Composites Part B 95 (2016) 355e373 Contents lists available at ScienceDirect Composites Part B journal homepage: www.elsevier.com/locate/compositesb Nonlinear dynamical analyses of eccentrically stiffened functionally graded toroidal shell segments surrounded by elastic foundation in thermal environment Dao Huy Bich a, Dinh Gia Ninh b, *, Bui Huy Kien c, David Hui d a Vietnam National University, Hanoi, Viet Nam School of Mechanical Engineering, Hanoi University of Science and Technology, Hanoi, Viet Nam Faculty of Mechanical Engineering, Hanoi University of Industry, Hanoi, Viet Nam d Department of Mechanical Engineering, University of New Orleans, Louisiana, USA b c a r t i c l e i n f o a b s t r a c t Article history: Received 19 February 2016 Received in revised form 31 March 2016 Accepted April 2016 Available online April 2016 In this study, the nonlinear vibration and dynamic buckling of eccentrically stiffened functionally graded toroidal shell segments surrounded by an elastic medium in thermal environment are presented The governing equations of motion of eccentrically stiffened functionally graded toroidal shell segments are derived based on the classical shell theory with the geometrical nonlinear in von Karman-Donnell sense and the smeared stiffeners technique Furthermore, the dynamical characteristics of shells as natural frequencies, nonlinear frequencyeamplitude relation, nonlinear dynamic responses and the nonlinear dynamic critical buckling loads evaluated by Budiansky-Roth criterion are considered The effects of characteristics of functionally graded materials, geometrical ratios, elastic foundation, pre-loaded axial compression and temperature on the dynamical behavior of shells are investigated © 2016 Elsevier Ltd All rights reserved Keywords: Toroidal shell segments A Discontinuous reinforcement B Thermomechanical B Vibration C Analytical modelling Introduction In the past, the dynamic problems of laminated composite structures studied by many authors Hui [1,2] presented the effects of shear loads on vibration and buckling of typical antisymmetric cross-ply thin cylindrical panels under combined loads; effects of geometrical imperfections on the large amplitude vibration of shallow spherical shells as well as effects of structural damping Dynamic fracture and delamination of unidirectional graphite/ epoxy composites for end-notched flexure center-noted flexure pure mode II loading configurations using a modified split Hopkinson pressure bar were investigated by Nwosu et al [3] Functionally graded materials (FGMs) were invented by Japanese scientists in 1984 [4] This composite material is a mixture of ceramic and metallic constituent materials by continuously changing in the volume fractions of their components Mechanical * Corresponding author Tel.: ỵ84 988 287 789 E-mail addresses: ninhdinhgia@gmail.com, (D.G Ninh) http://dx.doi.org/10.1016/j.compositesb.2016.04.004 1359-8368/© 2016 Elsevier Ltd All rights reserved ninh.dinhgia@hust.edu.vn and physical behavior of FGMs are better than fiber reinforced laminated composite materials with advantages of no stress concentration, high toughness, oxidation resistance and heatresistance so FGMs are applied to heat-resistant, lightweight structures in aerospace, mechanical, and medical industry and so forth Therefore, the nonlinear vibration and dynamic buckling problems of FGM structures have been attracted a vast amount attention of researchers Pradhan et al [5] studied vibration of FGM cylindrical shells under various boundary conditions with the strain-displacement relations form Love's shell theory 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 general formulation for free, steadystate and transient vibration analyses of FGM shells of revolution subjected to arbitrary boundary conditions was presented by Qu et al [6] The formulation was derived by means of a modified variational principle in conjunction with a multi-segment partitioning procedure on the basis of the first order shear deformation shell theory G G Sheng and X Wang [7] investigated the nonlinear vibrations control of FGM laminated cylindrical shell based on 356 D.H Bich et al / Composites Part B 95 (2016) 355e373 Hamilton's principle, von Karman nonlinear theory and constantgain negative velocity feedback approach The thin piezoelectric layers embedded on inner and outer surfaces of the smart FG laminated cylindrical shell were acted as distributed sensor and actuator, which is used to control nonlinear vibration of the smart FG laminated cylindrical shell The free vibration analysis of functionally graded cylindrical panels with cut-out and under temperature condition using the three-dimensional Chebyshev-Ritz method was noticed by Malekzadeh et al [8] Chen and Babcock [9] gave the large amplitude vibration of a thin-walled cylindrical shell using the perturbation method the steady-state forced vibration problem Furthermore, the simply-supported boundary conditions and the circumferential periodicity condition were satisfied The unified solution for the vibration analysis of functionally graded material (FGM) doubly-curved shells of revolution with arbitrary boundary conditions was given by Jin et al [10] The solution was derived by means of the modified Fourier series method on the basis of the first order shear deformation shell theory considering the effects of the deepness terms Kim [11] performed free vibration characteristics of FGM cylindrical shells partially resting on elastic foundation with an oblique edge using an analytical method The motion of shell was represented based on the first order shear deformation theory to account for rotary inertia and transverse shear strains The nonlinear dynamic buckling and pre-buckling deformation of FGM truncated conical shells under axial compressive load varying as a linear function of time using the Superposition principle, Galerkin and Runge-Kutta methods were studied by Deniz and Sofiyev [12,13] Shen and Yang [14] carried out the free vibration and dynamic instability of functionally graded cylindrical panels subjected to combined static and periodic axial forces and in thermal environment with theoretical formulations based on Reddy's higher order shear deformation shell theory taking into account rotary inertia and the parabolic distribution of the transverse shear strains through the panel thickness The characteristics of free vibration and nonlinear responses were investigated 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 by Bich et al [15] Moreover, Bich and Nguyen [16] 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 dynamic behavior of moderately thick functionally graded conical, cylindrical shells and annular plates with a four-parameter power law distribution based on the First order Shear Deformation Theory were focused by Tornabene [17] The materials were assumed to be isotropic and inhomogeneous though the thickness direction Sofiyev [18e20] studied the dynamic behavior of FGM structures such as: the vibration of FGM conical shells under a compressive axial load using the shear deformation theory using Donnell shell theory; the parametric vibration of shear deformable functionally graded truncated conical shells subjected to static and time dependent periodic uniform lateral pressures based on the first order shear deformation theory; the theoretical approach to solve vibration problems of FGM truncated conical shells under mixed boundary conditions using means of the Airy stress function method to derive the fundamental relations, motion and strain compatibility equations The vibration of thin cylindrical shells extracted from Ref Függe's three equations of motion was investigated by Warburton [21] This solution required the assumption of a natural frequency and the determination of the corresponding shell length for the prescribed end conditions Based on straindisplacement relations from the Love's shell theory and the eigenvalue governing equation using Rayleigh-Ritz method, Loy et al [22] gave the study at vibration filed of functionally cylindrical shells 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 modification of Donnell type equations of motion were examined by Sofiyev et al [23] The basic equations were reduced to an algebraic equation of the sixth order and numerically solving this algebraic equation gave the dimensionless fundamental frequency Noda [24], Praveen et al [25] first discovered the heat-resistant FGM structures and studied material properties dependent on temperature in thermo elastic analyses Heydarpour and Malekzadeh [26] pointed out the free vibration analysis of rotating functionally graded cylindrical shells in temperature environment with the equations of motion and related boundary conditions derived to Hamilton's principle The initial thermo-mechanical stresses were obtained by solving the thermo elastic equilibrium equations Sheng and Wang [27] 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 [28] 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 shellefoundation interaction The large amplitude vibration behavior of a shear deformable FGM cylindrical panel resting on elastic foundations in thermal environments based on a higher order shear deformation shell theory was investigated by Shen and Wang [29] The thermal effects are also included and the material properties of FGMs are assumed to be temperature-dependent The equations of motion are solved by a two step perturbation technique to determine the nonlinear frequencies of the FGM cylindrical panel Toroidal shell segment has been used in such applications as satellite support structures, rocket fuel tanks, fusion reactor vessels, diver's oxygen tanks and underwater toroidal pressure hull Today, FGMs have received mentionable attention in structural applications The smooth and continuous change in material properties enables FGMs to avoid interface problems and unexpected thermal stress concentrations Some components of the above-mentioned structures may be made of FGM Stein and McElman [30] 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 [31] Parnell [32] 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 [33] has studied the buckling of eccentrically stiffened functionally graded toroidal shell segment under axial compression, lateral pressure and hydrostatic pressure based on the classical thin shell theory, the smeared stiffeners technique and the adjacent equilibrium criterion Furthermore, the nonlinear buckling and postbuckling of ES-FGM toroidal shell segments surrounded by an elastic medium under torsional load based on the analytical approach are investigated by Ninh et al [34,35] Bich et al [36,37] studied the post-buckling of FGM and S-FGM toroidal shell segment under external pressure loads by an analytical approach using the Galerkin method To the best of the authors' knowledge, there has not been any study to the nonlinear dynamical analysis of eccentrically stiffened FGM toroidal shell segments surrounded by an elastic foundation including temperature effects D.H Bich et al / Composites Part B 95 (2016) 355e373 In the present paper, the dynamic buckling behavior and nonlinear vibration of eccentrically stiffened FGM toroidal shell segments on elastic medium in thermal environment are investigated Based on the classical shell theory with the nonlinear straindisplacement relation of large deflection, the Galerkin method, 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 357 and the stiffeners and easier to manufacture, the homogeneous stiffeners can be used Because the pure ceramic ones are brittleness the metal stiffeners are used and arranged at metal-rich side of the shell With the law indicated in (1) the outer surface of the shell is metal-rich and the external metal stiffeners are arranged at this side The strains across the shell thickness at a distance z from the mid-surface are: ε1 ¼ ε01 À zc1 ; ε2 ¼ ε02 À zc2 ; g12 ¼ g012 À 2zc12 ; Governing equations 2.1 Functionally graded material (FGM) Suppose that the material composition of the shell varies smoothly along the thickness is such a way inner surface is ceramicrich and the outer surface is metal-rich by a simple power law in terms of the volume fractions of the constituents We denote Vm and Vc being volume e fractions of metal and ceramic phases respectively, which are related by Vm ỵ Vc ẳ and  k , where h is the thickness of thin Vc is expressed as Vc zị ẳ 2zỵh 2h e walled structure, k is the volume e fraction exponent (k ! 0); z is the thickness coordinate and varies from Àh/2 to h/2; the subscripts m and c refer to the metal and ceramic constituents respectively According to the mentioned law, the Young modulus E(z), the mass density r(z) and the thermal expansion coefficient a(z) can be expressed in the form  2z ỵ h k Ezị ẳ Em Vm ỵ Ec Vc ẳ Em ỵ Ec Em ị ; 2h   2z ỵ h k azị ẳ am Vm ỵ ac Vc ẳ am ỵ ac am ị ; 2h   2z ỵ h k rzị ẳ rm Vm ỵ rc Vc ẳ rm ỵ rc rm Þ : 2h where ε01 and ε02 are normal strains, g012 is the shear strain at the middle surface of the shell and cij are the curvatures 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 [38]: ε01 ¼     vu w vw vv w vw ỵ ; ẳ ỵ ; vx1 R vx1 vx2 a vx2 g012 ¼ The constitutive stressestrain equations by Hooke law for the shell material are given ssh ¼ (1) vu vv vw vw v2 w v2 w v2 w ỵ ỵ ; c ẳ ; c2 ẳ ; c12 ¼ vx2 vx1 vx1 vx2 vx vx1 vx2 vx2 (3) ssh ¼  (2) ssh 12 Ezị n2 Ezị ỵ n2 ị Ezịazị DT; DT ẳ T T0 ; 1n ỵ n1 ị Ezịazị DT; 1n 1n Ezị ẳ g 21 ỵ nị 12 (4) and for metal stiffeners the Poisson's ratio n is assumed to be constant st sst ¼ Em ε1 À Em am DT; s2 ¼ Em ε2 À Em am DT: 2.2 Constitutive relations and governing equations Integrating the stressestrain 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 toroidal shell segment are obtained [38]: Consider a functionally graded 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 The geometry and coordinate system of a stiffened FGM toroidal shell segments are depicted in Fig For the middle surface of a toroidal shell segment, from the figures we have: r ẳ a R1 sin4ị; where a is the equator radius and f 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 f is approximately equal to p/2, thus sinf z 1; cosf z and r ¼ a [30] The form of governing equation is simplified by putting: dx1 ¼ Rd4; dx2 ¼ adq 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 FGM toroidal shell segment is reinforced by string and ring stiffeners In order to provide continuity within the shell   Em A1 ε1 þ A12 ε02 À ðB11 þ C1 Þc1 À B12 c2 Fa F*a ; N1 ẳ A11 ỵ s1   Em A2 N2 ¼ A12 ε01 þ A22 þ ε2 À B12 c1 À ðB22 þ C2 Þc2 À Fa À F** a ; s2 N12 ¼ A66 g012 À 2B66 c12 ; (5)   Em I1 c1 À D12 c2 À Fm À F*m ; M1 ẳ B11 ỵ C1 ị01 ỵ B12 02 D11 ỵ s1   Em I2 M2 ẳ B12 01 ỵ B22 ỵ C2 ị02 D12 c1 D22 ỵ c2 Fm F** m; s2 M12 ¼ B66 g012 À 2D66 c12 ; (6) where Aij,Bij,Dij (i, j ¼ 1, 2, 6) are extensional, coupling and bending stiffnesses of the shell without stiffeners 358 D.H Bich et al / Composites Part B 95 (2016) 355e373 Fig Configuration of toroidal shell segments A11 ¼ A22 ¼ B11 ¼ B22 ¼ D11 ¼ D22 ¼ E1 ; A12 ¼ ; B12 ¼ 1Àn E2 1Àn E3 À n2 ; E1 n ; A66 ẳ E1 ; 21 ỵ nị ; B66 ẳ E2 ; 21 ỵ nị 1n E2 n 1n D12 ¼ E3 n À n2 ; D66 E3 ; ẳ 21 ỵ nị where  Ec Em ịkh2 Ec Em h; E2 ẳ ; kỵ1 2k þ 1Þðk þ 2Þ    Em 1 ỵ h3 ; E3 ẳ ỵ Ec Em ị k ỵ k ỵ 4k ỵ 12  (7) E1 ẳ Em ỵ (8) D.H Bich et al / Composites Part B 95 (2016) 355e373 359 vN1 vN12 v2 u ỵ ẳ r1 ; vx1 vx2 vt vN12 vN2 v2 v ỵ ẳ r1 ; vx1 vx2 vt v2 M1 vx21 ỵ2 ỵN2 þK2 v2 M12 v2 M2 v2 w v2 w þ þ N þ 2N 12 vx1 vx2 vx1 vx2 vx21 vx22 v2 w vx22 À ph v2 w vx21 þ v2 w vx21 ! v2 w vx22 þ (11) N1 N2 ỵ ỵ q K1 w R a ẳ r1 v2 w vt ỵ r1 vw ; vt where K1 (N/m3) is linear stiffness of foundation, K2 (N/m) is the shear modulus of the sub-grade, is damping coefcient and  r1 ẳ rm ỵ Fig Geometry and coordinate system of a stiffened FGM toroidal shell segments (a) stringer stiffeners; (b) ring stiffeners Em A1 z1 Em A2 z1 C1 ¼ À ; C2 ¼ À : s1 s2 (9) Fa ¼ 1Àn F** a ¼ Fm ¼ F** m ¼ d2 s2 h=2 h=2 Z d EzịazịDTdz; F*a ẳ s1 h=2 Z Em am DTdz; Àh=2Àh1 Em am DTdz; Zh=2 EðzÞaðzÞDTzdz; F*m ¼ Àh=2 Àh=2 Z d1 s1 Àh=2 Z Em am DTzdz; Àh=2Àh1 Em am DTzdz: v2 u vt ; v2 v ; vt H31 uị ỵ H32 vị ỵ H33 wị ỵ P3 wị ỵ Q3 u; wị  v2 w 1 Fa ỵ F*a ỵR3 v; wị ph ỵ q R vx1 (13) where the linear operators Hij( )(i,j ¼ 1, 2, 3) and the nonlinear operators Pi( )(i ¼ 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 eccentrically stiffened functionally graded toroidal shell segments surrounded by elastic foundation in thermal environment Àh=2Àh2 (10) If DT ẳ const F*a ẳ H11 uị ỵ H12 vị ỵ H13 wị ỵ P1 wị ẳ r1  1 v2 w vw ; ẳ r1 ỵ r1 Fa ỵ F** a a vt vt h=2h2 1Àn d2 s2 Zh=2 (12) By substituting Eq (3) into Eqs (5) and (6) and then into Eq (11), the term of displacement components are expressed as follows: H21 uị ỵ H22 vị ỵ H23 wị ỵ P2 wị ¼ r1 and Fa ¼    ðrc À rm ị A A h ỵ rm ỵ kỵ1 s1 s2 Em acm ỵ Ecm am Ecm acm ỵ PhDT; P ẳ Em am ỵ 1n kỵ1 2k ỵ d1 h1 d2 h2 Em am DT; F** Em am DT: a ¼ s1 s2 where Ecm ¼ Ec À Em; acm ¼ ac À am 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 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 Ref [38]: Nonlinear analysis In the present paper, the simply-supported boundary conditions are considered w ¼ 0; v ¼ 0; M1 ¼ 0: at x1 ¼ and x1 ¼ L (14) The approximate solutions of the system of Eq (13) satisfying the conditions Eq (14) can be expressed as: mpx1 nx mpx1 nx sin ; v ẳ Vtịsin cos ; L 2a L 2a mpx1 nx2 sin ; w ¼ WðtÞsin L 2a u ¼ UðtÞcos (15) where U, V, W 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: 360 D.H Bich et al / Composites Part B 95 (2016) 355e373 h11 U ỵ h12 V ỵ h13 W þ n1 W ¼ r1 h21 U þ h22 V ỵ h23 W ỵ n2 W ẳ r1 h31 U ỵ h32 V ỵ h33 W ỵ p hp2 m2 d2 U dt d2 V dt Solving Eq (20) leads to three angular frequencies of the toroidal shell in the axial, circumferential and radial directions, and the smallest one is being considered On the other hand, the fundamental frequencies of the shell can be approximately determined by explicit expression in Eq (18) ; ; W ỵ n3 W ỵ n4 W L2   1  4d d 1 ỵn5 UW ỵ n6 VW þ 22 q À Fa þ F*a À Fa þ F** a R a mnp ¼ r1 d2 W dt ỵ 2r1 dW ; dt (16) umn ¼ rffiffiffiffiffi a1 (21) r1 Solving Eq (20) leads to exact solution but implicit expression while Eq (21) performs approximate frequencies but explicit expression and simpler 3.2 Frequency-amplitude curve where hij; ni are given in Appendix B Otherwise, the Volmir's assumption [39] can be used in the dynamic analysis Taking the inertia forces r1(d2U/dt2) / and r1(d2V/dt2) / into consideration because of u