a mnπ (31) Using the fourth-order Runge-Kutta method into Eq (30) or Eq (31) combined with initial conditions, the nonlinear vibration responses of ES-FGM toroidal shell segment can be investigated Results and discussion 4.1 Validation Up to now, to the best of the authors' knowledge, there is no publication about nonlinear vibration of ES-C-FGM-M toroidal shell segment, that is reason to compare the results in this paper with homogenous and FGM cylindrical shell (i.e a toroidal shell segment with R-1 and hc ¼hm ¼0) Firstly, the results of natural frequencies in present will be compared with results for the un-stiffened isotropic cylindrical shell studied by Lam and Loy [50], Li [51] and Shen [43] and can be seen in Table As can be seen in Table that good agreements are obtained in this comparison Moreover the frequencies calculated by Eq (20) (the full order equation ODE) and Eq (21) (the Volmir's assumption) are quite close to each other Secondly, the natural frequencies of FGM cylindrical shell illustrated in Table are computed and compared with the results of Loy et al [35] using Rayleigh–Ritz method and Shen [43] with two kinds of micromechanics models: Voigt model and Mori–Tanaka model based on a higher order shear deformation shell theory A FGM cylindrical shell is made of stainless steel and nickel material in initial temperature T0 ¼300 K is considered with the following material properties Table Comparison of dimensionless frequencies ω¯ = Ω(h/π ) 2(1 + ν )ρ /E for an isotropic cylindrical shell (a/L ¼ 2; h/a ¼ 0.06, E ¼ 210 GPa, υ¼ 0.3, ρ = 7850 kg/m3) (m, n) Lam and Loy [50] Li [51] Shen [43] Present (Eq 21) Present (Eq 20) (1, 1) (1, 2) (1, 3) (1, 4) 0.03739 0.03666 0.03634 0.03723 0.03712 0.03648 0.03620 0.03700 0.03755 0.03751 0.03717 0.03684 0.03748 0.03671 0.03635 0.03720 ρc = 3800 kg/m3; αc = 5.4 × 10 C −1 and Poisson's ratio is chosen to be 0.3 The elastic foundation parameters are taken as K1 ¼2.5 Â 108 N/m3, K2 ¼5 Â 105 N/m with Pasternak foundation The parameters n1 ¼50 and n2 ¼50 are the number of stringer and ring stiffeners, respectively , ⎤ 4δ1δ ⎡ 1 ⎢ Q sin Ωt − (Φa + Φa*) − (Φa + Φa**)⎥⎦ R a mnπ ⎣ + 2ρ1ε Ec = 380 × 109N /m2; −6 d Vmn ENi ¼205.09 GPa; υNi ¼0.31; ρNi ¼8900 kg/m3; ESS ¼ 207.7877 GPa; υ ¼0.32; ρNi ¼ 8166 kg/m3 As can be seen, a very good agreement is obtained in the comparison with the results of Ref [35], but there is a little difference with those of Ref [43] because the author used other theories In the following sections, the materials consist of Aluminum Em = 70 × 109N /m2; ρm = 2702 kg/m3; and Alumina with 0.03781 0.03761 0.03733 0.03705 4.2 The fundamental frequencies The natural frequencies of ES-C-FGM-M toroidal shell segment in three cases using Eq (20) are illustrated in Table The datum of problem: h¼0.01 m; hc ¼0.3 h; hm ¼ 0.1 h; a¼300 h; R ¼500 h; L¼ 2a; d1 ¼ d2 ¼ h/2; h1 ¼h2 ¼ h/2; n1 ¼n2 ¼ 50; k ¼1 It can be seen that the natural frequencies of the shell on Pasternak foundation are the highest while the natural frequencies with pre-loaded axial compression (p ¼1 GPa) are the lowest It means that when the shell is subjected to pre-loaded axial compression, the natural frequencies will lessen 4.3 Frequency-amplitude curve For investigating the dynamic responses, we can use with arbitrary mode (m, n), for instance mode number (m, n) ¼(1, 7) The frequency-amplitude curve of nonlinear free vibration of the shell and the effects of pre-loaded axial compression, elastic foundation are indicated in Fig It can be observed that the lowest frequency will increase when the shell is on elastic foundation Whereas, it will decrease when the shell bears the pre-loaded axial compression without elastic foundation The effect of amplitude of external force on the frequencyamplitude curve in case of forced vibration is illustrated in Fig The line is corresponding to the free vibration case (F ¼0, p ¼0) of the shell without elastic foundation The lines and correspond to the forced vibration cases of the shell with pre-loaded axial compression (p ¼2.5 GPa) and without elastic foundation under excited forces with F¼5 Â 105 and F¼ Â 105, respectively Finally, the lines and correspond to the free vibration and the forced vibration cases of the shell on Pasternak foundation, respectively It can be seen, the frequency-amplitude curve trend further from the curve of the free vibration case when the amplitude of external force increases The frequency-amplitude curves of the shell on elastic foundation move ahead in the increasing frequency direction in comparison with those curves of the shell without elastic foundation 4.4 Nonlinear vibration responses The comparison of the nonlinear response of the shell calculated by the approximate Eq (31) (Volmir's assumption) and the full order system Eq (30) is shown in Fig From Tables and and Fig 6, we conclude that the Volmir's assumption can be used to investigate nonlinear dynamical analysis with an acceptable accuracy In the next sections, the full order system Eq (30) is used to investigate nonlinear vibration responses As following the effects of the characteristics of functionally graded materials, the preloaded axial compression, the dimensional ratios, the elastic foundation and thermal loads on the nonlinear dynamic responses D.G Ninh, D.H Bich / Thin-Walled Structures 104 (2016) 198–210 205 Table Comparisons of natural frequencies f = (m, n) SUS304/Ni (1, 7) (1, 8) Ni/SUS304 (1, 7) (1, 8) Ω (Hz) 2π Source for FGM cylindrical shells (L/a ¼20, a/h¼ 20, h ¼ 0.05 m, T ¼ 300 K) k 0.0 0.5 1.0 2.0 5.0 15.0 Loy et al [35] Shen [43] Present (Eq 21) Present (Eq 20) Loy et al [35] Shen [43] Present (Eq 21) Present (Eq 20) 580.78 585.788 601.267 578.011 763.98 759.914 784.929 761.367 570.25 575.266 591.025 568.187 750.12 746.278 771.557 748.427 565.46 570.471 586.279 563.630 743.82 740.065 765.362 742.425 560.93 565.925 581.741 559.267 737.86 734.176 759.437 736.677 556.45 561.399 577.242 554.934 731.97 728.311 753.565 730.971 553.37 558.274 574.228 552.027 737.92 724.262 749.631 727.140 Loy et al [35] Shen [43] Present (Eq 21) Present (Eq 20) Loy et al [35] Shen [43] Present (Eq 21) Present (Eq 20) 551.22 556.073 572.189 550.057 725.08 721.406 746.968 724.546 560.94 565.780 581.633 559.113 737.87 733.988 759.297 736.474 565.63 570.478 586.279 563.574 744.04 740.074 765.362 742.349 570.25 575.119 590.913 568.028 750.13 746.088 771.411 748.217 575.03 579.936 595.704 572.642 756.41 752.327 777.666 754.294 578.40 583.355 599.008 575.829 760.84 756.757 781.979 758.492 Table The fundamental frequencies (s À 1) using Eq (20) in various cases of ES-C-FGM-M toroidal shell segment Cases ω1 (1, 1)* ω2 (1, 2) Natural frequencies Natural frequencies of shell on Pasternak foundation Natural frequencies of shell with pre-loaded axial compression (p¼ GPa) 2362.216 2932.162 2214.457 2096.708 2805.679 2727.048 3037.544 3116.408 3846.529 3788.811 n ω3 (1, 3) 2351.385 2201.753 2081.748 ω4 (2, 1) 2750.674 ω5 (2, 2) 2670.454 The numbers in brackets indicate the vibration buckling mode (m, n) Fig Effects of elastic foundation and pre-loaded axial compression on frequency-amplitude curve of ES-C-FGM-M toroidal shell segment in case of free vibration and no damping of the ES-FGM sandwich toroidal shell segments are analyzed The effects of material and geometric parameters, elastic foundation, thermal environment and the beating vibration phenomenon on the non-linear vibration of FGM sandwich toroidal shell are considered in Figs 7–21 Figs and depict the effect of R/h ratio on nonlinear vibration of convex and concave stiffened FGM sandwich toroidal shell segment, respectively It can be seen that when increasing R/h ratio, the amplitudes of nonlinear vibration of both stiffened Fig The frequency-amplitude curve in case of forced vibration Fig The comparison of the nonlinear dynamical response on Pasternak foundation calculated by Eq (31) (Volmir's assumption) and Eq (30) (the full order equation system) C-FGM-M shell also increase and the frequency does not modify much Furthermore, the amplitudes of nonlinear vibration of convex ES-FGM sandwich shell are smaller than ones of concave ES-FGM sandwich shell The effect of L/R ratio is described in Figs and 10 It can be seen that when L/R ratios increase, the amplitudes of nonlinear vibration of convex ES-FGM core shell also go up while those of concave ES-C-FGM-M shell decrease It means that the amplitudes 206 D.G Ninh, D.H Bich / Thin-Walled Structures 104 (2016) 198–210 Fig Effect of R/h ratio on nonlinear vibration response of convex ES-FGM sandwich toroidal shell segment on elastic medium Fig 11 Effect of L/a on nonlinear vibration response of convex ES-FGM sandwich toroidal shell segment on elastic medium Fig Effect of R/h ratio on nonlinear vibration response of concave ES-FGM sandwich toroidal shell segment on elastic medium Fig 12 Effect of L/a on nonlinear vibration response of concave ES-FGM sandwich toroidal shell segment on elastic medium Fig Effect of L/R ratio on nonlinear vibration response of convex ES-FGM sandwich toroidal shell segment on elastic medium Fig 10 Effect of L/R ratio on nonlinear vibration response of concave ES-FGM sandwich toroidal shell segment on elastic medium Fig 13 Effect of thickness of ceramic and metal layer on nonlinear vibration response of convex ES-FGM sandwich toroidal shell segment on elastic medium Fig 14 Effect of thickness of ceramic and metal layer on nonlinear vibration response of concave ES-FGM sandwich toroidal shell segment on elastic medium D.G Ninh, D.H Bich / Thin-Walled Structures 104 (2016) 198–210 207 Fig 15 Effect of volume-fraction k on nonlinear vibration response of ES-FGM sandwich toroidal shell segment on elastic medium Fig 19 Effect of thermal environment on nonlinear vibration response of ES-FGM sandwich concave toroidal shell segment Fig 16 Effect of elastic foundation on nonlinear vibration response of ES-FGM sandwich toroidal shell segment Fig 20 Nonlinear response of FGM convex toroidal shell segment on elastic medium with k ¼ 1( ω1 = 3089.009 rad/s) Fig 17 Effect of pre-loaded axial compression on nonlinear vibration response of ES-FGM sandwich toroidal shell segment Fig 21 Nonlinear response of FGM concave toroidal shell segment on elastic medium with k ¼ ( ω1 = 2701.644 rad/s) Fig 18 Effect of thermal environment on nonlinear vibration response of ES-FGM sandwich convex toroidal shell segment of the more convex shells are greater than that of the less convex ones whereas this feature of concave shells is completely on the contrary On the other hand, the amplitudes of nonlinear vibration of convex shell are lower than ones of concave shell As can be observed in Figs 11 and 12, the influence of ratio L/a on the nonlinear response of the shell is similar as one of ratio L/R In addition, the amplitude of nonlinear vibration response of ESFGM sandwich concave shell is unequal Based on Figs 13 and 14, as can be seen, the amplitude of nonlinear vibration of both ES-FGM core shells decrease when the thickness of ceramic layer increases It means that the sandwich structures will be stiffer than FGM structures with the same geometry parameters Thus, the amplitudes of nonlinear vibration of sandwich structure will be lower than those of FGM structure 208 D.G Ninh, D.H Bich / Thin-Walled Structures 104 (2016) 198–210 Furthermore, the amplitudes of nonlinear vibration of convex shell are lower than those of concave shell The effect of volume-fraction k on nonlinear vibration of ESFGM core shell is illustrated in Fig 15 The amplitudes of nonlinear vibration of ES-FGM toroidal shell segment increase when the value of volume fraction index increases This property evidently appropriates to the real characteristic of material, because the higher value of k corresponds to a metalricher shell which has less stiffness than a ceramic-richer one Based on Fig 16, the amplitudes of nonlinear vibration of shell without elastic foundation are the highest and those on Pasternak foundation are lowest In addition, the amplitudes of nonlinear vibration of shell without elastic foundation in this example are about times more than ones on Pasternak foundation The effect of pre-loaded axial compression on the nonlinear responses of ES-FGM core shell is indicated in Fig 17 As can be seen, the amplitude of nonlinear vibration of the shell increases when the value of axial compression load increases The pre-loaded axial compression makes the load bearing capacity of the shell reduce under dynamical loads Based on Figs 18 and 19, we can see that the temperature increases, the stiffness of the structure will reduce and the amplitude will rise Therefore, the load bearing capacity of the structure will mitigate The shell is preheated, the temperature field makes the shell be deflected outward (negative deflection) before it is affected by lateral load It means that the amplitude of structure will be negative Furthermore, the amplitude of nonlinear vibration of convex shell is about times higher than one of concave shell This can be explained as follows: The geometry of the shell is concave, thus when temperature field affects, it is expanded and has trend as cylindrical shell In contrarily, the shape of convex shell is convex thus it is expanded so much In the previous sections nonlinear dynamic responses of ESFGM core toroidal shell segments are investigated when the frequency of excited force is far from the natural frequency of the shell The picture of nonlinear dynamic response of the shell is quite different when the frequency of the excitation is near to the natural frequency of the shell Fig 20 describes the nonlinear response of ES-C-FGM-M convex toroidal shell segment with the natural frequency ω1 = 3239.2094 rad/s under excitation q ¼1000sin (3150t) Similarly, Fig 18 demonstrates the nonlinear response of ESFGM sandwich concave shell with the same datum of problem excepted only R/h¼ À 500 and in this case the natural frequency of the shell ω1 = 2748.4063 rad/s and the external frequency Ω¼2650 rad/s From obtained figures, the interesting phenomenon is observed like harmonic beat of linear vibration, where the number of the beats of ES-FGM sandwich convex shell is smaller than those of concave shell Concluding remarks The nonlinear thermal vibration of eccentrically stiffened Ceramic-FGM-Metal layer toroidal shell segment subjected to mechanical loads using the displacement method is investigated Based on the classical shell theory with the geometrical nonlinear in von Karman-Donnell sense and the smeared stiffeners technique, the governing equations of motion of eccentrically stiffened functionally graded toroidal shell segments are derived in this paper Furthermore, the Galerkin method is used for the vibration analysis of shells to perform natural frequencies, nonlinear frequency-amplitude relation and nonlinear dynamic responses Some remarkable conclusions are obtained as follows: – The obtained results of natural frequencies are compared with the results of the other authors to validate Moreover, elastic foundation and pre-loaded axial compression also remarkably influence to the fundamental frequencies – The lowest frequency in frequency-amplitude curve increase when the toroidal shell segment is on elastic foundation while it will decrease if the shell bears the pre-loaded axial compression – The effects of characteristic of functionally graded materials, geometrical ratios, elastic foundation, pre-loaded axial compression, ceramic and metal layer and thermal environment on the nonlinear vibration behavior of shells are indicated The trend of the nonlinear vibration responses of ES-FGM sandwich concave shell is contrast with those of ES-FGM sandwich convex shell when changing L/R and L/a ratios Furthermore, the thermal environment significantly impacts on the nonlinear vibration response to make the amplitude of structural shell negative – The beating vibration phenomenon is shown when the frequency of the external force is near to the natural frequency of the shell Moreover, the number of the beats of ES-FGM sandwich concave shell is more than those of ES-FGM sandwich convex shell Appendix A Operators in Eq (13) ⎛ E A ⎞ ∂ 2u ∂ 2u Y11(u) = ⎜ A11 + ⎟ + A66 , s1 ⎠ ∂x1 ⎝ ∂x2 Y12(v) = ( A12 + A66 ) ∂ 2v , ∂x1∂x2 ⎛ A + E0A1/s1 A ⎞ ∂w ∂ 3w ∂ 3w + 12 ⎟ − (B11 + C1) − (B12 + 2B66 ) Y13(w ) = − ⎜ 11 , ⎝ R a ⎠ ∂x1 ∂x1 ∂x1∂x22 ⎛ E A ⎞ ∂w ∂ 2w ∂w ∂ 2w ∂w ∂ 2w P1(w ) = ⎜ A11 + ⎟ + ( A12 + A66 ) + A66 , ∂x2 ∂x1∂x2 ∂x1 ∂x22 s1 ⎠ ∂x1 ∂x12 ⎝ Y21(u) = ( A66 + A12 ) Y22(v) = A66 ∂ 2u , ∂x1∂x2 E A ⎞ ∂ 3v ∂ 2v ⎛ + ⎜ A22 + ⎟ , ∂x1 s2 ⎠ ∂x2 ⎝ Y23(w ) = − ( B22 + C2) ∂ 3w ∂x23 − ( B12 + 2B66) ∂ 3w ∂x12∂x2 ⎛A A + E0A2 /s2 ⎞ ∂w − ⎜ 12 + 22 , ⎟ ⎝ R ⎠ ∂x2 a P2(w ) = ( A66 + A12 ) Y31(u) = ( B11 + C1) E A ⎞ ∂w ∂ 2w ∂w ∂ 2w ∂w ∂ 2w ⎛ + A66 + ⎜ A22 + ⎟ , ∂x1 ∂x1∂x2 ∂x2 ∂x1 s2 ⎠ ∂x2 ∂x22 ⎝ ∂ 3u ∂x13 + ( 2B66 + B12) ⎛A A + E0A1/s1 ⎞ ∂u + ⎜ 12 + 11 , ⎟ ⎠ ∂x1 ⎝ a R ∂ 3u ∂x1∂x22 D.G Ninh, D.H Bich / Thin-Walled Structures 104 (2016) 198–210 ∂ 3v Y32(v) = ( B12 + 2B66 ) ⎛ Y33(w ) = − ⎜ D11 + ⎝ + ( B22 + C 2) ∂x12∂x2 E0I1 ⎞ ∂ 4w ⎟ s1 ⎠ ∂x14 ∂ 3v ∂x23 ⎛A A + E0A2 /s2 ⎞ ∂v , + ⎜ 12 + 22 ⎟ ⎝ R ⎠ ∂x2 a y23 = ( B22 + C2) ⎛ 2( B11 + C1) ⎞ ∂ 2w 2B12 − ⎜⎜ + − K2 + (Φa + Φa⁎)⎟⎟ R a ⎝ ⎠ ∂x1 y31 = − ( B11 + C1) ⎛ ∂ 2w ⎞2 ⎛ A + E A /s A ⎞⎛ ∂w ⎞ 1 ⎟⎟ + ⎜ 11 + 12 ⎟⎜ P3(w ) = ( 2B12 − 2B66 )⎜⎜ ⎟ 2 ∂ ∂ ∂ x x R a x ⎝ ⎠ ⎝ ⎠ ⎝ 2⎠ + ( 2B66 − 2B12) ∂ 2w ∂ 2w ∂x12 ∂x22 ⎛ A + E0A1/s1 A ⎞ ∂ 2w − ⎜ 11 + 12 ⎟w R a ⎠ ∂x1 ⎝ + ( B11 + C1) ⎛ ∂w ∂ 3w ∂w ∂ 3w ∂w ∂ 3w ⎞⎟ + + ( B12 + 2B66 )⎜⎜ ∂x2 ∂x12∂x2 ⎟⎠ ∂ ∂x2 ∂x23 x ⎝ ∂x1∂x2 ∂w ∂ 3w ∂x1 ∂x13 2⎤ ⎡ ⎛ ⎛ A + E0A1/s1 ⎞ ∂ 2w ⎛ ∂w ⎞ A ∂ w ∂w ⎞ ∂ 2w ⎛ ∂w ⎞ ⎥ + ⎜ 11 ⎟ + 12 ⎢⎢ ⎜ ⎟ + ⎜ ⎟ ⎥ ⎟ 2⎜ 2 ⎣ ∂x1 ⎝ ∂x2 ⎠ ⎝ ⎠ ∂x1 ⎝ ∂x1 ⎠ ∂x2 ⎝ ∂x1 ⎠ ⎦ ⎛ A + E0A2 /s2 ⎞ ∂ 2w ⎛ ∂w ⎞ + ⎜ 22 ⎟ ⎟ 2⎜ ⎝ ⎠ ∂x2 ⎝ ∂x2 ⎠ ∂w ∂w ∂ 2w , ∂x1 ∂x2 ∂x1∂x2 ⎛ E A ⎞ ∂u ∂ 2w ∂u ∂ 2w ∂u ∂ 2w Q 3(u, w ) = ⎜ A11 + ⎟ + 2A 66 + A12 , s1 ⎠ ∂x1 ∂x12 ∂x2 ∂x1∂x2 ∂x1 ∂x22 ⎝ R3(v, w ) = A12 ⎛ E A ⎞ ∂v ∂ 2w ∂v ∂ 2w ∂v ∂ 2w + ⎜ A22 + ⎟ + 2A66 ∂x2 ∂x12 ∂x1 ∂x1∂x2 ⎝ s2 ⎠ ∂x2 ∂x22 Appendix B Coefficients in system of Eq (16) ⎛ E A ⎞ m2π n2 y11 = − ⎜ A11 + ⎟ − A66 , s1 ⎠ L ⎝ 4a y12 = − ( A12 + A66 ) mnπ , 2La y22 = − A66 mnπ , 2La E A ⎞ n2 m2π ⎛ − ⎜ A22 + ⎟ , s2 ⎠ 4a ⎝ L mπ n2 L 4a2 ⎛ 2( B11 + C1) ⎞ m2 π 2B12 + ⎜⎜ + − K2 + (Φa + Φa⁎)⎟⎟ R a ⎝ ⎠ L ⎛ 2( B22 + C 2) ⎞ n2 2B12 m2 π n + ⎜⎜ + − K2 + (Φa + Φa⁎⁎)⎟⎟ − (2D12 + 4D66 ) a R ⎝ ⎠ 4a L 4a ⎛ A + E0A1/s1 ⎞ 2A12 A + E0A2 /s2 − ⎜ 11 + + 22 + K1⎟ , ⎝ ⎠ Ra R2 a2 ⎡⎛ E0A1 ⎞ m3π mπ n ⎤ ⎥ n1 = + A 66 δ1δ 2⎢ ⎜ A11 + ⎟ ⎢⎣ ⎝ s1 ⎠ L3 9mnπ 4La ⎥⎦ + n2 = n3 = 9mnπ ( A12 + A66 ) δδ 2 mπ n 4La m2π 2n δ δ A + A12 ) 2( 66 9mnπ 2L2a ⎡ E A ⎞ n3 ⎤ m2π 2n ⎛ − δ δ ⎢ A66 + ⎜ A22 + ⎟ ⎥ 2 s2 ⎠ 8a ⎦ ⎝ 9mnπ 2L a ⎣ 9aLmnπ δ1δ 2( 2B12 − 2B 66 ) m2π 2n2 4L2a2 ⎡ ⎛ A + E 0A2 /s2 ⎛ A E A A ⎞ m2π A ⎞ n2 + 16 1/s1 + δ1δ 2⎢ ⎜ 11 + 12 ⎟ + ⎜ 22 + 12 ⎟ ⎝ ⎢⎣ ⎝ 2R 2a ⎠ L2 2a 2R ⎠ 4a 9aLmnπ 2 4⎤ n m π n m π ⎥ − ( B 22 + C2) − ( 4B 66 + 2B12) + ( B11 + C1) L4 ⎥⎦ 16a4 4L2a2 ⎛ A11 + E 0A1/s1 A ⎞ m2π m2π 2n2 128 ⎡⎢ + + 12 ⎟ ( 2B 66 − 2B12) 2 + ⎜⎝ R a ⎠ L2 9aLmnπ ⎢⎣ 4L a ⎛ A22 + E 0A2 /s2 A12 ⎞ n2 ⎤ ⎥, +⎜ + ⎟ ⎝ a R ⎠ 4a2 ⎥⎦ ⎛ ⎞ m2π 2n2 A + E 0A2 /s2 n4 ⎞ ⎛⎜ A11 + E 0A1/s1 m4 π ⎟⎟, + ⎜ A12 − A 66 ⎟ + 22 n4 = − ⎜ ⎝ ⎠ 4L2a2 8πaL ⎝ 16a4 ⎠ L4 n5 = E A ⎞ m3π 64 ⎡ ⎛ mπn2 ⎤ 16La mπn2 ⎥+ ⎢ ⎜ A11 + ⎟ + A12 δ1δ2A66 , 2 π s mn 4La2 9mnπ ⎣ ⎝ 4La ⎦ ⎠ L n6 = E A ⎞ n3 64 ⎡ ⎛ m2π 2n ⎤ m2π 2n 16La ⎥+ ⎢ ⎜ A22 + ⎟ + A12 δ1δ2A66 s2 ⎠ 8a 2La ⎦ 9mnπ 9mnπ ⎣ ⎝ 2L2a ⎛ A + E0A1/s1 A ⎞ mπ m3 π mπ n + 12 ⎟ + (B11 + C1) + (B12 + 2B66 ) y13 = − ⎜ 11 , ⎝ R a ⎠ L L 4a L y21 = − ( A66 + A12 ) + ( 2B66 + B12) ⎛ ⎛ E I ⎞ m4 π E I ⎞ n4 y33 = − ⎜ D11 + ⎟ − ⎜ D 22 + ⎟ s1 ⎠ L s2 ⎠ 16a ⎝ ⎝ + 2A 66 m3π m2π 2n 2L2a ⎛A A + E0A2 /s2 ⎞ n n3 n + ( B22 + C2) − ⎜ 12 + 22 ⎟ , ⎝ R ⎠ 2a 2a a 8a ⎛ A + E0A2 /s2 A ⎞ ∂ 2w − ⎜ 22 + 12 ⎟w a R ⎠ ∂x2 ⎝ + ( B22 + C 2) + ( B12 + 2B66) L3 ⎛ A12 A11 + E0A1/s1 ⎞ mπ m2π −⎜ + , y32 = ( B12 + 2B66) ⎟ ⎠ L ⎝ a R L ⎛ A + E0A1/s1 ⎞ 2A12 A + E0A2 /s2 − ⎜ 11 + + 22 − K1⎟w , ⎝ ⎠ Ra R2 a2 ⎛ A + E0A2 /s2 A ⎞⎛ ∂w ⎞ + ⎜ 22 + 12 ⎟⎜ ⎟ 2a 2R ⎠⎝ ∂x2 ⎠ ⎝ 8a ⎛ A12 A + E0A2 /s2 ⎞ n −⎜ + 22 ⎟ , ⎝ R ⎠ 2a a ⎛ E I ⎞ ∂ 4w − ⎜ D 22 + ⎟ s2 ⎠ ∂x2 ⎝ ⎛ 2( B22 + C 2) ⎞ ∂ 2w 2B12 ∂ 4w − ⎜⎜ + − K2 + (Φa + Φa⁎⁎)⎟⎟ − (2D12 + 4D66 ) 2 a R ∂x1 ∂x2 ⎝ ⎠ ∂x2 n3 209 where δ1 = ( − 1)m − and δ2 = ( − 1)n − References [1] W Jiang, D Redekop, Static and vibration analysis of orthotropic toroidal shells of variable thickness by differential quadrature, Thin-Walled Struct 41 210 D.G Ninh, D.H Bich / Thin-Walled Structures 104 (2016) 198–210 (1995) 461–478 [2] G.R Buchanan, Y.J Liu, An analysis of the free vibration of 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analysis of circular cylindrical shells using wave propagation, J Sound Vib 311 (2008) 667–682  ... best of the authors'' knowledge, there has not been any study to the nonlinear thermal vibration of eccentrically stiffened Ceramic-FGM-Metal layer toroidal shell segments surrounded by an elastic. .. harmonic beat of linear vibration, where the number of the beats of ES-FGM sandwich convex shell is smaller than those of concave shell Concluding remarks The nonlinear thermal vibration of eccentrically. .. nonlinear vibration of convex shell are lower than those of concave shell The effect of volume-fraction k on nonlinear vibration of ESFGM core shell is illustrated in Fig 15 The amplitudes of nonlinear