DSpace at VNU: Nonlinear static and dynamic buckling analysis of imperfect eccentrically stiffened functionally graded circular cylindrical thin shells under axial compression

International Journal of Mechanical Sciences 74 (2013) 190–200 Contents lists available at SciVerse ScienceDirect International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci Nonlinear static and dynamic buckling analysis of imperfect eccentrically stiffened functionally graded circular cylindrical thin shells under axial compression Dao Huy Bich a, Dao Van Dung a, Vu Hoai Nam b,n, Nguyen Thi Phuong b a b Vietnam National University, Ha Noi, Vietnam University of Transport Technology, Ha Noi, Vietnam art ic l e i nf o a b s t r a c t Article history: Received 24 November 2012 Received in revised form 20 May 2013 Accepted June 2013 Available online 12 June 2013 An analytical approach is presented to investigate the nonlinear static and dynamic buckling of imperfect eccentrically stiffened functionally graded thin circular cylindrical shells subjected to axial compression Based on the classical thin shell theory with the geometrical nonlinearity in von Karman–Donnell sense, initial geometrical imperfection and the smeared stiffeners technique, the governing equations of motion of eccentrically stiffened functionally graded circular cylindrical shells are derived The functionally graded cylindrical shells with simply supported edges are reinforced by ring and stringer stiffeners system on internal and (or) external surface The resulting equations are solved by the Galerkin procedure to obtain the explicit expression of static critical buckling load, post-buckling load–deflection curve and nonlinear dynamic motion equation The nonlinear dynamic responses are found by using fourth-order Runge–Kutta method The dynamic critical buckling loads of shells under step loading of infinite duration are found corresponding to the load value of sudden jump in the average deflection and those of shells under linear-time compression are investigated according to Budiansky–Roth criterion The obtained results show the effects of stiffeners and input factors on the static and dynamic buckling behavior of these structures & 2013 Elsevier Ltd All rights reserved Keywords: Static and dynamic buckling analysis Stiffener Functionally graded material Stiffened circular cylindrical shell Critical buckling load Introduction Functionally graded (FGM) plate and shell structures have became popular in engineering designs of coating of nuclear reactors and space shuttle The static and dynamic behavior of FGM cylindrical shell attracts special attention of a lot of scientists in the world In static analysis of FGM cylindrical shells, many studies have been focused on the buckling and post-buckling of shells under mechanic and thermal loading Shen [1] presented the nonlinear post-buckling of perfect and imperfect FGM cylindrical thin shells in thermal environments under lateral pressure by using the classical shell theory with the geometrical nonlinearity in von Karman–Donnell sense By using higher order shear deformation theory; this author [2] continued to investigate the post-buckling of FGM hybrid cylindrical shells in thermal environments under axial loading Bahtui and Eslami [3] investigated the coupled thermo-elasticity of FGM cylindrical shells Huang and Han [4–7] studied the buckling and post-buckling of unstiffened FGM cylindrical shells under axial compression, radial pressure and combined axial compression and radial pressure based on the Donnell shell theory and the nonlinear strain–displacement n Corresponding author Tel.: +84 98 3843 387 E-mail address: hoainam.vu@utt.edu.vn (V.H Nam) 0020-7403/$ - see front matter & 2013 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.ijmecsci.2013.06.002 relations of large deformation The post-buckling of shear deformable FGM cylindrical shells surrounded by an elastic medium was studied by Shen [8] Sofiyev [9] analyzed the buckling of FGM circular shells under combined loads and resting on the Pasternak type elastic foundation Zozulya and Zhang [10] studied the behavior of functionally graded axisymmetric cylindrical shells based on the high order theory For dynamic analysis of FGM cylindrical shells, Ng et al [11] and Darabi et al [12] presented respectively linear and nonlinear parametric resonance analyses for un-stiffened FGM cylindrical shells Three-dimensional vibration analysis of fluid-filled orthotropic FGM cylindrical shells was investigated by Chen et al [13] Sofiyev and Schnack [14] and Sofiyev [15] obtained critical parameters for unstiffened cylindrical thin shells under linearly increasing dynamic torsional loading and under a periodic axial impulsive loading by using the Galerkin technique together with Ritz type variation method Shariyat [16,17] investigated the nonlinear dynamic buckling problems of axially and laterally preloaded FGM cylindrical shells under transient thermal shocks and dynamic buckling analysis for unstiffened FGM cylindrical shells under complex combinations of thermo–electro-mechanical loads Geometrical imperfection effects were also included in his research Li et al [18] studied the free vibration of three-layer circular cylindrical shells with functionally graded middle layer Huang and Han [19] presented the nonlinear D Huy Bich et al / International Journal of Mechanical Sciences 74 (2013) 190–200 Nomenclature h thickness of the shell m number of half waves in axial direction n number of wave in circumferential direction k volume-fraction index z coordinate in thickness direction EðzÞ; Em ; Ec Young's modulus of shell, metal, ceramic, respectively ρðzÞ; ρm ; ρc mass density of shell, metal, ceramic, respectively L length of the shell R radius of the shell Es ; Er Young's modulus of stringer and ring stiffeners, respectively Aij ; Bij ; Dij extensional, coupling and bending stiffness of the unstiffened shell, respectively Cs; Cr coupling parameters ss ; sr spacing of the stringer and ring stiffeners, respectively dynamic buckling problems of un-stiffened functionally graded cylindrical shells subjected to time-dependent axial load by using the Budiansky–Roth dynamic buckling criterion [20] Various effects of the inhomogeneous parameter, loading speed, dimension parameters; environmental temperature rise and initial geometrical imperfection on nonlinear dynamic buckling were discussed Shariyat [21] analyzed the nonlinear transient stress and wave propagation analyses of the FGM thick cylinders, employing a unified generalized thermoelasticity theory Recently, idea of eccentrically stiffened FGM structures has been proposed by Najafizadeh et al [22] and Bich et al [23,24] Najafizadeh et al [22] have studied linear static buckling of FGM axially loaded cylindrical shell reinforced by ring and stringer FGM stiffeners In order to provide material continuity and easily to manufacture, the FGM shells are reinforced by an eccentrically homogeneous stiffener system; Bich et al [23] have investigated the nonlinear static postbuckling of functionally graded plates and shallow shells and nonlinear dynamic buckling of functionally graded cylindrical panels [24] Literature on the nonlinear static and dynamic analysis of imperfect FGM stiffened circular cylindrical shells is still very limited In this paper, the mentioned just problem is investigated by analytical approach The nonlinear dynamic equations of eccentrically stiffened FGM circular cylindrical shells are derived based on the classical shell theory with the nonlinear strain– displacement relation of large deflection and the smeared stiffeners technique By using the Galerkin method, the closed-form expression to determine the static critical buckling load and load– deflection curves are obtained The nonlinear dynamic responses are found by using fourth-order Runge–Kutta method The dynamic buckling loads of shells under step loading of infinite duration are found corresponding to the load value of sudden jump in the average deflection and those of shells under lineartime compression are investigated according to Budiansky–Roth criterion The results show that the stiffener, volume-fractions index, initial imperfection and geometrical parameters influence strongly to the static and dynamic buckling of shells Eccentrically stiffened FGM (ES-FGM) circular cylindrical shells 2.1 Functionally graded material In this paper, functionally graded material is assumed to be made from a mixture of ceramic and metal with the volume- 191 As ; Ar Is ; Ir cross-section areas of stiffeners moments of inertia of stiffener cross sections relative to the shell middle surface zs ; zr eccentricities of stiffeners with respect to the middle surface of shell ds ; dr width of the stringer and ring stiffeners, respectively hs ; hr height of the stringer and ring stiffeners, respectively f ¼ f ðtÞ time dependent total amplitude f0 known imperfect amplitude r0 compressive load per unit length r ¼ r =h compressive stress r sbu static buckling stress r scr static critical buckling stress r scr static critical buckling loads per unit length t; t cr time and dynamic critical time c loading speed r dcr dynamic critical buckling stress τcr dynamic coefficient fractions given by a power law V m ỵ V c ẳ 1; V c ẳ V c zị ẳ  k 2z ỵ h ; 2h 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 According to the mentioned law, the Young modulus and the mass density can be expressed in the form  k 2z ỵ h Ezị ẳ Em V m ỵ Ec V c ẳ Em ỵ Ec Em ị ; 2h  k 2z ỵ h ; 1ị zị ẳ m V m ỵ c V c ẳ m ỵ c m ị 2h Poisson’s ratio ν is assumed to be constant 2.2 Constitutive relations and governing equations Consider a functionally graded thin circular cylindrical shell with length L, mean radius R This shell is assumed to be reinforced by closely spaced [22,25,29] homogeneous ring and stringer stiffener systems (see Fig 1) Stiffener is pure-ceramic if it is located at ceramic-rich side and is pure-metal if is located at metal-rich side, such FGM stiffened circular cylindrical shells provide continuity within shell and stiffeners and can be easier manufactured The origin of the coordinate locates on the middle plane and at the left end of the shell, x; y (y ¼ Rθ) and z axes are in the axial, circumferential, and inward radial directions, respectively According to the von Karman nonlinear strain–displacement relations [25], the strain components at the middle plane of imperfect circular cylindrical shells are of the form   ∂u w w w0 0x ẳ ỵ ; ỵ ∂x ∂x ∂x ∂x  2 ∂v w w w w0 ỵ 0y ẳ ; ỵ y R ∂y ∂y ∂y ∂u ∂v ∂w ∂w ∂w w0 w w0 ỵ ỵ ỵ ỵ ; 0xy ¼ ∂y ∂x ∂x ∂y ∂y ∂x ∂x ∂y χx ¼ ∂2 w ; ∂x2 χy ¼ ∂2 w ; y2 xy ẳ w ; xy 2ị 192 D Huy Bich et al / International Journal of Mechanical Sciences 74 (2013) 190–200 Fig Configuration of an eccentrically stiffened cylindrical shell where u ẳ ux; yị, v ẳ vx; yị and w ẳ wx; yị are displacements along x; y and z axes respectively and w0 ¼ w0 ðx; yÞ denotes initial imperfection of shell, which is very small compared with the shell dimensions, but may be compared with the shell wall thickness The strains across the shell thickness at a distance z from the mid-surface are given by   Er I r M y ¼ B12 0x ỵ B22 ỵ C r ị0y D12 x D22 ỵ y; sr M xy ẳ B66 0xy −2D66 χ xy ; where Aij ; Bij ; Dij i; j ẳ 1; 2; 6ị are extensional, coupling and bending stiffness of the un-stiffened FGM cylindrical shell E1 E1 ν E1 ; ; A12 ¼ ; A66 ¼ 21 ỵ ị 12 12 E2 E2 E2 B11 ¼ B22 ¼ ; ; B12 ¼ ; B66 ¼ 21 ỵ ị 12 12 E3 E3 E3 ; D11 ¼ D22 ¼ ; D12 ¼ ; D66 ¼ 21 ỵ ị 12 12 A11 ẳ A22 ẳ x ¼ ε0x −zχ x ; εy ¼ ε0y −zχ y ; xy ẳ 0xy 2z xy : 3ị From Eq (2) the strains must be relative in the deformation compatibility equation  2 2 ∂2 ε0x ∂ ε0y ∂ γ 0xy ∂2 w ∂ w w0 ỵ ẳ ỵ ỵ R x2 ∂x∂y ∂x∂y ∂y2 ∂x2 ∂x∂y    2 ∂ w ∂ w0 ∂ w ∂ w0 ỵ ỵ : x2 y2 x2 y2 4ị Ezị x ỵ y ị; 12 ssh y ẳ Ezị y ỵ x ị; 12 ssh xy ẳ Ezị ; 21 ỵ ị xy   Ec Em Ec Em ịkh h; E2 ẳ ; E1 ẳ E m þ kþ1 2ðk þ 1Þðk þ 2Þ    Em 1 ỵ h ; E3 ẳ ỵ Ec Em ị k ỵ k ỵ 4k ỵ 12 sry ẳ Er y ; ð5Þ ð6Þ where Es ; Er is Young's modulus of stringer and ring stiffeners, respectively The force and moment of an un-stiffened FGM circular cylindrical shell can be determine by Z h=2 fðN x ; N y ; N xy Þ; ðM x ; M y ; M xy Þg ¼ fsx ; sy ; sxy gð1; zÞ dz: ð7Þ −h=2 Taking into account the contribution of stiffeners by the smeared stiffeners technique and omitting the twist of stiffeners and integrating the stress–strain equations and their moments through the thickness of shell, we obtain the expressions for force and moment resultants of an ES-FGM circular cylindrical shell   Es As N x ẳ A11 ỵ x ỵ A12 0y B11 ỵ C s ị x B12 y ; ss   Er Ar εy −B12 χ x B22 ỵ C r ị y ; Ny ẳ A12 0x ỵ A22 ỵ sr Nxy ẳ A66 0xy −2B66 χ xy ;   Es I s χ x −D12 χ y ; M x ¼ ðB11 þ C s Þε0x þ B12 ε0y − D11 þ ss ds hs dr h r ỵ As z2s ; I r ẳ ỵ Ar z2r ; 12 12 Es As zs E r Ar z r Cs ¼ ; Cr ẳ ; ss sr hs ỵ h hr ỵ h ; zr ẳ ; zs ẳ 2 Is ¼ and for stiffeners ssx ¼ Es εx ; ð10Þ with Hooke's stress–strain relation is applied for the shell ssh x ẳ 9ị 8ị 11ị where the coupling parameters C s and C r are negative for outside stiffeners and positive for inside ones The spacing of the stringer and ring stiffeners is denoted by ss and sr respectively The quantities As , Ar are the cross-section areas of stiffeners and I s , I r , zs , zr are the second moments of cross section areas and the eccentricities of stiffeners with respect to the middle surface of shell, respectively The width and thickness of the stringer and ring stiffeners are denoted by ds ; hs and dr ; hr respectively The Young modulus of stiffeners Es , Er take the value Em if the full metal stiffeners are put at the metal-rich side of the shell and conversely, Ec if the full ceramic ones are put at the ceramic-rich side From the constitutive relations (8), one can write inversely ε0x ¼ An22 N x An12 N y ỵ Bn11 x ỵ Bn12 χ y ; ε0y ¼ An11 N y −An12 N x ỵ Bn21 x ỵ Bn22 y ; 0xy ẳ An66 ỵ 2Bn66 xy ; in which     Es As A12 Er Ar A11 ỵ A22 ỵ An11 ẳ ; An22 ¼ ; An12 ¼ ; Δ Δ ss Δ sr    Es As Er Ar A22 ỵ A212 : ; ẳ A11 ỵ An66 ẳ A66 ss sr ð12Þ D Huy Bich et al / International Journal of Mechanical Sciences 74 (2013) 190–200 Bn11 ¼ An22 B11 ỵ C s ịAn12 B12 ; n n n B12 ẳ A22 B12 A12 B22 ỵ C r ị; B66 Bn66 ẳ ; A66 Bn22 ẳ An11 B22 ỵ C r ịAn12 B12 ; n n ỵ2 n B21 ẳ A11 B12 A12 B11 ỵ C s ị; 193     ∂2 φ ∂2 w w0 w w0 ỵ ỵ ẳ 0: 2 xy xy xy x ∂y ∂y ð20Þ ð13Þ Eqs (19) and (20) are a nonlinear equation system in terms of two dependent unknowns w and φ They are used to investigate the dynamic characteristics of imperfect ES-FGM circular cylindrical shells ð14Þ Nonlinear static and dynamic buckling analysis Substituting Eq (12) into Eq (9) leads to M x ẳ Bn11 Nx ỵ Bn21 N y −Dn11 χ x −Dn12 χ y ; M y ẳ Bn12 Nx ỵ Bn22 N y Dn21 x −Dn22 χ y ; M xy ¼ Bn66 Nxy −2Dn66 χ xy ; in which Suppose that an imperfect ES-FGM cylindrical shell is simply supported and subjected to axial compressive load r ¼ r h where r is the average axial stress on the shell's end sections, positive when the shells subjected to axial compression (in N/m2) Thus, the boundary conditions considered in the current study are Es I s B11 ỵ C s ịBn11 B12 Bn21 ; ss Er I r B12 Bn12 B22 ỵ C r ịBn22 ; Dn22 ẳ D22 ỵ sr Dn12 ẳ D12 B11 ỵ C s ịBn12 B12 Bn22 ; Dn11 ẳ D11 ỵ w ẳ 0; Dn21 ẳ D12 B12 Bn11 B22 ỵ C r ịBn21 ; n n D66 ¼ D66 −B66 B66 : ð15Þ The nonlinear equations of motion of a cylindrical thin shell based on the classical shell theory and the assumption [12,14,26] u o o w and v o o w, ρ1 ð∂2 u=∂t Þ-0, ρ1 ð∂2 v=∂t Þ-0 are given by [4,14] ∂Nxy N y N x N xy ỵ ẳ 0; ỵ ¼ 0; ∂x ∂y ∂x ∂y   2 ∂ M xy ∂ M y ∂ Mx w w0 ỵ ỵ2 ỵ Nx ỵ 2 2 ∂x∂y ∂x ∂x ∂y ∂x     ∂ w ∂2 w0 ∂ w w0 w ỵ ỵ Ny ỵ þ N y ¼ ρ1 ; þ2Nxy 2 xy xy R y y t h=2 ỵ r Ar ; sr w ẳ f tị sin 17ị s ẳ ρm ; ρr ¼ ρm for metal stiffeners; ρs ¼ ρc ; ρr ¼ ρc for ceramic stiffeners: Considering the first two of Eq (16), a stress function φ may be defined as ∂ φ ; ∂y2 Ny ¼ ∂ φ ; ∂x2 N xy ¼ − ∂ φ : ∂x∂y ð18Þ Substituting Eq (11) into the compatibility Eqs (4) and (14) into the third of Eq (16), taking into account Eqs (2) and (18) neglecting small terms of higher second order with respect to w0 , yields ∂4 w ỵ An66 2An12 ị 2 ỵ An22 ỵ Bn21 4 ∂x ∂x ∂y ∂y ∂x ∂4 w ∂2 w n n n n w ỵB11 ỵ B22 2B66 ị 2 ỵ B12 ỵ R x2 ∂x ∂y ∂y " # 2 2 2 ∂ w ∂ w∂ w ∂ w ∂ w0 − − −2 ∂x∂y ∂x∂y ∂x∂y ∂x ∂y2 ρ1 w w0 w w0 ỵ ¼ 0; ∂x2 ∂y2 ∂y ∂x2 ð22Þ mπx ny sin ; L R ð23Þ where f is the known imperfect amplitude Parameters Present R/h ¼500, L/R ¼2, c ¼ 100 MPa/s k ¼0.2 194.94(2,11) k ¼1.0 169.94(2,11) k ¼5.0 149.98(2,11) R/h ¼500, L/R ¼2, k ¼ 0.5 c¼ 100 MPa/s 181.68(2,11) c¼ 50 MPa/s 179.38(2,11) c¼ 10 MPa/s 177.02(2,11) L/R ¼2, k¼ 0.2, c ¼ 100 MPa/s R/h ¼ 800 124.67(2,12) R/h ¼ 600 162.18(3,14) R/h ¼ 400 239.56(5,15) Huang and Han [19] r dcr m; nị cr ẳ 1.030 1.034 1.041 194.94(2,11) 169.94(2,11) 150.25(2,11) 1.030 1.034 1.040 1.032 1.019 1.006 181.67(2,11) 179.37(2,11) 177.97(1,8) 1.032 1.019 1.009 1.049 1.026 1.013 124.91(2,12) 162.25(3,14) 239.18(5,15) 1.051 1.027 1.011 τcr ¼ r dcr r scr r dcr r scr Table An11 þ ð21Þ L: mπx ny sin ; L R r dcr m; nị at x ẳ 0; where f ðtÞ is the time dependent total amplitude, m is the number of half waves in axial direction and n is the number of wave in circumferential direction The initial-imperfection w0 is assumed to be the same form of the deflection as with Nxy ¼ 0; Table Comparisons of dynamic critical buckling stress r dcr (MPa) and dynamic coefficient τcr ¼ r dcr =r scr of perfect un-stiffened FGM cylindrical shells under linear-time compression   As Ar ρ As zịdz ỵ s ỵ r ẳ m ỵ c m h ỵ s ss sr kỵ1 ss h=2 Nx ¼ N x ¼ −r h; The deflection of shell is satisfying the mentioned condition (21) is represented by w0 ẳ f sin 16ị where Z ¼ M x ¼ 0; Comparisons of static critical buckling load per unit length r scr ¼ r scr h (Â106 N/m) of perfect stiffened homogeneous cylindrical shells under axial compression Present Brush and Almroth [25] Difference (%) 50 rings, 50 stringers, L ¼ m, R¼ 0.5 m, E ¼ Â 1010 N=m2 , υ ¼ 0:3, dr ¼ ds ¼ 0:0025 m, hr ¼ hs ¼ 0:01 m ð19Þ ∂2 w ∂4 w ∂4 w ∂4 w ỵ Dn11 ỵ Dn12 ỵ Dn21 ỵ 4Dn66 ị 2 ỵ Dn22 Bn21 ∂x ∂x ∂y ∂y ∂x ∂t   ∂4 φ ∂2 φ ∂2 φ ∂2 w ∂ w0 n n n n B11 ỵ B22 2B66 ị 2 B12 ỵ R ∂x2 ∂y2 ∂x2 ∂x ∂y ∂y ∂x2 Internal stiffeners R/h ¼ 100 3.0725(6,7) R/h ¼ 200 1.4147(6,7) R/h ¼ 500 0.6924(5,6) External stiffeners R/h ¼ 100 3,9529(9,3) R/h ¼ 200 2.1410(9,4) R/h ¼ 500 1.2764(6,6) 3.0906(6,7) 1.4328(6,7) 0.7057(5,6) 0.59 1.28 1.92 3.9551(9,2) 2.1469(9,4) 1.2897(6,6) 0.06 0.28 1.04 194 D Huy Bich et al / International Journal of Mechanical Sciences 74 (2013) 190–200 Substituting Eqs (22) and (23) into Eq (19) and solving obtained equation for unknown φ lead to φ ẳ cos 2mx 2ny mx ny y2 ỵ cos −φ3 sin sin −r h ; L R L R 24ị ỵ Gf f ỵ f ịf ỵ 2f ịL2 m2 hr f ỵ f ị ẳ 0; where denote ẳ ẳ A ẳ An11 m4 ỵ An66 2An12 ịm2 n2 2 ỵ An22 n4 ; m2 f f ỵ 2f Þ; 32n2 λ2 An22 h i Bn21 m4 π ỵ Bn11 ỵ Bn22 2Bn66 ịm2 n2 2 ỵ Bn12 n4 LR m2 f ẳ f tị; An11 m4 ỵ An66 2An12 ịm2 n2 2 ỵ An22 n4 ẳ L : R 26ị where n2 f f ỵ 2f ị; 32m2 An11 ẳ Galerkin method to the resulting equation yield ! B2 4€ L f ỵ D ỵ f A L2 2 m π ; R D ¼ Dn11 m4 π ỵ Dn12 ỵ Dn21 ỵ 4Dn66 ịm2 n2 2 ỵ Dn22 n4 ; ! n4 m4 ỵ Gẳ : 27ị 16An11 16An22 B ẳ Bn21 m4 ỵ Bn11 ỵ Bn22 2Bn66 ịm2 n2 2 ỵ Bn12 n4 − f; ð25Þ Substituting the expressions (22)–(24) into Eq (20) and applying Fig Effect of k on the static post-buckling of un-stiffened shells Fig Effect of k on the static post-buckling of external ring and stringer stiffened shells Fig Effect of k on the static post-buckling of internal ring and stringer stiffened shells Introduce parameters D¼ D h ; B¼ B ; h A ¼ Ah; G¼ G ; h ξ¼ f ; h ξ0 ¼ f0 ; h ð28Þ Fig Dynamic responses of un-stiffened shell under step loading of infinite duration Fig Dynamic response of external rings and stringers stiffened shell under step loading of infinite duration Fig Dynamic response of internal rings and stringers stiffened shell under step loading of infinite duration D Huy Bich et al / International Journal of Mechanical Sciences 74 (2013) 190–200 195 3.2 Dynamic buckling analysis For dynamic buckling analysis, this paper investigates two cases as following Fig Effect of k on the dynamic responses of un-stiffened shells under linear-time compression Case Consider a cylindrical shell subjected to the axial compression linearly varying on time r ¼ ct in which c is a loading speed By using the Runge–Kutta method, the responses of ES-FGM cylindrical shells can be determined from Eq (29) The dynamic critical time t cr can be obtained according to Budiansky–Roth criterion [20]: for large value of loading speed, the amplitude– time curve of obtained displacement response increases sharply and this curve obtain a maximum by passing from the slope point and at the corresponding time t ¼ t cr the stability loss occurs Here, t cr is called critical time and the corresponding dynamic critical buckling stress r dcr ¼ ct cr and dynamic coefficient τcr ¼ r dcr =r scr Case Assume that a shell is conducted for step loading of infinite duration r ¼ const; ∀t The dynamic critical load is found based on the criterion mentioned in [27]: the load corresponding to a sudden jump in the maximum average deflection in the time history of the shell is taken as the critical buckling step load Numerical results and discussions Fig Effect of k on the dynamic responses of external ring and stringer stiffened shells under linear-time compression the non-dimension form of Eq (26) is written as ! ρ1 L4 € B ỵ Dỵ A h  2 L m2 ỵ ịr ẳ 0: þ Gξðξ þ ξ0 Þðξ þ 2ξ0 Þ− h ð29Þ 3.1 Static buckling and post-buckling analysis Omitting the term of inertia, Eq (29) leads to ! 2 h B h ỵ Dỵ G ỵ 20 ị: r0 ẳ A ỵ ị L m2 π L m2 π Putting ξ0 ¼ in Eq (30), yields ! 2 h B h ỵ Dỵ G2 : r0 ẳ 2 A L m π L m2 π ð30Þ To validate the present formulation, two comparisons on critical load are carried out with results from open literatures First, the dynamic buckling of perfect un-stiffened FGM cylindrical shells under linear-time compression is given in Table 1, which was also analyzed by Huang and Han [19] using the energy method and classical shell theory As can be seen, the good agreements are observed Second, the present static buckling load (Table 2) of stiffened homogeneous cylindrical shells under axial compression is compared with the results in the monograph of Brush and Almroth [25] (based on equations in page 180) where the smeared stiffeners technique, equilibrium path and classical shell theory are used This comparison once again also shows that the good agreements are obtained To illustrate the proposed approach of eccentrically stiffened FGM cylindrical shells, the stiffened and un-stiffened FGM cylindrical shells are considered with R ¼ 0:5 m, L ¼ 0:75 m, R=h ¼ 250 The combination of materials consists of aluminum Em ¼ 7Â 1010 N/m2, ρm ¼ 2702 kg/m3 and alumina Ec ¼ 38 Â 1010 N/m2, ρc ¼ 3800 kg/m3 The compressive stress of dynamic analysis is taken to be r ¼ 1010 t Poisson's ratio ν is chosen to be 0.3 for simplicity The height of stiffeners is equal to 0:01 m, its width 0:0025 m The material properties are Es ¼ Ec and Er ¼ Ec , ρs ¼ ρc and ρr ¼ ρc with internal stringer stiffeners and internal ring ð31Þ From Eq (31), by taking ξ ¼ the buckling stress of perfect ESFGM cylindrical shells can be determined as ! 2 h B r sbu ẳ Dỵ : 32ị A L m2 π The static critical buckling stress of perfect ES-FGM cylindrical shells are determined by condition r scr ¼ minr sbu vs ðm; nÞ and the static post-buckling curves of perfect and imperfect shells may be traced by using Eqs (30) and (31) with the same buckling mode shape of critical buckling stress for evaluate static behavior of these structures Fig 10 Effect of k on the dynamic responses of internal ring and stringer stiffened shells under linear-time compression 196 D Huy Bich et al / International Journal of Mechanical Sciences 74 (2013) 190–200 Table Effect of k on critical static and dynamic buckling stress r ( Â 108 N/m2) 0.2 k Un-stiffened Static Dynamic r ¼ const Dynamic r ¼ ct τcr External rings and stringers Static Dynamic r ¼ const Dynamic r ¼ ct τcr Internal rings and stringers Static Dynamic r ¼ const Dynamic r ¼ ct τcr 7.743(12,10) 7.743(12,10) 8.002(12,10) 1.033 4.998(7,15) 4.998(7,15) 5.310(7,15) 1.062 10 2.985(13,6) 2.986(13,6) 3.185(13,6) 1.067 2.560(12,8) 2.560(12,8) 2.770(12,8) 1.082 17.814(5,10) 17.815(5,10) 18.315(5,10) 1.028 14.658(4,9) 14.658(4,9) 15.245(4,9) 1.040 10.926(4,8) 10.927(4,8) 11.570(4,8) 1.059 9.815(4,8) 9.816(4,8) 10.361(4,8) 1.056 26.660(3,7) 26.660(3,7) 27.466(3,7) 1.030 20.350(3,7) 20.350(3,7) 21.081(3,7) 1.036 13.181(3,6) 13.182(3,6) 13.912(3,6) 1.055 11.480(3,6) 11.480(3,6) 12.455(3,6) 1.085 Table Effects of number, type and position of stiffeners on critical static and dynamic buckling stress r ( Â 108 N/m2) 15 rings and 63 stringers Static Un-stiffened ER IR ES IS IR and IS ER and ES IR and ES ER and IS 4.998(7,15) 5.201(15,1) 5.184(13,8) 5.603(1,8) 5.222(1,8) 20.350(3,7) 14.658(4,9) 12.345(6,8) 16.448(2,8) 20 rings and 84 stringers Dynamic Static r ¼ const r ¼ ct 4.998(7,15) 5.201(15,1) 5.184(13,8) 5.604(1,8) 5.223(1,8) 20.350(3,7) 14.658(4,9) 12.345(6,8) 16.449(2,8) 5.310(7,15) 5.397(15,1) 5.378(14,8) 6.903(1,8) 6.785(1,8) 21.081(3,7) 15.245(4,9) 12.917(6,8) 17.305(2,8) 4.998(7,15) 5.261(15,1) 5.238(14,8) 5.779(1,8) 5.273(1,8) 22.351(3,6) 16.850(4,9) 13.099(5,7) 18.805(2,7) Dynamic r ¼ const r ¼ ct 4.998(7,15) 5.261(15,1) 5.239(14,8) 5.779(1,8) 5.273(1,8) 22.352(3,6) 16.850(4,9) 13.099(5,7) 18.806(2,7) 5.310(7,15) 5.441(16,1) 5.447(14,8) 7.273(1,8) 6.670(1,8) 23.108(3,6) 17.414(4,9) 13.754(5,7) 19.727(2,7) ER, external rings; IR, internal rings; ES, external stringers; IS, internal stringers Fig 11 Effect of external ring and external stringer stiffeners on the static postbuckling curves Fig 12 Effect of internal ring and internal stringer stiffeners on the static postbuckling curves stiffeners; Es ¼ Em , Er ¼ Em , ρs ¼ ρm and ρr ¼ ρm with external stringer stiffeners and external ring stiffeners, respectively The stiffener system includes 15 ring stiffeners and 63 stringer stiffeners distributed regularly in the axial and circumferential directions, respectively In Figs 2–4, the static post-buckling curves of un-stiffened and stiffened shells are traced by Eqs (30) and (31) of perfect (ξ0 ¼ 0) and imperfect (ξ0 ¼ 0:1) cases versus three different values of volume fraction index k (¼0.2, 1, 5) As can be seen, the postbuckling curves are lower with increasing values of k Furthermore, the post-buckling curves of imperfect shells are lower than those of perfect shells when deflection is small and post-buckling curves of imperfect shells is higher than that of perfect shells when the deflection is sufficiently large By using the fourth-order Runge–Kutta method, Eq (29) is solved to obtain the dynamic responses of perfect (ξ0 ¼ 0) shells under step loading of infinite duration Dynamic responses of unstiffened and stiffened shells are presented in Figs 5–7 As can be seen, there is a sudden jump in the value of the average deflection when the axial compression reaches the critical value In addition, the dynamic critical step load corresponding to internal ring and stinger stiffened shell is biggest This value is bigger than about 1.4 times in comparison with the external ring and stinger stiffened shell Figs 8–10 show the effect of k on the dynamic responses of perfect and imperfect un-stiffened and stiffened shells under linear-time compression These figures also show that there is no definite point of instability as in static analysis Rather, there is a D Huy Bich et al / International Journal of Mechanical Sciences 74 (2013) 190–200 Fig 13 Effect of internal stiffeners and external stiffeners on the static postbuckling curves Fig 14 Effect of position of stiffeners on the static post-buckling of stiffened shells 197 Fig 17 Effect of internal stiffeners and external stiffeners on the dynamic responses of shells under linear-time compression Fig 18 Effect of stiffeners position on the dynamic responses of shells under linear-time compression Table Effect of R/h on critical static and dynamic buckling load per unit length r ( Â 106 N/m) R/h Fig 15 Effect of external ring and external stringer stiffeners on the dynamic responses of shells under linear-time compression 100 Un-stiffened Static 6.247(6,9) Dynamic r ¼ const 6.247(6,9) Dynamic r ¼ ct 6.457(6,9) External rings and stringers Static 8.341(4,8) Dynamic r ¼ const 8.341(4,8) Dynamic r ¼ ct 8.607(4,8) Internal rings and stringers Static 9.964(3,7) Dynamic r ¼ const 9.964(3,7) Dynamic r ¼ ct 10.288(3,7) 250 500 1000 0.999(7,15) 0.999(7,15) 1.062(7,15) 0.250(9,21) 0.250(9,21) 0.277(9,21) 0.062(25,20) 0.062(25,20) 0.069(25,20) 2.932(4,9) 2.932(4,9) 3.049(4,9) 1.859(4,9) 1.859(4,9) 1.920(4,9) 1.292(4,8) 1.293(4,8) 1.326(4,8) 4.070(3,7) 4.070(3,7) 4.216(3,7) 2.380(3,6) 2.381(3,6) 2.457(3,6) 1.480(3,6) 1.480(3,6) 1.528(3,6) Fig 16 Effect of internal ring and internal stringer stiffeners on the dynamic responses of shells under linear-time compression Fig 19 Effect of R/h on the static post-buckling of un-stiffened shells region of instability where the slope of ξ vs t curve increases rapidly (in perfect shell cases) According to the Budiansky–Roth criterion [20], the critical time t cr can be taken as an intermediate value of this region Therefore, one can choose the inflexion point of curve i.e d ξ=dt jt ¼ t cr ¼ as Huang and Han [19] This region is clearly recognized with perfect shells but it is very difficult to 198 D Huy Bich et al / International Journal of Mechanical Sciences 74 (2013) 190–200 Fig 20 Effect of R/h on the static post-buckling of external stiffened shells Fig 24 Effect of R/h on the dynamic responses of internal ring and stringer stiffened shells under linear-time compression Fig 21 Effect of R/h on the static post-buckling of internal stiffened shells Fig 25 Effect of loading speed on the dynamic responses of un-stiffened shells Fig 22 Effect of R/h on the dynamic responses of un-stiffened shells under lineartime compression Fig 26 Effect of loading speed on the dynamic responses of internal stiffened shells Fig 23 Effect of R/h on the dynamic responses of internal ring and stringer stiffened shells under linear-time compression define that with imperfect shells Therefore, critical dynamic buckling compressions of imperfect un-stiffened and stiffened cylindrical shells cannot accurately predict by Budiansky–Roth criterion (like a remark given by Huang and Han [19] for FGM un-stiffened shells) This figure also shows that a sudden jump in the value of deflection occurs earlier when k increases and it corresponds a smaller dynamic buckling compression In the next figures, the dynamic response is traced by relation of deflection ratio ξ versus excited load r (where r ¼ ct) Table shows the critical static and dynamic buckling stresses of stiffened and un-stiffened cylindrical shells vs four different values of volume fraction index k ¼(0.2,1,5,10) With the same input parameters, the effectiveness of stiffeners is obviously proven; the critical buckling stress of stiffened shell is greater than one of un-stiffened shell Table also shows that the dynamic critical stress decreases with the increase of the volume fraction index k and the buckling modes ðm; nÞ seem smaller with stiffened shells The critical parameter τcr is larger than 1, it denotes that the dynamic critical buckling stress of linear-time compression case is larger than static buckling stress The largest value of and τcr is equal to 1.085 for the internal rings and stringers stiffened shell with k ¼10 and the smallest τcr ¼ 1:028 corresponds to external D Huy Bich et al / International Journal of Mechanical Sciences 74 (2013) 190–200 rings and stringers stiffened shell with k ¼0.2 In addition, when the shell subjected to the step loading of infinite duration, it seems that the dynamic critical buckling compression is approximately equal to the static critical buckling compression Effect of the stiffener number, type and position of stiffeners on the nonlinear critical buckling stress is given in Table Clearly, the ring or stringer stiffeners lightly influence to the critical buckling stress of shells But, the combination of ring and stringer stiffeners has a considerable effect on the stability of shell Especially, the critical buckling stress of internal rings and stringers stiffened shell is greatest and the critical buckling stress of internal rings stiffened shell is smallest When the number of stiffeners increases, it is evident that critical buckling stresses increase Figs 11–14 show the effect of type and position of stiffeners on the static post-buckling of stiffened and un-stiffened shells (k ¼1, 15 rings and 63 stringers) According to the critical buckling values, the post-buckling of un-stiffened shells is lower than one of stiffened shells For stiffened shells, the post-buckling of internal rings stiffened shell is the lowest and one of internal rings and stringers stiffened shell is the highest Figs 15–18 show the effect of type and position of stiffeners on the dynamic response of stiffened and un-stiffened shells under linear-time compression (k¼ 1, 15 rings and 63 stringers) For one type of stiffeners shells (Figs 15 and 16), it seem that the amplitude responses of perfect rings stiffened shells are smallest and those of perfect stringers stiffened shells are the biggest In the results considered, the slope of instability region of stringer stiffened shells is smaller The effects of R/h on the behavior of buckling loads per unit length (r ¼ r h) are illustrated in Table Clearly, the critical buckling load decreases when the R/h ratio increases, the stiffeners are more effective with thinner shells When R/h ratio increases, the critical buckling loads of un-stiffened shells strongly decrease (about 100 times for variation of R/h from 100 to 1000) but lightly with stiffened shells (about times for variation of R/h from 100 to 1000) Figs 19–21 show the static post-buckling of un-stiffened and stiffened shells The post-buckling curves of shells are much higher when R/h ratio decreases The dynamic responses of un-stiffened and stiffened shells under linear-time compression are presented in Figs 22–24 As can be observed, maximal amplitude responses of instability region increase when R/h ratio increases These figures also show that the slope of instability region of thinner shells is greater Effects of the loading speed on the dynamic responses of unstiffened and internal stiffened shells under linear-time compression are shown in Figs 25 and 26 Three values of loading speed are used, i.e c ¼1010 , c ¼2 Â 1010 , c ¼5 Â 1010 Clearly, the critical dynamic buckling loads and amplitude response increase when the loading speed increases It mean that rapidly compressed cylindrical shell will buckle at a higher critical stress than a very slowly compressed cylindrical shell Conclusions A formulation of governing equations of eccentrically stiffened functionally graded circular cylindrical thin shells based on the classical shell theory and the smeared stiffeners technique with von Karman–Donnell nonlinear terms is presented in this paper By using the Galerkin method the explicit expressions of static buckling compression, post-buckling load–deflection curve and the nonlinear dynamic equation of ES-FGM circular cylindrical shells are obtained, the later is solved by using the Runge–Kutta method and the criteria for determining critical dynamic compressions are applied 199 Some conclusions can be obtained from the present analysis: (i) Stiffeners enhance the static and dynamic stability and loadcarrying capacity of FGM circular cylindrical shells (ii) Combination of ring and stringer stiffeners has a large effect on the stability of shell The critical buckling compression of internal rings and stringers stiffened shell is greatest (iii) In dynamic linear-time load case, there is no definite point of instability as in static analysis Rather, there is a region of instability where the slope of ξ vs t curve increases rapidly in perfect shell cases (iv) For imperfect FGM shell, it is difficult to accurately predict the critical buckling compression (v) The dynamic critical buckling compressions of linear-time compression case is larger than static critical buckling compressions (τcr is about 1.028–1.085) and the dynamic critical buckling compression of step loading of infinite duration is approximately equal to the static critical buckling compression (vi) Initial geometrical imperfection, radius-to-thickness ratio, position, type and number of stiffeners significantly influence on the static and dynamic behavior of cylindrical shell Major purpose of this study is to analyze the global buckling and post-buckling behavior of FGM cylindrical shells reinforced by closely spaced stiffener system For local buckling analysis, the approach of Stamatelos et al [28] may be used Acknowledgments This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.01-2012.02 References [1] Shen HS Postbuckling analysis of pressure-loaded functionally graded cylindrical shells in thermal environments Eng Struct 2003;25(4):487–97 [2] Shen HS Postbuckling of axially-loaded FGM hybrid cylindrical shells in thermal environments Compos Sci Technol 2005;65(11–12):1675–90 [3] Bahtui A, Eslami MR Coupled thermoelasticity of functionally graded cylindrical shells Mech Res Commun 2007;34(1):1–18 [4] Huang H, Han Q Buckling of imperfect functionally graded cylindrical shells under axial compression Eur J Mech—A/Solids 2008;27(6):1026–36 [5] Huang H, Han Q Nonlinear elastic buckling and postbuckling of axially compressed functionally graded cylindrical shells Int J Mech Sci 2009;51 (7):500–7 [6] Huang H, Han Q Nonlinear buckling and postbuckling of heated functionally graded cylindrical shells under combined axial 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of shells Eccentrically stiffened FGM (ES-FGM) circular cylindrical shells 2.1 Functionally graded material In this paper, functionally graded material is... (20) and applying Fig Effect of k on the static post -buckling of un -stiffened shells Fig Effect of k on the static post -buckling of external ring and stringer stiffened shells Fig Effect of k