An analytical approach to analyze nonlinear dynamic response of eccentrically stiffened functionally graded circular cylindrical shells subjected to time dependent axial compression and external pressure - Part 1: Governing equations establishment

Volume 36 Number 3 3 2014 Vietnam Journal of Mechanics, VAST, Vol. 36, No. 3 (2014), pp. 201 – 214 AN ANALYTICAL APPROACH TO ANALYZE NONLINEAR DYNAMIC RESPONSE OF ECCENTRICALLY STIFFENED FUNCTIONALLY GRADED CIRCULAR CYLINDRICAL SHELLS SUBJECTED TO TIME DEPENDENT AXIAL COMPRESSION AND EXTERNAL PRESSURE. PART 1: GOVERNING EQUATIONS ESTABLISHMENT Dao Van Dung1, Vu Hoai Nam2,∗ 1Hanoi University of Science, VNU, Vietnam 2University of Transport Technology, Hanoi, Vietnam ∗E-mail: hoainam.vu@utt.edu.vn Received December 25, 2013 Abstract. Based on the classical thin shell theory with the geometrical nonlinearity in von Karman-Donnell sense, the smeared stiffener technique and Galerkin method, this paper deals with the nonlinear dynamic problem of eccentrically stiffened functionally graded circular cylindrical shells subjected to time dependent axial compression and external pressure by analytical approach. The present novelty is that an approximate three-term solution of deflection taking into account the nonlinear buckling shape is cho- sen, the nonlinear dynamic second-order differential three equations system is established and the frequency-amplitude relation of nonlinear vibration is obtained in explicit form. Keywords: Functionally graded material, discontinuous reinforcement, buckling, elastic- ity, analytical modelling. 1. INTRODUCTION Many authors studied the static buckling and postbuckling of FGM cylindrical shells subjected to the mechanic and thermal loading. Shen [1, 2] investigated the non- linear postbuckling of thin FGM cylindrical shells and FGM hybrid cylindrical shells in thermal environments under lateral pressure and axial loading, respectively. Bahtui and Eslami [3] investigated the coupled thermo-elasticity of FGM cylindrical shells. Batra and Iaccarino [4] presented the exact solutions for radial deformations of a functionally graded isotropic and incompressible second-order elastic cylinder. Huang and Han [5–7] studied the buckling and postbuckling of un-stiffened FGM cylindrical shells under axial compres- sion, radial pressure and combined axial compression and radial pressure based on the Donnell shell theory and the nonlinear strain-displacement relations and the nonlinear 202 Dao Van Dung, Vu Hoai Nam three term solution form is used. Sun et al. [8] proposed the accurate symplectic space solutions for thermal buckling of functionally graded cylindrical shells. The postbuckling of shear deformable FGM cylindrical shells surrounded by an elastic medium was studied by Shen [9]. Dung and Hoa [10] investigated the nonlinear torsional buckling and post- buckling of eccentrically stiffened functionally graded thin circular cylindrical shells. Liew et al. [11] studied postbuckling responses of functionally graded cylindrical shells under axial compression and thermal loads. Sofiyev [12] analyzed the buckling of FGM circular shells under combined loads and resting on the Pasternak type elastic foundation. The non-linear static buckling of FGM conical shells which is more general than cylindrical shells, were studied by Sofiyev [13, 14]. Torabi et al. [15] studied the linear thermal buck- ling analysis of truncated hybrid FGM conical shells. For dynamic analysis of FGM cylindrical shells, Singh et al. [16] investigated tor- sional vibrations of functionally graded finite cylinders. Darabi et al. [17] presented respec- tively linear and nonlinear parametric resonance analyses for un-stiffened FGM cylindri- cal shells. Three-dimensional vibration analysis of fluid-filled orthotropic FGM cylindrical shells was investigated by Chen et al. [18]. Sofiyev and Schnack [19] and Sofiyev [20] obtained critical parameters for un-stiffened cylindrical thin shells under linearly increas- ing dynamic torsional loading and under a periodic axial impulsive loading by using the Galerkin technique together with Ritz type variation method. Sofiyev [21–24] and Deniz and Sofiyev [25] investigated the vibration and dynamic instability of FGM conical shells. Torsional vibration and stability of functionally graded orthotropic cylindrical shells on elastic foundations is presented by Najafov et al. [26]. Sofiyev and Kuruoglu [27] investi- gated the torsional vibration and buckling of the cylindrical shell with functionally graded coatings surrounded by an elastic medium. Tornabene and Viola [28] studied free vibra- tion analysis of functionally graded panels and shells of revolution. Huang and Han [29] presented the nonlinear dynamic buckling problems of un-stiffened functionally graded cylindrical shells subjected to time-dependent axial load by using the Budiansky-Roth dynamic buckling criterion [30]. Various effects of the inhomogeneous parameter, loading speed, dimension parameters; environmental temperature rise and initial geometrical im- perfection on nonlinear dynamic buckling were discussed. Dynamic analysis of thick short length FGM cylinders was investigated by Asemi et al. [31]. In engineering design, plates and shells are usually reinforced by stiffeners for the benefit of added load carrying capability with a relatively small additional weight. How- ever, the investigation on this field has received comparatively little attention. Najafizadeh et al. [32] have studied linear static buckling of FGM cylindrical shell under axial compres- sion reinforced by FGM stiffeners. Bich et al. [33–36] investigated the nonlinear static and dynamic analysis of FGM plates, cylindrical panels, shallow shells and cylindrical shells with eccentrically homogeneous stiffener system. Dung and Hoa [37] presented an analyt- ical study of nonlinear static buckling and post-buckling analysis of eccentrically stiffened functionally graded circular cylindrical shells under external pressure with FGM stiffen- ers and approximate three-term solution of deflection taking into account the nonlinear buckling shape. To best of authors’ knowledge, there is no analytical approach on the nonlinear dy- namic analysis of stiffened FGM shells subjected to time dependent external pressure and An analytical approach to analyze nonlinear dynamic response of eccentrically stiffened . . . 203 axial compression by analytical approach. In addition, the nonlinear three term solution of deflection is popular used to investigate the nonlinear static analysis of shell [5-8 and 37], but there are a mathematical difficulty on the nonlinear dynamic analysis. This paper studies the dynamic behavior of stiffened FGM cylindrical circular shells under mechanic loads. The nonlinear dynamic equations are derived by using the classical shell theory with the nonlinear strain-displacement relation of large deflection, the smeared stiffeners technique and Galerkin method. The three-term solution of deflection is used and the frequency-amplitude relation of nonlinear vibration is obtained in explicit form. 2. ECCENTRICALLY STIFFENED FGM CYLINDRICAL SHELLS (ES-FGM CYLINDRICAL SHELLS) An ES-FGM cylindrical shell as shown in Fig. 1 is assumed to be thin with length L, mean radius R, reinforced by homogeneous ring and stringer stiffener systems. Stiffener material is similarly designed with Refs. [33–36] is full ceramic if it is located at ceramic-rich surface and is pure-metal if is located at metal-rich surface. The origin of the coordinate 0 locates on the middle plane of the shell, x, y = Rθ, z axes are in the axial, circumferential, and inward radial directions, respectively. Fig. 1. Geometric and the coordinate system of an eccentrically stiffened FGM cylindrical shell Functionally graded material is assumed to be made from a mixture of ceramic and metal with the simple power law exponent of volume fraction distribution Vc = Vc(z) = ( 2z + h 2h )k , Vm = Vm(z) = 1− Vc(z), where h is the thickness of shell; k ≥ 0 is the volume fraction index; z is the thickness coordinate and varies from −h/2 to h/2; the subscripts m and c refer to the metal and ceramic constituents, respectively. 204 Dao Van Dung, Vu Hoai Nam Effective properties Preff of functionally graded material are determined by linear rule of mixture Preff = Prm(z)Vm(z) + Prc(z)Vc(z). The Young’s modulus and mass density can be written by according to the men- tioned law E(z) = EmVm + EcVc = Em + (Ec − Em) ( 2z + h 2h )k , ρ(z) = ρmVm + ρcVc = ρm + (ρc − ρm) ( 2z + h 2h )k , (1) and the Poisson’s ratio ν is assumed to be constant for simplicity. According to the von Karman nonlinear strain-displacement relations of cylindrical shell [38], the mid-surface strain components are ε0x = ∂u ∂x + 1 2 ( ∂w ∂x )2 , ε0y = ∂v ∂y − w R + 1 2 ( ∂w ∂y )2 , γ0xy = ∂u ∂y + ∂v ∂x + ∂w ∂x ∂w ∂y , χx = ∂2w ∂x2 , χy = ∂2w ∂y2 , χxy = ∂2w ∂x∂y , (2) where ε0x and ε 0 y are normal strains, γ 0 xy is the shear strain at the middle surface of shell, χx, χy, χxy are the change of curvatures and twist of shell, and u = u (x, y), v = v (x, y), w = w (x, y) are displacements along x, y and z axes, respectively. The strains components can be written as the form εx = ε0x − zχx, εy = ε0y − zχy, γxy = γ0xy − 2zχxy. (3) The deformation compatibility equation is deduced from Eq. (2) ∂2ε0x ∂y2 + ∂2ε0y ∂x2 − ∂ 2γ0xy ∂x∂y = − 1 R ∂2w ∂x2 + ( ∂2w ∂x∂y )2 − ∂ 2w ∂x2 ∂2w ∂y2 . (4) Hook’s stress-strain relation for the shell is presented σshx = E(z) 1− ν2 (εx + νεy) , σ sh y = E(z) 1− ν2 (εy + νεx) , τ sh xy = E(z) 2 (1 + ν) γxy, (5) and for stiffeners σsts = Esεx, σ st r = Erεy, (6) where Es, Er are Young’s modulus of stringer and ring stiffeners, respectively. Taking into account the contribution of stiffeners by the smeared stiffeners technique and omitting the twist of stiffeners [38] and integrating the stress-strain relations and their moments through the thickness of shell, the force and moment of an ES-FGM cylindrical shell are Nx = ( A11 + EsAs ss ) ε0x +A12ε 0 y − (B11 + Cs)χx −B12χy, Ny = A12ε0x + ( A22 + ErAr sr ) ε0y −B12χx − (B22 + Cr)χy, Nxy = A66γ0xy − 2B66χxy, (7) An analytical approach to analyze nonlinear dynamic response of eccentrically stiffened . . . 205 Mx = (B11 + Cs) ε0x +B12ε 0 y − ( D11 + EsIs ss ) χx −D12χy, My = B12ε0x + (B22 + Cr) ε 0 y −D12χx − ( D22 + ErIr sr ) χy, Mxy = B66γ0xy − 2D66χxy, (8) where Aij , Bij , Dij (i, j = 1, 2, 6) are extensional, coupling and bending stiffness of the un- stiffened FGM cylindrical shell. They are defined as A11 = A22 = E1 1− ν2 , A12 = E1ν 1− ν2 , A66 = E1 2 (1 + ν) , B11 = B22 = E2 1− ν2 , B12 = E2ν 1− ν2 , B66 = E2 2 (1 + ν) , D11 = D22 = E3 1− ν2 , D12 = E3ν 1− ν2 , D66 = E3 2 (1 + ν) , (9) in which E1 = ( Em + Ec − Em k + 1 ) h, E2 = (Ec − Em) kh2 2 (k + 1) (k + 2) , E3 = [ Em 12 + (Ec − Em) ( 1 k + 3 − 1 k + 2 + 1 4k + 4 )] h3, Is = dsh 3 s 12 +Asz2s , Ir = drh 3 r 12 +Arz2r , Cs = ±EsAszs ss , Cr = ±ErArzr sr , zs = hs + h 2 , zr = hr + h 2 , ss = 2piR ns , sr = L nr , (10) where Cs and Cr are negative for outside stiffeners and positive for inside ones. ss and sr are the spacing of the stringer and ring stiffeners, respectively. ds, hs and dr, hr are the width and thickness of the stringer and ring stiffeners, respectively. As, Ar are the cross-section areas of stiffeners. Is, Ir, zs, zr are the second moments of cross section areas and the eccentricities of stiffeners with respect to the middle surface of shell, respectively. Es, Er are Young’s modulus of stringer and ring stiffeners, respectively. From the Eq. (7), one can obtain inversely ε0x = A ∗ 22Nx −A∗12Ny +B∗11χx +B∗12χy, ε0y = A ∗ 11Ny −A∗12Nx +B∗21χx +B∗22χy, γ0xy = A ∗ 66Nxy + 2B ∗ 66χxy, (11) 206 Dao Van Dung, Vu Hoai Nam where A∗11 = 1 ∆ ( A11 + EsAs ss ) , A∗22 = 1 ∆ ( A22 + ErAr sr ) , A∗12 = A12 ∆ , A∗66 = 1 A66 , ∆ = ( A11 + EsAs ss )( A22 + ErAr sr ) −A212, B∗11 = A ∗ 22 (B11 + Cs)−A∗12B12, B∗22 = A ∗ 11 (B22 + Cr)−A∗12B12, B∗12 = A ∗ 22B12 −A∗12 (B22 + Cr) , B∗21 = A ∗ 11B12 −A∗12 (B11 + Cs) , B∗66 = B66 A66 . (12) Substituting Eq. (11) into Eq. (8) leads to Mx = B∗11Nx +B ∗ 21Ny −D∗11χx −D∗12χy, My = B∗12Nx +B ∗ 22Ny −D∗21χx −D∗22χy, Mxy = B∗66Nxy − 2D∗66χxy, (13) where D∗11 = D11 + EsIs ss − (B11 + Cs)B∗11 −B12B∗21, D∗22 = D22 + ErIr sr −B12B∗12 − (B22 + Cr)B∗22, D∗12 = D12 − (B11 + Cs)B∗12 −B12B∗22, D∗21 = D12 −B12B∗11 − (B22 + Cr)B∗21, D∗66 = D66 −B66B∗66. (14) The nonlinear equations of motion of a thin circular cylindrical shell based on the assumption u w and v  w, ρ1∂ 2u ∂t2 → 0, ρ1∂ 2v ∂t2 → 0 [17, 19,39] are given by ∂Nx ∂x + ∂Nxy ∂y = 0, ∂Nxy ∂x + ∂Ny ∂y = 0, ∂2Mx ∂x2 + 2 ∂2Mxy ∂x∂y + ∂2My ∂y2 +Nx ∂2w ∂x2 + 2Nxy ∂2w ∂x∂y +Ny ∂2w ∂y2 + + 1 R Ny + q0 = ρ1 ∂2w ∂t2 + 2ρ1µ ∂w ∂t , (15) An analytical approach to analyze nonlinear dynamic response of eccentrically stiffened . . . 207 where µ is the linear damping coefficient, and ρ1 = h 2∫ −h 2 ρ(z)dz + ρs As ss + ρr Ar sr = ( ρm + ρc − ρm k + 1 ) h+ ρs As ss + ρr Ar sr , (16) with ρr = ρm, ρs = ρm if stiffeners are full metal and ρr = ρc, ρs = ρc if stiffeners are full ceramic. The first two of Eq. (15) are identically satisfied by introducing a stress function ϕ Nx = ∂2ϕ ∂y2 , Ny = ∂2ϕ ∂x2 , Nxy = − ∂ 2ϕ ∂x∂y . (17) Substituting Eq. (11) into Eqs. (4) and (13) and the third of Eq. (15), taking into account Eqs. (2) and (17), we have A∗11 ∂4ϕ ∂x4 + (A∗66 − 2A∗12) ∂4ϕ ∂x2∂y2 +A∗22 ∂4ϕ ∂y4 +B∗21 ∂4w ∂x4 + + (B∗11 +B ∗ 22 − 2B∗66) ∂4w ∂x2∂y2 +B∗12 ∂4w ∂y4 + 1 R ∂2w ∂x2 − [( ∂2w ∂x∂y )2 − ∂ 2w ∂x2 ∂2w ∂y2 ] = 0, (18) ρ1 ∂2w ∂t2 + 2ρ1µ ∂w ∂t +D∗11 ∂4w ∂x4 + (D∗12 +D ∗ 21 + 4D ∗ 66) ∂4w ∂x2∂y2 + +D∗22 ∂4w ∂y4 −B∗21 ∂4ϕ ∂x4 − (B∗11 +B∗22 − 2B∗66) ∂4ϕ ∂x2∂y2 −B∗12 ∂4ϕ ∂y4 − − 1 R ∂2ϕ ∂x2 − ∂ 2ϕ ∂y2 ∂2w ∂x2 + 2 ∂2ϕ ∂x∂y ∂2w ∂x∂y − ∂ 2ϕ ∂x2 ∂2w ∂y2 − q0 = 0. (19) Eqs. (18) and (19) are a nonlinear governing equation system. They are used to investigate the dynamic characteristics of ES-FGM circular cylindrical shells. 3. NONLINEAR DYNAMIC BUCKLING ANALYSIS Suppose that an ES-FGM cylindrical shell is simply supported and subjected to uniformly distributed pressure of intensity q0 and axial compression of intensity r0 respec- tively at its cross-section (in N/m2). The boundary conditions of this study are w = 0, Mx = 0, Nx = −r0h, Nxy = 0, at x = 0, L. (20) Assume the buckling mode shape is represented by the popular form [5–8, 39], the simply supported boundary condition Eq. (20) is fulfilled on the average sense w = f0 + f1 sin mpix L sin ny R + f2 sin2 mpix L , (21) where f0 = f0(t) is time dependent unknown uniform deflection of pre-buckling state, f1 = f1(t) is time dependent unknown linear buckling deflection, f2 = f2(t) is time dependent unknown nonlinear deflection, m is numbers of half waves and n is numbers of full wave in axial and circumferential directions, respectively. 208 Dao Van Dung, Vu Hoai Nam Substituting Eq. (21) into Eq. (18) and solving obtained equation, leads to ϕ = ϕ1 cos 2mpix L + ϕ2 cos 2ny R − ϕ3 sin mpix L sin ny R + + ϕ4 sin 3mpix L sin ny R − σ0yhx 2 2 − r0hy 2 2 , (22) where denote ϕ1 = n2λ2 32A∗11m2pi2 f21 − ( 4λL− 16B∗21m2pi2 ) 32A∗11m2pi2 f2, ϕ2 = m2pi2 32A∗22n2λ2 f21 , ϕ3 = B A f1 + m2n2pi2λ2 A f1f2, ϕ4 = m2n2pi2λ2 G f1f2, (23) A = A∗11m 4pi4 + (A∗66 − 2A∗12)m2n2pi2λ2 +A∗22n4λ4, B = B∗21m 4pi4 + (B∗11 +B ∗ 22 − 2B∗66)m2n2pi2λ2 +B∗12n4λ4 − L2 R m2pi2, D = D∗11m 4pi4 + (D∗12 +D ∗ 21 + 4D ∗ 66)m 2n2pi2λ2 +D∗22n 4λ4, G = 81A∗11m 4pi4 + 9 (A∗66 − 2A∗12)m2n2pi2λ2 +A∗22n4λ4, λ = L R . (24) Substituting the expressions (21) and (22) into Eq. (19) and then applying Galerkin method lead to σ0yh = Rq0 −Rρ1d 2f0 dt2 −Rρ1 2 d2f2 dt2 − 2Rρ1µdf0 dt −Rρ1µdf2 dt , (25) L4ρ1 d2f1 dt2 + 2L4ρ1µ df1 dt + ( D + B2 A ) f1 + ( m4pi4 16A∗22 + n4λ4 16A∗11 ) f31+ + [ 2Bm2n2pi2λ2 A − n 2λ2 ( λL− 4B∗21m2pi2 ) 4A∗11 ] f1f2+ + ( m4n4pi4λ4 A + m4n4pi4λ4 G ) f1f 2 2 −m2pi2L2hr0f1 − σ0yhn2L2λ2f1 = 0, (26) ρ1 d2f0 dt2 + 2ρ1µ df0 dt + ρ1 3 4 d2f2 dt2 + 3 2 ρ1µ df2 dt + + {[ 4B∗21 (mpi L )4 − 1 R (mpi L )2] n2λ2 16A∗11m2pi2 + 1 2 B A (mpi L )2 ( n R )2} f21+ + 1 2 m2n2pi2λ2 (mpi L )2 ( n R )2( 1 A − 1 G ) f21 f2+ + { 4D∗11 (mpi L )4 − [4B∗21 (mpiL )4 − 1R (mpiL )2 ] λL− 4B∗21m2pi2 4A∗11m2pi2 } f2− − r0h (mpi L )2 f2 + σ0yh R = q0. (27) An analytical approach to analyze nonlinear dynamic response of eccentrically stiffened . . . 209 For circular cylindrical shell, the circumferential closed condition must be considered as [29,39] 2piR∫ 0 L∫ 0 ∂v ∂y dxdy = 2piR∫ 0 L∫ 0 [ ε0y + w R − 1 2 ( ∂w ∂y )2] dxdy = 0. (28) From Eqs. (11), (17), (21) and (22), this integral becomes −2A∗11σ0yh+ 2A∗12r0h+ 1 R (f2 + 2f0)− 14 ( n R )2 f21 = 0. (29) Eliminating σ0y from Eqs. (25)-(27) and the condition of closed form (29), lead to( d2f0 dt2 + 2µ df0 dt ) + 1 2 ( d2f2 dt2 + 2µ df2 dt ) + α11 (f2 + 2f0)− α12f21 − α13q0 + α14r0 = 0, (30) α21f1 ( d2f0 dt2 + 2µ df0 dt ) + ( d2f1 dt2 + 2µ df1 dt ) + α21 2 f1 ( d2f2 dt2 + 2µ df2 dt ) + + α22f1 + α23f1f2 + α24f31 + α25f1f 2 2 − α26q0f1 − α27r0f1 = 0, (31) ( d2f2 dt2 + 2µ df2 dt ) + α31f21 + α32f 2 1 f2 + α33f2 − α34r0f2 = 0, (32) in which α11 = 1 2A∗11R2ρ1 , α12 = n2 8A∗11R3ρ1 , α13 = 1 ρ1 , α14 = A∗12h A∗11Rρ1 , (33) α21 = Rn2λ2 L2 , α22 = 1 L4ρ1 ( D + B2 A ) , α23 = 1 L4ρ1 [ 2Bm2n2pi2λ2 A − n 2λ2 ( λL− 4B∗21m2pi2 ) 4A∗11 ] , α24 = 1 L4ρ1 ( m4pi4 16A∗22 + n4λ4 16A∗11 ) , α25 = 1 L4ρ1 ( m4n4pi4λ4 A + m4n4pi4λ4 G ) , α26 = Rn2λ2 L2ρ1 , α27 = m2pi2h L2ρ1 , (34) α31 = 1 ρ1 {[ 4B∗21 (mpi L )4 − 1 R (mpi L )2] n2λ2 4A∗11m2pi2 + 2 B A (mpi L )2 ( n R )2} , α32 = 2 ρ1 m2n2pi2λ2 (mpi L )2 ( n R )2( 1 A − 1 G ) , α33 = 1 ρ1 { 16D∗11 (mpi L )4 − [4B∗21 (mpiL )4 − 1R (mpiL )2 ] λL− 4B∗21m2pi2 A∗11m2pi2 } , α34 = 4 ρ1 h (mpi L )2 . (35) 210 Dao Van Dung, Vu Hoai Nam Simplifying Eqs. (30)-(32), we have( d2f0 dt2 + 2µ df0 dt ) + β11f0 − β12f21 − β13f21 f2 + β14f2 + β15r0f2 − α13q0 + α14r0 = 0, (36)( d2f1 dt2 + 2µ df1 dt ) + α22f1 + β21f1f0 + β22f1f2 + α25f1f22 + β23f 3 1 − β24f1r0 = 0, (37)( d2f2 dt2 + 2µ df2 dt ) + α31f21 + α32f 2 1 f2 + α33f2 − α34r0f2 = 0, (38) where β11 = 2α11, β12 = 1 2 α31 + α12, β13 = 1 2 α32, β14 = α11 − 12α33, β15 = 1 2 α34, (39) β21 = −β11α21, β22 = −β14α21 − α33α212 + α23, β23 = β12α21 − α31α212 + α24, β24 = α27 + α14α21. (40) Denoting f = wmax, from Eq. (21), the maximal deflection of the shells f = f0 + f1 + f2, (41) locates at x = iL 2m , y = jpiR 2n where i, j are odd integer numbers. Note that f0 = f0(t), f1 = f1(t), f2 = f2(t) and f = f(t) in Eq. (41). From Eqs. (36)-(38) and (41), the effects of input parameters on the dynamic re- sponse of shells are investigated. 3.1. Nonlinear vibration analysis This section considers an ES-FGM cylindrical thin shell under uniformly lateral pressure q0 = Q sinΩt and r0 = 0, Eqs. (36)-(38) become( d2f0 dt2 + 2µ df0 dt ) + β11f0 − β12f21 − β13f21 f2 + β14f2 − α13Q sinΩt = 0, (42)( d2f1 dt2 + 2µ df1 dt ) + α22f1 + β21f1f0 + β22f1f2 + α25f1f22 + β23f 3 1 = 0, (43)( d2f2 dt2 + 2µ df2 dt ) + α31f21 + α32f 2 1 f2 + α33f2 = 0, (44) where Q is excitation force and Ω is excitation frequency. From these equations, nonlinear response of ES-FGM shell is investigated by using the fourth order Runge-Kutta iteration method. It is difficult to determine the fundamental frequencies of natural vibration, frequency- amplitude relation of nonlinear vibration of shell. In this paper, ones can be investigated by ignoring the uniform buckling shape and nonlinear buckling shape, Eq. (31) becomes( d2f1 dt2 + 2µ df1 dt ) + α22f1 + α24f31 − α26f1Q sinΩt = 0. (45) An analytical approach to analyze nonlinear dynamic response of eccentrically stiffened . . . 211 For the free and linear vibration without damping, the Eq. (45) becomes d2f1 dt2 + α22f1 = 0. (46) The fundamental frequency of natural vibration can be determined by ωmn = √ α22. (47) where ωmn is fundamental frequency of natural vibration of shell. Using the solution f1(t) = η sin (Ωt) and applying the procedure like Galerkin method to Eq. (45), the frequency-amplitude relation of nonlinear vibration is obtained Ω2 − 4 pi µΩ = α22 + 3 4 α24η 2 − 8 3pi α26Q. (48) By introducing the non-dimension frequency parameter ξ = Ω ωmn , Eq. (48) becomes ξ2 − 4µ piωmn ξ = 1 + 3 4 α24 ω2mn η2 − 8 3pi α26 ω2mn Q. (49) The frequency-amplitude relation of nonlinear free vibration Q = 0 is determined by ξ2 − 4µ piωmn ξ = 1 + 3 4 α24 ω2mn η2. (50) 3.2. Buckling analysis 3.2.1. Linear static buckling analysis of ES-FGM cylindrical shells Omitting the linear damping, uniform buckling shape, nonlinear buckling shape and putting f˙1 = 0, f¨1 = 0, and taking f1 6= 0 Eq. (31) becomes α22 + α24f21 − α26q0 − α27r0 = 0. (51) By ignoring the nonlinear term of f1 and r0 = 0 in Eq. (51), the linear upper static buckling load of ES-FGM cylindrical shells under only external pressure can be determined by qsbu = α22 α26 . (52) Similarly, the linear upper static buckling load of ES-FGM cylindrical shells under only axial compression (q0 = 0) leads to rsbu = α22 α27 . (53) From Eqs. (52)-(53), the linear static critical buckling loads of shells are determined by rscr = min rsbu ∀ (m,n) and qscr = min qsbu ∀ (m,n). 212 Dao Van Dung, Vu Hoai Nam 3.2.2. Dynamic buckling analysis of ES-FGM cylindrical shells The nonlinear dynamic critical buckling analysis of ES-FGM circular cylindrical shells based on Eqs. (36)-(38), is investigated for two load types as follows. Firstly, ES-FGM cylindrical shell is subjected to only lateral pressure varying as linear function of time q0 = cqt in which cq (N/m2s) is the loading speed of external pressure. Secondly, ES-FGM cylindrical shell is subjected to only axial compression varying as linear function of time r0 = crt where cr (N/m2s) is the loading speed of axial compression. Eqs. (36)-(38) are the nonlinear second-order differential three equations system. This equation system may be numerically solved. 4. CONCLUSIONS A formulation of governing equations of eccentrically stiffened functionally graded circular cylindrical thin shells subjected to time dependent axial compression and external pressure based upon the classical shell theory and the smeared stiffeners technique with von Karman-Donnell nonlinear terms is proposed in this paper. An approximate three- term solution of deflection taking into account the nonlinear buckling shape is used. 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Dao Van Dung, Vu Hoai Nam, An analytical approach to analyze nonlin- ear dynamic response of eccentrically stiffened functionally graded circular cylindrical shells subjected to time dependent axial compression and external pressure. Part 1: Governing equations establishment. 201 5. Manh Duong Phung, Thuan Hoang Tran, Quang Vinh Tran, Stable control of networked robot subject to communication delay, packet loss, and out-of- order delivery. 215 6. Phan Anh Tuan, Vu Duy Quang, Estimation of car air resistance by CFD method. 235