Nonlinear buckling and post-Buckling of eccentrically stiffened functionally graded cylindrical shells surrounded by an elastic medium based on the first order shear deformation theory

Volume 35 Number 4 4 Vietnam Journal of Mechanics, VAST, Vol. 35, No. 4 (2013), pp. 285 – 298 NONLINEAR BUCKLING AND POST-BUCKLING OF ECCENTRICALLY STIFFENED FUNCTIONALLY GRADED CYLINDRICAL SHELLS SURROUNDED BY AN ELASTIC MEDIUM BASED ON THE FIRST ORDER SHEAR DEFORMATION THEORY Dao Van Dung∗, Nguyen Thi Nga Hanoi University of Science, VNU, Vietnam ∗E-mail: dungdv90@gmail.com Abstract. In this paper, the nonlinear buckling and post-buckling of an eccentrically stiffened cylindrical shell made of functionally graded materials, surrounded by an elastic medium and subjected to mechanical compressive loads and external pressures are inves- tigated by an analytical approach. The cylindrical shells are reinforced by longitudinal and circumferential stiffeners. The material properties of cylindrical shells are graded in the thickness direction according to a volume fraction power-law distribution. The non- linear stability equations for stiffened cylindrical shells are derived by using the first order shear deformation theory and smeared stiffeners technique. Closed-form expressions for determining the buckling load and load-deflection curves are obtained. The effectiveness of stiffeners in enhancing the stability of cylindrical shells is shown. The effects of volume fraction indexes, material properties, geometrical parameters and foundation parameters are analyzed in detail. Keywords: Stiffened cylindrical shells, nonlinear buckling and post-buckling, function- ally graded, foundations. 1. INTRODUCTION Functionally graded material (FGM) cylindrical shells in recent years, are exten- sively used in many modern engineering applications. These structures are usually rested on or placed in a soil medium modelled as an elastic foundation. Thus, their stability analysis is an important problem and has received considerable interest by researchers. Bagherizadeh et al. [1] investigated the mechanical buckling of FGM cylindrical shell surrounded by Pasternak elastic foundation and subjected to combined axial and radial compressive loads based on a higher-order shear deformation shell theory (HSDT). The elastic foundation is modelled by two-parameter Pasternak model, which is obtained by adding a shear layer to the Winkler model. Shen and Wang [2] presented thermal buckling and post-buckling behavior for fiber reinforced composite (FRC) laminated cylindrical 286 Dao Van Dung, Nguyen Thi Nga shells embedded in a large outer elastic medium and subjected to a uniform tempera- ture rise. The surrounding elastic medium is modeled as a Pasternak foundation. Shen [3] presented the post-buckling response of a shear deformable functionally graded (FG) cylin- drical shell of finite length embedded in a large outer elastic medium and subjected to axial compressive loads in thermal environments based on a higher order shear defor- mation shell theory with von Kármán–Donnell-type of kinematic nonlinearity. Shen et al. [4] studied post-buckling of internal pressure loaded FG cylindrical shells surrounded by an elastic medium. This work employed a higher order shear deformation shell the- ory with von Kármán–Donnell-type of kinematic nonlinearity and a singular perturbation technique. Sofiyev and Kuruoglu [5] investigated torsional vibration and buckling of the cylindrical shell with functionally graded coatings surrounded by an elastic medium. Recently, idea of eccentrically stiffened FGM structures has been proposed by Bich et al. [6–8] and Najafizadeh et al. [9]. Bich et al. [6–8] studied the nonlinear static buck- ling behavior of eccentrically stiffened imperfect FGM plates and shallow shells and the nonlinear dynamic response of eccentrically stiffened FGM imperfect panels and doubly curved thin shallow shells on the basis of the classical plate and shell theory. Stiffeners are assumed to be homogenous. By considering FGM stiffeners, Najafizadeh et al. [9] investi- gated the mechanical buckling behavior of axially loaded FGM stiffened cylindrical shells reinforced by rings and stringers based on the classical shell theory (CST). Following this direction, Dung and Hoa [10, 11] researched on nonlinear buckling and post-buckling of eccentrically stiffened FGM thin circular cylindrical shells under torsional load or external pressure. Approximate three-term solution of deflection taking into account the nonlinear buckling shape is chosen and the Galerkin method is used in those works. In this paper, following the direction of work [6–8], the nonlinear buckling and post- buckling analysis of eccentrically stiffened FGM thin circular cylindrical shells surrounded by an elastic medium and under axial load and external pressure based on the first or- der shear deformation theory (FSDT) is studied. Governing equations using the smeared stiffeners technique and FSDT are derived. Applying Galerkin’s method, the closed-form expression to determine the buckling loads and post-buckling equilibrium paths are found. The effects of various parameters as stiffeners and foundations, volume fraction index of material and geometrical parameters on the nonlinear stability of cylindrical shells are shown. 2. FUNCTIONALLY GRADED SHELLS AND THEORETICAL FORMULATIONS 2.1. Functionally graded material shells Consider a functionally graded material cylindrical shell of length L, mean radius R, and uniform thickness h, as depicted in Fig. 1. An orthogonal Descartes coordinate system xyz is chosen so that the axes x, y (y = Rθ) are in the longitudinal, circumferential directions, respectively, and the axis z is perpendicular to the middle surface and in inward thickness direction ( − h 2 ≤ z ≤ h 2 ) . Nonlinear buckling and post-buckling of eccentrically stiffened functionally graded cylindrical shells ... 287 Fig. 1. Configuration of a cylindrical shell surrounded by an elastic medium The cylindrical shell is assumed to be made from a mixture of ceramic and metal with the volume-fractions given by a power-law distribution as Vm + Vc = 1, Vc = Vc(z) = ( z h + 1 2 )k , (1) where k is the volume fraction exponent and takes only non-negative values, and the subscripts m and c refer to the metal and ceramic constituents, respectively. According to the mixture rule, the effective Young’s modulus can be expressed by E = E(z) = EmVm + EcVc = Em + (Ec − Em) ( z h + 1 2 )k (2) whereas the Poisson’s ratio is assumed to be a constant. As can be seen, from Eq. (2) that, the surface ( z = h 2 ) of a cylindrical shell is ceramic-rich whereas the surface ( z = − h 2 ) is metal-rich. 2.2. Theoretical formulations Using the first order shear deformation shell theory and geometrical nonlinearity in von Karman sense, the strains across the cylindrical shell thickness at a distance z from the middle surface are [12]   εxεy γxy   =   ε0x ε0y γ0xy  + z   κxκy κxy   , (3) [ γxz γyz ] = [ γ0xz γ0yz ] = [ w,x + φx w,y + φy ] , (4) where   ε0x ε0y γ0xy   =   u,x + 1 2 w2,x v,y + 1 2 w2,y − w R u,y + v,x +w,xw,y   ,   κxκy κxy   =   φx,xφy,y φx,y + φy,x   , (5) 288 Dao Van Dung, Nguyen Thi Nga in which εx, εy are normal strains, γxy is the in-plane shear strain and γxz, γyz are the transverse shear deformations. Also, u, v, w are the displacement components along the x, y, z directions, respectively, and φx, φy are the slope rotation in the (y, z) and (x, z) planes, respectively. From the relations (5), the compatibility equation of a cylindrical shell is obtained as ε0x,yy + ε 0 y,xx − γ 0 xy,xy = w 2 ,xy − w,xxw,yy − 1 R w,xx. (6) Assume that the cylindrical shell and stiffeners are treated as assembled shell and beams elements and shell is reinforced by closely spaced [13] homogeneous ring and stringer stiffener systems. Stiffener is pure-ceramic if it is located at ceramic-rich side and is pure- metal if is located at metal-rich side, and such FGM stiffened circular cylindrical shells provide continuity between shells and stiffeners. In this case, the stress-strain relationship is given by Hooke’s law as, for shells( σshx , σ sh y ) = E(z) 1− ν2 [(εx, εy) + ν (εy, εx)] , ( σshxy, σ sh xz, σ sh yz ) = E(z) 2 (1 + ν) (γxy, γxz, γyz) , (7) and for stiffeners [14] σsx = Esxεx, σ s y = Esyεy, σsxz = Gsxγxz, σ s yz = Gsyγyz. (8) where the superscripts “sh” and “s” denote shell and stiffener, respectively; Esx, Esy and Gsx, Gsy are Young’s moduli and shear moduli of longitudinal and circumferential stiffen- ers, respectively. Taking into account the contribution of stiffeners by the smeared stiffeners technique and omitting the twist of stiffeners and integrating the above stress-strain equations and their moments through the thickness of the cylindrical shell, we obtain the expressions for force and moment resultants of an eccentrically stiffened FGM cylindrical shell Nx = ( A11 + EsxA1 d1 ) ε0x +A12ε 0 y + (B11 +C1) φx,x + B12φy,y, Ny = A12ε 0 x + ( A22 + EsyA2 d2 ) ε0y + B12φx,x + (B22 + C2)φy,y , Nxy = A66γ 0 xy +B66 (φx,y + φy,x) , (9) Mx = (B11 + C1) ε 0 x +B12ε 0 y + ( D11 + EsxI1 d1 ) φx,x +D12φy,y, My = B12ε 0 x + (B22 +C2) ε 0 y + ( D22 + EsyI2 d2 ) φy,y +D12φx,x, Mxy = B66γ 0 xy +D66 (φx,y + φy,x) . (10) The transverse shear force resultants are Qx = A44w,x +A44φx, Qy = A55w,y + A55φy, (11) Nonlinear buckling and post-buckling of eccentrically stiffened functionally graded cylindrical shells ... 289 where the specific expressions of coefficients Aij , Bij, Dij are given as A11 = A22 = E1 1− ν2 , A12 = E1ν 1− ν2 , A66 = E1 2 (1 + ν) , A44 = χ1 [ E1 2 (1 + ν) + GsxA1 d1 ] , A55 = χ2 [ E1 2 (1 + ν) + GsyA2 d2 ] , 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) , (12) and E1 = ( Em + Ec − Em k + 1 ) h, E2 = (Ec −Em) kh 2 2 (k + 1) (k + 2) , E3 = [ Em 12 + (Ec − Em) ( 1 k + 3 − 1 k + 2 + 1 4k + 4 )] h3, (13) and I1 = b1h 3 1 12 + A1e 2 1 , I2 = b2h 3 2 12 + A2e 2 2 , C1 = ± EsxA1e1 d1 , C2 = ± EsyA2e2 d2 , e1 = h1 + h 2 , e2 = h2 + h 2 . (14) In the relation (12), χ1 and χ2 are the shear correction factors and are taken as χ1 = χ2 = 5/6. Note that Young’s modulus of stiffener takes the value being assigned to Em if the full metal stiffeners are put at the metal-rich side of the cylindrical shell and conversely being assigned to Ec if the full ceramic ones at the ceramic-rich side. In addition, the thickness and width for longitudinal stiffeners (x-direction) are respectively denoted by h1 and b1 and for circumferential stiffeners (y-direction) are h2 and b2. Also, d1 and d2 are the distance between two longitudinal and circumferential stiffeners, respectively. The quantities A1, A2 are the cross-section areas of stiffeners and I1, I2 are the second moments of inertia of the stiffener cross sections relative to the cylindrical shell middle surface; and the eccentricities e1 and e2 represent the distance from the cylindrical shell middle surface to the centroid of the longitudinal and circumferential stiffener cross section, respectively. The quantities C1 and C2 are taken plus sign if stiffeners are attached inside and taken minus sign if they are outside. The strain-force resultant relations reversely are obtained from Eq. (9) ε0x = A ∗ 22 Nx −A ∗ 12 Ny −B ∗ 11 φx,x −B ∗ 12 φy,y, ε0y = A ∗ 11 Ny − A ∗ 12 Nx −B ∗ 21 φx,x −B ∗ 22 φy,y, γ0xy = A ∗ 66 Nxy − B ∗ 66 (φx,y + φy,x) . (15) Substituting Eq. (15) into Eq. (10) yields Mx = B ∗ 11 Nx +B ∗ 21 Ny +D ∗ 11 φx,x +D ∗ 12 φy,y, My = B ∗ 12 Nx +B ∗ 22 Ny +D ∗ 21 φx,x +D ∗ 22 φy,y, Mxy = B ∗ 66 Nxy +D ∗ 66 (φx,y + φy,x) , (16) 290 Dao Van Dung, Nguyen Thi Nga where the coefficients A∗ij , B ∗ ij and D ∗ ij are found in Appendix I. The nonlinear equilibrium equations of FGM cylindrical shells surrounded by an elastic medium, based on the first order shear deformation theory, are [1, 12] Nx,x +Nxy,y = 0, (17) Nxy,x +Ny,y = 0, (18) Qx,x+Qy,y+Nxw,xx+2Nxyw,xy+Nyw,yy+q+ 1 R Ny−K1w+K2 (w,xx +w,yy) = 0, (19) Mx,x +Mxy,y −Qx = 0, (20) Mxy,x +My,y −Qy = 0. (21) where K1 (N/m 3) is modulus of subgrade reaction for foundation and K2 (N/m)-the shear modulus of the subgrade and q is uniform lateral pressure. Introduce Airy’s stress function f = f(x, y) so that Nx = f,yy , Ny = f,xx, Nxy = −f,xy. (22) It is easy to see that the first two equations (17) and (18) are automatically satisfied. Three equations (19), (20) and (21) become Mx,xx + 2Mxy,xy +My,yy + f,yyw,xx − 2f,xyw,xy + f,xxw,yy+ +q + 1 R Ny −K1w +K2 (w,xx + w,yy) = 0, Mx,x +Mxy,y −Qx = 0, Mxy,x +My,y −Qy = 0. (23) Substituting the expressions ofMij in Eq. (16) and Qx, Qy in Eq. (11) into Eq. (23), we obtain Ω ≡B∗21 ∂4f ∂x4 + (B∗11 + B ∗ 22 − 2B ∗ 66) ∂4f ∂x2∂y2 +B∗12 ∂4f ∂y4 +D∗11 ∂3φx ∂x3 + + (D∗12 + 2D ∗ 66) ∂3φy ∂x2∂y + (D∗21+ 2D ∗ 66) ∂3φx ∂x∂y2 +D∗22 ∂3φy ∂y3 + ∂2f ∂y2 ∂2w ∂x2 − − 2 ∂2f ∂x∂y ∂2w ∂x∂y + ∂2f ∂x2 ∂2w ∂y2 + q + 1 R ∂2f ∂x2 −K1w +K2 ( ∂2w ∂x2 + ∂2w ∂y2 ) = 0, (24) B∗21 ∂3f ∂x3 +(B∗11 − B ∗ 66) ∂3f ∂x∂y2 +D∗11 ∂2φx ∂x2 +(D∗12 +D ∗ 66) ∂2φy ∂x∂y +D∗66 ∂2φx ∂y2 −A44 ∂w ∂x −A44φx = 0, (25) B∗ 12 ∂3f ∂y3 + (B∗ 22 −B∗ 66 ) ∂3f ∂y∂x2 +D∗ 22 ∂2φy ∂y2 + (D∗ 21 +D∗ 66 ) ∂2φx ∂x∂y +D∗ 66 ∂2φy ∂x2 −A55 ∂w ∂y −A55φy = 0. (26) The system of Eqs. (24) ÷ (26) includes four unknown functions w, φx, φy and f so it is necessary to find the fourth equation relating to these functions by using the Nonlinear buckling and post-buckling of eccentrically stiffened functionally graded cylindrical shells ... 291 compatibility equation (6). For this aim, substituting the expressions of εij in Eq. (15) into Eq. (6), one can write as A∗ 11 ∂4f ∂x4 + (A∗ 66 − 2A∗ 12 ) ∂4f ∂x2∂y2 +A∗ 22 ∂4f ∂y4 − B∗ 21 ∂3φx ∂x3 − (B∗ 11 −B∗ 66 ) ∂3φx ∂x∂y2 − − (B∗ 22 − B∗ 66 ) ∂3φy ∂y∂x2 −B∗ 12 ∂3φy ∂y3 − ( ∂2w ∂x∂y )2 + ∂2w ∂x2 ∂2w ∂y2 + 1 R ∂2w ∂x2 = 0. (27) Eqs. (24) ÷ (27) are nonlinear equations in terms of four dependent unknown func- tions w, φx, φy and f and are used to investigate the stability of eccentrically stiffened functionally graded (ES-FGM) cylindrical shells surrounded by an elastic medium and under mechanical loads. 3. BUCKLING AND POST-BUCKLING ANALYSIS Assume that an ES-FGM cylindrical shell is simply supported at x = 0, x = L and subjected to in-plane compression load uniformly distributed of intensities Px along x-direction and uniform lateral pressure q. Thus the boundary conditions are given by w = φy = Nxy = Mx = 0, Nx = Nx0 = −hPx at x = 0, x = L. (28) The approximate solution of the system of Eqs. (24) ÷ (27) satisfying the boundary conditions (28) can be expressed by w = W sinαx sinβy, φx = φ10 cosαx sin βy + φ11 sin 2αx, φy = φ20 sinαx cosβy + φ21 sin 2βy, f = f1 cos 2αx+ f2 cos 2βy + f3 sinαx sinβy + 1 2 Nx0y 2, (29) where α = mpi L , β = n R and m is number of half waves in x-direction and n is number of wave in y-direction, respectively. Substituting Eq. (29) into Eqs. (25), (26) and (27) and carrying out some calcula- tions, yield f1 = 4α2D∗11 + A44 A∗ 11 (4α2D∗ 11 + A44) + 4α2B∗221 · β2 32α2 W 2 = L1.W 2 , f2 = 4β2D∗22 +A55 A∗ 22 (4β2D∗ 22 +A55) + 4β2B∗212 · α2 32β2 W 2 = L2.W 2 , f3 = (A44a22α− A55a12β) a13 + (A55a11β −A44a21α)a23 + D∗ R α2 D∗ [α4A∗ 11 + α2β2 (A∗ 66 − 2A∗ 12 ) + β4A∗ 22 ] + (a13a22 − a12a23) a13 + (a11a23 − a21a13)a23 ·W =L3.W, (30) and φ10 = L4.W, φ20 = L5.W, (31) φ11 = L6.W 2, φ21 = L7.W 2, (32) where D∗, Li are given in Appendix II. 292 Dao Van Dung, Nguyen Thi Nga Substituting the expression (29) into Eq. (24) and then applying Galerkin’s method to the resulting equation piR∫ 0 L∫ 0 Ω sin mpix L sin ny R dxdy = 0 yields α2β2 (L1 + L2) ( −LpiR 2 ) ·W 3 + ( 16α4B∗21L1 + 16β 4B∗12L2 − 2α 2β2L3 − 4α2 R L1 − 8α 3D∗11L6 −8β3D∗22L7 )( −4 3αβ δmδn ) ·W 2 + {[ α4B∗21 + α 2β2 (B∗11 +B ∗ 22 − 2B ∗ 66) + β 4B∗12 − α2 R ] L3 + [ α3D∗ 11 + αβ2 (D∗ 21 + 2D∗ 66 ) ] L4 + [ β3D∗ 22 + α2β (D∗ 12 + 2D∗ 66 ) ] L5 − α 2Nx0 −K1 − ( α2 + β2 ) K2 }(LpiR 4 ) ·W + q ( 4 αβ δmδn ) = 0 (33) in which δm = 1− (−1)m 2 , δn = 1− (−1)n 2 . Eq. (33) is governing equations used to determine critical buckling loads and post- buckling load-deflection curves of ES-FGM cylindrical shells surrounded by an elastic medium and subjected to mechanical compressive loads and external pressures. If ES-FGM cylindrical shells only subjects to uniform external pressures q, Nx0 = 0 and m, n are odd numbers, thus Eq. (33) becomes q = q(W ) = I 3 .W 3 − J.W 2 +K.W. (34) in which I = 3α3β3 (L1 + L2) ( LpiR 8 ) , J = − 1 3 ( 16α4B∗21L1 + 16β 4B∗12L2 − 2α 2β2L3 − 4α2 R L1 − 8α 3D∗11L6 − 8β 3D∗22L7 ) , K = {[ α4B∗ 21 + α2β2 (B∗ 11 +B∗ 22 − 2B∗ 66 ) + β4B∗ 12 − α2 R ] L3 + [ α3D∗ 11 + αβ2 (D∗ 21 + 2D∗ 66 ) ] L4 + [ β3D∗22 + α 2β (D∗12 + 2D ∗ 66) ] L5 −K1 − ( α2 + β2 ) K2 }(−LpiRαβ 16 ) (35) Eq. (34) leads to the equation from which the upper and lower buckling external pressures may be obtained as qu = 1 3I2 [ J ( 3IK − 2J2 ) + 2 ( J2 − IK )3/2] , ql = 1 3I2 [ J ( 3IK − 2J2 ) − 2 ( J2 − IK )3/2] , (36) The condition providing the existence of the upper and lower buckling loads is J2 − IK > 0. (37) Nonlinear buckling and post-buckling of eccentrically stiffened functionally graded cylindrical shells ... 293 If ES-FGM cylindrical shells only subjects to axial compressive loads, substituting q = 0, Nx0 = −h.Px into Eq. (33), obtains Px = ( −1 hα2 )( H2.W 2 +H1.W +H0 ) . (38) in which H0 = » α 4 B ∗ 21 + α 2 β 2 (B∗11 +B ∗ 22 − 2B ∗ 66) + β 4 B ∗ 12 − α2 R – L3+ + ˆ α 3 D ∗ 11 + αβ 2 (D∗21 + 2D ∗ 66) ˜ L4 + ˆ β 3 D ∗ 22 + α 2 β (D∗12 + 2D ∗ 66) ˜ L5 −K1 − ` α 2 + β2 ´ K2, H1 = „ 16α4B∗21L1 + 16β 4 B ∗ 12L2 − 2α 2 β 2 L3 − 4α2 R L1 − 8α 3 D ∗ 11L6 − 8β 3 D ∗ 22L7 «„ −16 3αβLpiR δmδn « , H2 =− 2α 2β2 (L1 + L2) . (39) Taking W → 0, Eq. (38) gives the upper buckling load as Pupper = ( −1 hα2 ) H0. (40) The lower buckling load can be obtained from Eq. (38) by using the condition dPx dW = 0 leads to W∗ = −H1 2H2 and obtains Plower = Px (W∗) = ( −1 hα2 )( H0 − H2 1 4H2 ) . (41) 4. NUMERICAL RESULTS AND DISCUSSION 4.1. Comparison results As part of the validation of the present approach, a simple supported un-stiffened FGM cylindrical shell without elastic foundations under axial compressive load is consid- ered. The geometrical and material properties of the cylindrical shell are taken by [15] L/R = 2; Ec = 168.08× 10 9 Pa; Em = 105.69× 10 9 Pa; ν =0.3, and k and R/h change. The buckling load of static-axial loaded FGM cylindrical shells Pscr = Pdcr/τcr in which Table 1. Comparisons with results of [15] for un-stiffened FGM cylindrical shells without elastic foundations under axial compressive load Huang and Han Present Difference (%) Critical load versus k R/h = 500 k = 0.2 189.262 (2,11)a 189.166 (3,13) 0.051 k = 1.0 164.352 (2,11) 164.248 (3,13) 0.063 k = 5.0 144.471 (2,11) 144.123 (3,13) 0.241 Critical load versus R/h k = 0.2 R/h = 400 236.578 (5,15) 236.245 (19,14) 0.141 R/h = 600 157.984 (3,14) 157.558 (22,19) 0.270 R/h = 800 118.898 (2,12) 118.1658 (33,1) 0.616 a Buckling mode (m, n). 294 Dao Van Dung, Nguyen Thi Nga the non-dimensional parameter τcr is the critical parameter and Pdcr is the corresponding dynamic buckling load, is calculated by Huang and Han [15] and Pupper is given by Eq. (40) in this paper. The comparison results are given in Tab. 1. It is seen that the present results are in good agreement to those from the reference [5]. 4.2. ES-FGM cylindrical shells surrounded by elastic foundations In this section the above formulations are used to analyze the effects of input param- eters on buckling and post-buckling behavior of cylindrical shells. The geometric properties of shells are h = 0.006m, R/h = 100, L/R= 2, h1 = h2 = 0.006 m, b1 = b2 = 0.003 m and d1 = 2piR/n1 m, d2 = L/n2 m. The material properties are Ec =380×10 9 Pa, Em =70×10 9 Pa, ν =0.3. Table 2. Critical buckling compressive load for different types of stiffeners (k = 1) Pupper cr(MPa) Inside stiffeners Outside stiffeners Without stiffeners 1244.0685 (12,2) 1244.0685 (12,2) Longitudinal stiffeners (n1 = 26) 1251.1232 (2,7) 1257.5814 (2,7) Circumferential stiffeners (n2 = 26) 1250.7971 (9,8) 1257.1162 (12,1) Orthogonal stiffeners (n1 = n2 = 13) 1305.0626 (7,9) 1279.5732 (12,1) a Buckling mode (m, n). Tab. 2 shows that the critical buckling loads of FG cylindrical shells without foun- dations only attached by x-stiffener or by y-stiffener are close each other. But, the combi- nation of longitudinal and circumferential stiffeners has strongly effect on the stability of cylindrical shells. The critical load in this case is biggest, but the critical load of cylindrical shells attached by circumferential stiffeners is smallest, for stiffened cylindrical shells. As results in Tab. 2, it can be seen that the critical buckling load of stiffened cylindrical shells is greater than one of un-stiffened cylindrical shells. It is reasonable because the present of stiffeners makes cylindrical shells to become more rigid. Table 3. Critical buckling compressive loads (MPa) for different types of foundation parameters (k = 1) Foundation K1 = K2 = 0 K1=2.5×10 8 N/m3, K1 = 0, K1=2.5×10 8 N/m3, parameters K2 = 5×10 5 N/m K2=0 K2=5×10 5 N/m Pupper cr 1244.0685 (12,2) 1286.2857(12,2) 1327.7614 (12,1) 1369.9786 (12,1) a Buckling mode (m, n). Tab. 3 shows the effect of the foundation on the critical buckling loads of FGM cylindrical shells without stiffeners in which the foundation parameters are K1 = 2.5×10 8 N/m3, K2 = 5 × 10 5 N/m. The critical load using two foundation parameters is biggest, whereas the critical load without foundations is smallest. This difference is considerable. Nonlinear buckling and post-buckling of eccentrically stiffened functionally graded cylindrical shells ... 295 For example Pupper cr = 1369.9786 MPa for shell with two foundation parameters is great than Pupper cr = 1244.0685 MPa for shell without about 9.2%. Tab. 4 shows the effect of both foundations and stiffeners on the critical buckling loads of FGM cylindrical shells. The number of stiffeners is 26 in all cases. If cylindri- cal shells have the same value of foundation parameters, the critical load is biggest for longitudinal stiffeners, but the critical load is smallest for circumferential stiffeners. The case of two foundation parameters is biggest, the case of the first foundation parameter is smallest for the same type of stiffener. Table 4. Critical buckling compressive loads (MPa) for different types of foundation parameters and stiffeners (k = 1) Pupper cr(MPa) K1 = 2.5×10 8 N/m3, K1 = 0, K1=2.5×10 8 N/m3, K2 = 0 K2 = 5×10 5 N/m K2=5×10 5 N/m Longitudinal stiffeners 1432.0105 (8,9) 1451.0503 (7,9) 1558.0887 (8,9) (n1 = 26) Circumferential stiffeners 1309.8784 (11,7) 1351.9671 (10,7) 1406.1543 (11,6) (n2 = 26) Orthogonal stiffeners 1389.0893 (9,8) 1424.0553 (9,8) 1495.6795 (10,7) (n1 = n2 = 13) Fig. 2. Effect of volume fraction index on the nonlinear response of ES-FGM cylindrical shells under axial compressive load Fig. 3. Effect of radius-to-thickness ratio on the nonlinear response of ES-FGM cylindrical shells under external pressure Fig. 2 shows the effect of material index k on the stability of inside-stiffened FG cylindrical shells. As can be seen, the critical buckling compressive loads decrease when the value of k increases. In Fig. 3, the effects of radius-to-thickness ratios are focused on with (m, n) = (1, 1), k = 1. Three values of R/h are 100, 150 and 300. As can be seen that the post-buckling load-deflection curves become lower when the values of R/h increase. 296 Dao Van Dung, Nguyen Thi Nga Fig. 4. Effect of volume fraction index on the nonlinear response of un-stiffened FG cylindri- cal shells under external pressure without elas- tic foundations Fig. 5. Effect of volume fraction index on the nonlinear response of inside-stiffened FG cylin- drical shells under external pressure with elastic foundations Figs. 4 and 5 show the effect of volume fraction index on the nonlinear response of FG cylindrical shells under external pressure. Fig. 4 is plotted with k = 0, 0.5, 1 and 5 and (m, n) = (1, 3), L = 2 m, L/R = 5, R/h = 100 for cylindrical shells without both stiffeners and foundations. Fig. 5 is plotted with k = 0, 1 and 5, (m, n) = (1, 1), h = 0.006 m, L/R = 2, R/h = 100 for cylindrical shells with orthogonal stiffeners and two-parameter foundation. As can be seen that the post-buckling load-deflection curves become lower when the values of k increase, i.e. the load carrying capacity of structures decreases with the greater percentage of metal. 5. CONCLUSIONS This paper presents an analytical solution to investigate the buckling and post- buckling behavior of ES-FGM cylindrical shells surrounded by an elastic medium subjected to mechanical compressive loads and external pressures. Theoretical formulations are based on the smeared stiffeners technique and the first-order shear deformation theory. The analytical expressions to determine the static critical buckling load and analyze the post- buckling load-deflection are obtained. Numerical results shows the effects of stiffeners, geometrical parameters and elastic foundations on the buckling and post-buckling response of ES-FGM cylindrical shells. Some following remarks are deduced from the present results: a. The stiffeners enhance the stability of cylindrical shells. Particularly, the combi- nation of longitudinal and circumferential stiffeners has strongly effect on the stability of shells. The critical load in this case is biggest. b. The foundation parameters affects strongly critical buckling load. Especially, the critical buckling load corresponding to the presence of the both foundation parameters K1 and K2 is biggest. c. The loading carrying capacity of shell is reduced considerably when R/h ratio or volume fraction index k increases. Nonlinear buckling and post-buckling of eccentrically stiffened functionally graded cylindrical shells ... 297 ACKNOWLEDGEMENT This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number 107.02-2013.02 and the project No.TN- 14. The authors are grateful for the financial support. REFERENCES [1] Bagherizadeh E, Kiani Y, Eslami MR, Mechanical buckling of functionally graded mate- rial cylindrical shells surrounded by Pasternak elastic foundation, Compos Struct, 93, (2011), pp. 3063–3071. 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Received June 18, 2013 298 Dao Van Dung, Nguyen Thi Nga APPENDIX I ∆ = ( A11 + EsxA1 d1 )( A22 + EsyA2 d2 ) − A212, A∗11 = 1 ∆ ( A11 + EsxA1 d1 ) , A∗22 = 1 ∆ ( A22 + EsyA2 d2 ) , A∗12 = A12 ∆ , A∗66 = 1 A66 , B∗ 11 =A∗ 22 (B11 +C1)− A ∗ 12 B12, B ∗ 22 = A∗ 11 (B22 +C2)−A ∗ 12 B12, B∗12 =A ∗ 22B12 − A ∗ 12 (B22 + C2) , B ∗ 21 = A ∗ 11B12 −A ∗ 12 (B11 +C1) , B ∗ 66 = B66 A66 , D∗11 = D11 + EsxI1 d1 −B∗11 (B11 +C1)−B ∗ 21B12, D∗22 = D22 + EsyI2 d2 −B∗22 (B22 + C2)−B ∗ 12B12, D∗12 = D12 −B ∗ 12 (B11 + C1)−B ∗ 22B12, D∗21 = D12 −B ∗ 21 (B22 + C2)−B ∗ 11B12, D∗66 = D66 −B ∗ 66B66. APPENDIX II L1 = 4α2D∗ 11 + A44 A∗ 11 (4α2D∗ 11 +A44) + 4α2B ∗2 21 · β2 32α2 , L2 = 4β2D∗ 22 + A55 A∗ 22 (4β2D∗ 22 +A55) + 4β2B∗212 · α2 32β2 , L3 = (A44a22α −A55a12β) a13 + (A55a11β −A44a21α) a23 + D∗ R α2 D∗ [α4A∗ 11 + α2β2 (A∗ 66 − 2A∗ 12 ) + β4A∗ 22 ] + (a13a22 − a12a23) a13 + (a11a23 − a21a13) a23 , L4 = L3 D∗ (a13a22 − a12a23)− A44a22α −A55a12β D∗ , L5 = L3 D∗ (a11a23 − a21a13)− A55a11β −A44a21α D∗ , L6 = 8α3B∗ 21 4α2D∗ 11 + A44 · L1, L7 = 8β3B∗ 12 4β2D∗ 22 + A55 · L2, in which a11 = α 2D∗ 11 + β2D∗ 66 + A44, a22 = β 2D∗ 22 + α2D∗ 66 + A55, a12 = αβ (D ∗ 12 +D∗ 66 ) , a21 = αβ (D ∗ 21 +D∗ 66 ) , a13 = − [ α3B∗ 21 + αβ2 (B∗ 11 − B∗ 66 ) ] , a23 = − [ β3B∗ 12 + α2β (B∗ 22 − B∗ 66 ) ] , D∗ = a11a22 − a12a21. VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY VIETNAM JOURNAL OF MECHANICS VOLUME 35, N. 4, 2013 CONTENTS Pages 1. Nguyen Manh Cuong, Tran Ich Thinh, Ta Thi Hien, Dinh Gia Ninh, Free vibration of thick composite plates on non-homogeneous elastic foundations by dynamic stiffness method. 257 2. Vu Lam Dong, Pham Duc Chinh, Construction of bounds on the effective shear modulus of isotropic multicomponent materials. 275 3. Dao Van Dung, Nguyen Thi Nga, Nonlinear buckling and post-buckling of eccentrically stiffened functionally graded cylindrical shells surrounded by an elastic medium based on the first order shear deformation theory. 285 4. N. T. Khiem, L. K. Toan, N. T. L. Khue, Change in mode shape nodes of multiple cracked bar: II. The numerical analysis. 299 5. Tran Van Lien, Trinh Anh Hao, Determination of mode shapes of a multiple cracked beam element and its application for free vibration analysis of a multi- span continuous beam. 313 6. Phan Anh Tuan, Pham Thi Thanh Huong, Vu Duy Quang, A method of skin frictional resistant reduction by creating small bubbles at bottom of ships. 325 7. Nguyen Thoi Trung, Bui Xuan Thang, Ho Huu Vinh, Lam Phat Thuan, Ngo Thanh Phong, An effective algorithm for reliability-based optimization of stiffened Mindlin plate. 335