Postbuckling of functionally graded cylindrical shells based on improved donnell equations

Vietnam Journal of Mechanics, VAST, Vol. 35, No. 1 (2013), pp. 1 – 15 POSTBUCKLING OF FUNCTIONALLY GRADED CYLINDRICAL SHELLS BASED ON IMPROVED DONNELL EQUATIONS Dao Huy Bich1, Nguyen Xuan Nguyen1, Hoang Van Tung2 1Hanoi University of Science, VNU, Vietnam 2Hanoi Architectural University, Vietnam Abstract. This paper presents an analytical approach to investigate the buckling and postbuckling of functionally graded cylindrical shells subjected to axial and transverse mechanical loads incorporating the effects of temperature. Material properties are as- sumed to be temperature independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. Equi- librium equations for perfect cylindrical shells are derived by using improved Donnell shell theory taking into account geometrical nonlinearity. One-term approximate solution is assumed to satisfy simply supported boundary conditions and closed-form expressions of buckling loads and load-deflection curves are determined by Galerkin method. Analysis shows the effects of material and the geometric parameters, buckling mode, pre-existent axial compressive and thermal loads on the nonlinear response of the shells. Keywords: Postbuckling, functionally graded materials, cylindrical shells, improved Don- nell theory, temperature effects. 1. INTRODUCTION Cylindrical shell is one of the most common structures found in many applications of various industries. As a result, problems relating to the stability including buckling and postbuckling behaviors of this type of shell have a major importance for safe and reliable design and attract attention of many researchers. Brush and Almroth [1] introduced an excellent work on buckling of bars, plates and shells in which linear stability of cylindrical shell structures under basic types of loading has been analyzed. However, the results were mainly presented for isotropic shallow cylindrical shells. Birman and Bert [2] investigated dynamic stability of reinforced composite cylindrical shells subjected to pulsating loads acting in the axial direction and in the presence of a thermal field on the basic of Donnell theory for laminated shells and a linear analysis. They then considered the buckling and postbuckling behaviors of reinforced cylindrical shells subjected to the simultaneous ac- tion of a thermal field and an axial loading by using improved version of Donnell theory ignoring the shallowness of cylindrical shells [3]. Their paper also formulated conditions for the snap-through of a cylindrical shell under thermomechanical loading. Eslami et al. [4] established improved stability equations for linear buckling analysis of isotropic short 2 Dao Huy Bich, Nguyen Xuan Nguyen, Hoang Van Tung and long cylindrical shells under thermal loadings. An analytical approach was used in above mentioned studies. Due to advanced characteristics in comparison with traditional metals and conven- tional composites, Functionally Graded Materials (FGMs) consisting of metal and ceramic constituents have received increasingly attention in structural applications in recent years. Smooth and continuous change in material properties enable FGMs to avoid interface problems and unexpected thermal stress concentrations. By high performance heat resis- tance capacity, FGMs are now chosen to use as structural components exposed to severe temperature conditions such as aircraft, aerospace structures, nuclear plants and other en- gineering applications. Shahsiah and Eslami [5] employed Donnell shell theory and coupled form of stability equations to study linear buckling of simply supported FGM shallow cylin- drical shells under thermal loads. Subsequently, they extended this analytical approach for FGM nonshallow long cylindrical shells [6] in which improved terms of the classical thin shell theory were incorporated. Lanhe et al. [7] utilized the Donnell theory, uncoupled form of stability equation and one-term approximate solution to determine closed-form expressions of critical temperatures for simply supported FGM cylindrical shells. Buckling behavior of cylindrical shells with FGM middle layer, imperfect FGM cylindrical shells and FGM stiffened cylindrical shells under axial compressive load were analytically in- vestigated in works [8-10], respectively. Postbuckling behavior of FGM cylindrical shells has been presented in some studies. Shen [11] investigated thermal postbuckling of simply supported FGM cylindrical shells under uniform temperature rise. He used the classical shell theory, boundary layer theory of shell buckling and asymptotic perturbation tech- nique to determine critical temperatures and postbuckling temperature-deflection curves with both geometric imperfection and temperature dependence of material properties are taken into consideration. Following this direction, thermomechanical postbuckling behav- iors of FGM cylindrical shells with and without piezoelectric layer were also reported in works [12-14]. By using analytical method and the classical theory, nonlinear buckling and postbuckling of FGM cylindrical shells under axial compression and combined mechanical loads have been considered by Huang and Han [15, 16]. Recently, Darabi et al. [17] pre- sented an analytical study on the nonlinear dynamic stability of simply supported FGM circular cylindrical shells under periodic axial loading. Also, nonlinear dynamic stability of FGM cylindrical shells with and without piezoelectric layers under thermomechanical and thermo-electro-mechanical loads has been treated by Shariyat [18,19]. He employed a high-order shell theory proposed by Shariyat and Eslami [20] in which transverse shear stress influences are also included, finite element method and a two-step iterative method to determine buckling loads and postbuckling curves. It is excepted for works [3, 4, 6], most of aforementioned investigations used the theories in which the shallowness of cylindrical shells is assumed. This results from the complexity of basic equations when assumption on the shallowness is ignored due to difficulty in defining a suitable stress function. How- ever, improved terms should be included in the shell theories for more exact predictions, especially nonshallow long cylindrical shells. In this paper, buckling and postbuckling behaviors of FGM cylindrical shells under mechanical loads with and without temperature effects are investigated by an analytical approach. Equilibrium equations are established by using improved Donnell shell theory Postbuckling of functionally graded cylindrical shells based on improved Donnell equations 3 with kinematic nonlinearity is taken into consideration. One-term approximate solution satisfying simply supported boundary conditions is assumed and closed-form expressions of buckling loads and nonlinear load-deflection curves are determined by Galerkin method. The effects of material and geometric parameters, buckling mode, pre-existent axial com- pressive and thermal loads on the stability of FGM cylindrical shells are considered and discussed. 2. FUNCTIONALLY CYLINDRICAL SHELLS Consider a functionally graded circular cylindrical shell of radius of curvature R, thickness h and length L as shown in Fig. 1. The shell is made from a mixture of ceramics and metals and is defined in coordinate system (x, θ, z), where x and θ are in the axial and circumferential directions of the shell, respectively, and z is perpendicular to the middle surface and points outwards (−h/2 ≤ z ≤ h/2). Fig. 1. Configuration and the coordinate system of a cylindrical shell Suppose that the material composition of the shell varies smoothly along the thick- ness is such a way the inner surface is metal-rich and the outer surface is ceramic-rich by following a simple power law in terms of the volume fractions of the constituents as Vc(z) = ( 2z + h 2h )k , Vm(z) = 1− Vc(z) (1) where Vc and Vm are the volume fractions of ceramic and metal constituents, respectively, and volume fraction index k is a nonnegative number that defines the material distribution. It is assumed that the effective properties Preff of FGM cylindrical shell change in the thickness direction z and can be determined by the linear rule of mixture as Preff (z) = PrcVc(z) + PrmVm(z) (2) where Pr denotes a temperature-independent material property, and subscripts m and c represent the metal and ceramic constituents, respectively. From Eqs. (1) and (2) the effective properties of FGM cylindrical shell such as modulus of elasticity E, the coefficient of thermal expansion α, and the coefficient of 4 Dao Huy Bich, Nguyen Xuan Nguyen, Hoang Van Tung thermal conduction K can be defined as [E(z), α(z), K(z)] = [Em, αm, Km] + [Ecm, αcm, Kcm] ( 2z + h 2h )k (3) whereas Poisson ratio ν is assumed to be constant and Ecm = Ec − Em , αcm = αc − αm , Kcm = Kc −Km. (4) It is evident that E = Ec , α = αc , K = Kc at z = h/2 and E = Em , α = αm , K = Km at z = −h/2. 3. GOVERNING EQUATIONS In the present study, the improved Donnell shell theory is used to obtain the equi- librium equations as well as expressions of buckling loads and nonlinear load-deflection curves of FGM cylindrical shells. The strains across the shell thickness at a distance z from the middle surface are εx = εx0 + zkx , εy = εy0 + zky , γxy = γxy0 + zkxy (5) where εx0 and εy0 are the normal strains, γxy0 is the shear strain at the middle surface of the shell, whereas kx, ky, kxy are the change of curvatures and twist. According to Sanders assumption, the strains at the middle surface and the change of curvatures and twist are related to the displacement components u, v, w in the x, y, z coordinate directions, respectively, as [1] εx0 = u,x + 1 2 w2,x , εy0 = v,y − w R + 1 2 w2,y , γxy0 = u,y + v,x + w,xw,y kx = −w,xx , ky = −w,yy − 1 R v,y , kxy = −w,xy − 1 2R v,x (6) where y = Rθ and subscript (, ) indicates the partial derivative. Hooke law for a functionally graded cylindrical shell including temperature effects is defined as (σx, σy) = E 1− ν2 [(εx, εy) + ν (εy, εx)− (1 + ν)α∆T (1, 1)] σxy = E 2(1 + ν) γxy , (7) where ∆T denotes the change of environment temperature from stress free initial state or temperature difference between the surfaces of FGM cylindrical shell. The force and moment resultants of an FGM cylindrical shell are expressed in terms of the stress components through the thickness as (Nij,Mij) = h/2∫ −h/2 σij(1, z)dz , ij = x, y, xy. (8) Postbuckling of functionally graded cylindrical shells based on improved Donnell equations 5 Introduction of Eqs. (3), (5) and (7) into Eqs. (8) gives the constitutive relations in the matrix form as  Nx Ny Nxy Mx My Mxy   =   A11 A12 0 B11 B12 0 A12 A22 0 B12 B22 0 0 0 A66 0 0 B66 B11 B12 0 D11 D12 0 B12 B22 0 D12 D22 0 0 0 B66 0 0 D66     εx0 εy0 γxy0 kx kxy 2kxy   −   Φ0/(1− ν) Φ0/(1− ν) 0 Φ1/(1− ν) Φ1/(1− ν) 0   (9) where A11 = A22 = E1 1− ν2 , A12 = νA11 , A66 = E1 2(1 + ν) B11 = B22 = E2 1− ν2 , B12 = νB11 , B66 = E2 2(1 + ν) D11 = D22 = E3 1− ν2 , D12 = νD11 , D66 = E3 2(1 + ν) (10) and E1 = Emh+Ecmh/(k+ 1) , E2 = Ecmh 2 [1/(k+ 2)− 1/(2k+ 2)] E3 = Emh 3/12 +Ecmh 3 [1/(k+ 3)− 1/(k + 2) + 1/(4k+ 4)] , (Φ0,Φ1) = h/2∫ −h/2 [ Em +Ecm ( 2z + h 2h )k][ αm + αcm ( 2z + h 2h )k] ∆T (1, z) dz (11) The nonlinear equilibrium equations of a perfect cylindrical shell based on the im- proved Donnell shell theory are Nx,x +Nxy,y = 0 Nxy,x +Ny,y − 1 R (Mxy,x +My,y) = 0 Mx,xx + 2Mxy,xy +My,yy + Ny R +Nxw,xx + 2Nxyw,xy +Nyw,yy + (Nxy,x +Ny,y)w,y − Pxhw,xx + q = 0 (12) where Px is axial uniform compressive force acting on two ends of the shell and q is external pressure uniformly distributed on the surface of the shell. Substituting of Eqs. (6) into Eqs. (9) and then into Eqs. (12), the system of equi- librium equations (12) is rewritten in terms of displacement components as follows L11(u) + L12(v)− L13(w) + P1(w) = 0 L21(u) + L22(v)− L23(w) + P2(w) = 0 L31(u) + L32(v)− L33(w) + P3(w) +Q3(u, w) +R3(v, w) − Φ0 1− ν ( w,xx +w,yy + 1 R ) − Pxhw,xx + q = 0 (13) 6 Dao Huy Bich, Nguyen Xuan Nguyen, Hoang Van Tung where linear operatorsLij() (i, j = 1, 2, 3) and nonlinear operators Pi() (i = 1, 2, 3), Q3(, ), R3(, ) are defined as follows L11() = A11 ∂2 ∂x2 + A66 ∂2 ∂y2 L12() = L21() = ( A12 +A66 − B12 +B66 R ) ∂2 ∂x∂y L13() = L31() = A12 R ∂ ∂x +B11 ∂3 ∂x3 + (B12 + 2B66) ∂3 ∂x∂y2 L22() = ( A66 − 2B66 R + D66 R2 ) ∂2 ∂x2 + ( A11 − 2B11 R + D11 R2 ) ∂2 ∂y2 L23() = L32() = ( A11 R − B11 R2 ) ∂ ∂y + ( B11 − D11 R ) ∂3 ∂y3 + ( B12 + 2B66 − D12 + 2D66 R ) ∂3 ∂x2∂y L33() = A11 R2 () + 2B12 R ∂2 ∂x2 + 2B11 R ∂2 ∂y2 +D11 ( ∂4 ∂x4 + ∂4 ∂y4 ) + 2 (D12 + 2D66) ∂4 ∂x2∂y2 P1() = A11 ∂ ∂x ∂2 ∂x2 + (A12 + A66) ∂ ∂y ∂2 ∂x∂y +A66 ∂ ∂x ∂2 ∂y2 P2() = ( A66 − B66 R ) ∂2 ∂x2 ∂ ∂y + ( A11 − B11 R ) ∂ ∂y ∂2 ∂y2 + ( A12 + A66 − B12 +B66 R ) ∂ ∂x ∂2 ∂x∂y P3(w) = − w R ( A12 ∂2w ∂x2 +A11 ∂2w ∂y2 ) + 2 (B66 − B12) ∂2 ∂x2 ∂2 ∂y2 + 2 (B12 −B66) ( ∂2w ∂x∂y )2 − A12 2R ( ∂w ∂x )2 − A11 2R ( ∂w ∂y )2 + 3A11 2 [ ∂2w ∂x2 ( ∂w ∂x )2 + ∂2w ∂y2 ( ∂w ∂y )2] + 2 (A12 + 2A66) ∂w ∂x ∂w ∂y ∂2 ∂x∂y + ( A12 2 + A66 )[ ∂2w ∂y2 ( ∂w ∂x )2 + ∂2w ∂x2 ( ∂w ∂y )2] Q3(u, w) = A11 ( ∂u ∂x ∂2w ∂x2 + ∂2u ∂x2 ∂w ∂x ) +A12 ∂u ∂x ∂2w ∂y2 + A66 ∂2u ∂y2 ∂w ∂x + (A12 + A66) ∂2u ∂x∂y ∂w ∂y + 2A66 ∂u ∂y ∂2w ∂x∂y (14) Postbuckling of functionally graded cylindrical shells based on improved Donnell equations 7 R3(v, w) = ( A12 − B12 R ) ∂v ∂y ∂2w ∂x2 + ( A66 − B66 R )[ 2 ∂v ∂x ∂2w ∂x∂y + ∂2v ∂x2 ∂w ∂y ] + ( A11 − B11 R )[ ∂v ∂y ∂2w ∂y2 + ∂2v ∂y2 ∂w ∂y ] + ( A12 + A66 − B12 +B66 R ) ∂2v ∂x∂y ∂w ∂x . In what follows, specific expressions of thermal parameter Φ0 for two types of ther- mal loads will be determined. 3.1. Uniform temperature rise Environment temperature can be raised from initial value Ti to final one Tf and temperature difference ∆T = Tf − Ti is a constant. In this case, the thermal parameter Φ0 can be expressed in terms of the ∆T from Eqs. (11) as follows Φ0 = I∆Th , I = Emαm + Emαcm +Ecmαm k + 1 + Ecmαcm 2k + 1 (15) 3.2. Through the thickness temperature gradient In this case, the temperature through the thickness is governed by the one-dimensional Fourier equation of steady-state heat conduction established in cylindrical coordinate sys- tem whose origin is on the symmetric axis of cylinder rather than on the middle surface of cylindrical shell d dz¯ [ K(z¯) dT dz¯ ] + K(z¯) z¯ dT dz¯ = 0 , T (z¯ = R− h/2) = Tm, T (z¯ = R+ h/2) = Tc (16) where Tc and Tm are temperatures at ceramic-rich and metal-rich surfaces, respectively. In Eq. (16), z¯ is radial coordinate of a point which is distant z from the shell middle surface with respect to the symmetric axis of cylinder, i.e. z¯ = R+ z and R−h/2 ≤ z¯ ≤ R+h/2. The solution of Eq. (16) can be expressed as follows T (z¯) = Tm + ∆T R+h/2∫ R−h/2 dz¯ z¯K(z¯) z¯∫ R−h/2 dζ ζK(ζ) (17) where, in this case, ∆T = Tc − Tm is defined as the temperature difference between ceramic-rich and metal-rich surfaces of the FGM shell. Due to mathematical difficulty, this section only considers linear distribution of metal and ceramic constituents, i.e. k = 1 and K(z¯) = Km +Kcm [ 2(z¯ − R) + h 2h ] . (18) Introduction of Eq. (18) into Eq. (17) gives temperature distribution across the shell thickness as T (z) = Tm + ∆T ln Km(R+h/2)Kc(R−h/2) [ ln R+ z R− h/2 − ln (Kc +Km) /2 +Kcmz/h Km ] (19) 8 Dao Huy Bich, Nguyen Xuan Nguyen, Hoang Van Tung where z¯ has been replaced by z +R after integration. Assuming the metal surface temperature as reference temperature and substituting Eq. (19) into Eqs. (11) give Φ0 = H∆Th, where H = 1 ξ − η { Emαm [ ξ (Rh + 1/2)− ηKc Kcm ] + Emαcm +Ecmα 2 [(Rh − 1) (1− Rhξ) + 3 4 ξ + 1 K2cm ( 3K2m +K 2 c 2 + 2KmKc (η − 1)− ηK 2 c )] +Ecmαcm [−5/8 + Rh/2− R2h 3 + ξ ( 7 24 + R3h 3 − R2h 2 + Rh 4 ) − 1 18K3cm ( 11K3m + 2K 3 c (3η − 1) −18ηKmKcKcm + 9KmKc(Kc − 2Km))]} (20) and Rh = R/h , ξ = ln 2Rh + 1 2Rh − 1 , η = ln Kc Km . (21) 4. STABILITY ANALYSIS In this section, an analytical approach is used to investigate the nonlinear stability of FGM cylindrical shells under mechanical and thermomechanical loads. Consider a perfect cylindrical shell with simply supported edge conditions. The boundary conditions at x = 0, L are w = w,xx = v = u,x = 0. (22) The approximate solution of the system of Eqs. (13) satisfying the boundary con- ditions (22) may be assumed as u = U cosλmx sinµny v = V sinλmx cosµny w = W sinλmx sinµny (23) where λm = mpi/L , µn = n/R andm, n are number of half waves in x direction and waves in y direction, respectively, and U, V,W are the amplitudes of displacements. Substitution of Eqs. (23) into Eqs. (13) and then applying Galerkin method for the resulting equations yield l11U + l12V + l13W + n1W 2 = 0 l21U + l22V + l23W + n2W 2 = 0 l31U + l32V + l33W + n3W 2 + n4W 3 + n5UW + n6V W − Pxhpi 2m2 L2 W − Φ0 1− ν ( m2pi2 L2 + n2 R2 ) W + 16Φ0 R(1− ν)pi2mn − 16q pi2mn = 0 (24) Postbuckling of functionally graded cylindrical shells based on improved Donnell equations 9 where l11 = A11 pi2m2 L2 + A66 n2 R2 , l12 = l21 = pimn LR [ A12 +A66 − 1 R (B12 +B66) ] , l13 = l31 = A12 pim LR − B11 pi3m3 L3 − (B12 + 2B66)pimn 2 LR2 , l22 = ( A11 − 2B11 R + D11 R2 ) n2 R2 + ( A66 − 2B66 R + D66 R2 ) pi2m2 L2 , l23 = l32 = ( A11 − B11 R ) n R2 + ( D11 R − B11 ) n3 R3 + ( D12 + 2D66 R − B12 − 2B66 ) pi2m2n L2R , l33 = D11 ( pi4m4 L4 + n4 R4 ) + (D12 + 2D66) 2pi2m2n2 L2R2 + A11 R2 − 2B12pi 2m2 L2R − 2B11n 2 R3 , n1 = A11 32pim2 9L3n − 16 (A12 −A66)n 9piLR2 , n2 = ( A11 − B11 R ) 32n2 9pi2mR3 + ( A66 − A12 − B66 − B12 R ) 16m 9L2R , n3 = 32 (B12 − B66)mn 3L2R2 − 16A12m 3L2Rn − 16A11n 3pi2R3m , n4 = 9A11 32 ( pi4m4 L4 + n4 R4 ) + (A12 + 2A66) pi2m2n2 16L2R2 , n5 = −A11 32pim2 9L3n − 32 (A12 −A66)n 9piLR2 , n6 = ( A66 −A12 + B12 − B66 R ) 32m 9L2R + ( B11 R −A11 ) 32n2 9pi2R3m (25) and m, n are odd numbers. Solving the first two of Eqs. (24) for U and V yields U = (l12l23 − l22l13)W + (l12n2 − l22n1)W 2 l11l22 − l 2 12 V = (l12l13 − l11l23)W + (l12n1 − l11n2)W 2 l11l22 − l212 (26) Substituting Eqs. (26) into the third of Eqs. (24) we obtain 16q pi2mn = a1W+a2W 2 +a3W 3 − Pxpi 2m2 R2hL 2 R W+ [ 16 Rhpi2mn − ( pi2m2 R2hL 2 R + n2 R2h ) W ] I∆T 1− ν (27) 10 Dao Huy Bich, Nguyen Xuan Nguyen, Hoang Van Tung where a1 = l¯33 + 2l¯12l¯23l¯13 − ( l¯11l¯ 2 23 + l¯22l¯ 2 13 ) l¯11l¯22 − l¯212 a2 = n¯3 + 1 χ [ l¯13 ( l¯12n¯2 − l¯22n¯1 ) + l¯23 ( l¯12n¯1 − l¯11n¯2 ) + n¯5 ( l¯12l¯23 − l¯22l¯13 ) + n¯6 ( l¯12l¯13 − l¯11l¯23 )] a3 = n¯4 + 1 χ [ n¯5 ( l¯12n¯2 − l¯22n¯1 ) + n¯6 ( l¯12n¯1 − l¯11n¯2 )] (28) in which χ = l¯11l¯22 − l¯ 2 12 , l¯11 = A¯11 pi2m2 R2hL 2 R − B¯11 pi3m3 R3hL 3 R − ( B¯12 + 2B¯66 ) pimn2 R3hLR , l¯22 = ( A¯11 − 2B¯11 Rh + D¯11 R2h ) n2 R2h + ( A¯66 − 2B¯66 Rh + D¯66 R2h ) pi2m2 R2hL 2 R , l¯23 = ( A¯11 − B¯11 Rh ) n R2h + ( D¯11 Rh − B¯11 ) n3 R3h + ( D¯12 + 2D¯66 Rh − B¯12 − 2B¯66 ) pi2m2n R3hL 2 R , l¯33 = D¯11 ( pi4m4 R4hL 4 R + n4 R4h ) + ( D¯12 + 2D¯66 ) 2pi2m2n2 R4hL 2 R + A¯11 R2h − 2B¯12pi 2m2 R3hL 2 R − 2B¯11n 2 R3h , n¯1 = A¯11 32pim2 9R3hL 3 Rn − 16n ( A¯12 − A¯66 ) 9piR3hLR , n¯2 = ( A¯11 − B¯11 Rh ) 32n2 9pi2mR3h + ( A¯66 − A¯12 − B¯66 − B¯12 Rh ) 16m 9R3hL 2 R , n¯3 = 32mn(B¯12 − B¯66) 3R4hL 2 R − 16mA¯12 3nR3hL 2 R − 16nA¯12 3mpi2R3h , n¯4 = 9A¯11 32 ( pi4m4 R4hL 4 R + n4 R4h ) + ( A¯12 + 2A¯66 ) pi2m2n2 16R4hL 2 R , n¯5 = −A¯11 32pim2 9nR3hL 3 R − 32n ( A¯12 − A¯66 ) 9piR3hLR , n¯6 = ( A¯66 − A¯12 + B¯12 − B¯66 Rh ) 32m 9R3hL 2 R + ( B¯11 Rh − A¯11 ) 32n2 9pi2mR3h (29) and LR = L/R , W = W/h ,[ A¯11, A¯12, A¯66 ] = [A11, A12, A66] h , [ B¯11, B¯12, B¯66 ] = [B11, B12, B66] h2 , [ D¯11, D¯12, D¯66 ] = [D11, D12, D66] h3 (30) Postbuckling of functionally graded cylindrical shells based on improved Donnell equations 11 Eq. (27) is explicit expression of external pressure-deflection curves accounting for pre-existent edge compressive and thermal loads. It is predicted that due to the presence of temperature conditions FGM cylindrical shells experience a bifurcation-type buckling behavior with buckling pressure qb = I∆T/ (Rh(1− ν)) (I is replaced by H in case of thermal gradient) which is independent of buckling mode (postbuckling behavior, however, is sensitive to buckling mode). In contrast, in the absence of the temperature and edge compressive force the q(W ) curves originate from coordinate origin and the shell undergoes bending at the onset of loading. In a particular case which the cylindrical shell is only subjected to axial compression, Eq. (27) leads to Px = R2hL 2 R pi2m2 ( a1 + a2W + a3W 2 ) (31) from which bifurcation compressive load Pxb is determined as Pxb = a1R 2 hL 2 R pi2m2 (32) whereas lower buckling compressive load may be obtained at W 0 = −a2/(2a3) as Pxl = Px(W 0) = R2hL 2 R pi2m2 ( a1 − a22 4a3 ) (33) and the intensity of well-known snap-through of compressed cylindrical shells is measured by difference between bifurcation and lower buckling loads, i.e. by a22R 2 hL 2 R/(4a3pi 2m2). 5. RESULTS AND DISCUSSION As part of the validation of the present approach, the buckling behavior of an isotropic thin cylindrical shell under uniform axial compressive load is analyzed, which was considered by Brush and Almroth [1] using adjacent equilibrium criterion and Don- nell shallow shell theory. The dimensionless buckling axial compressive loads of a simply supported cylindrical shell are compared in Tab. 1 with result of Ref. [1]. As can be seen, a good agreement is achieved in this comparison study. Brush and Almroth’s results are slightly higher than our results because the shallow shell theory, instead of improved theory, was used in their work. Table 1. Comparison of buckling loads Pxcr×10 3/E for simply supported isotropic perfect cylindrical shell under axial compression (ν = 0.3). R/h L/R = 0.5 L/R = 1.0 L/R = 1.5 100 150 100 150 100 150 Present 6.033(1,9)e 4.043(3,9) 5.954(1,7) 4.018(3,11) 6.033(3,9) 4.043(9,9) Ref. [1] 6.087 4.047 6.063 4.035 6.087 4.047 e The numbers in brackets indicate the buckling mode (m, n) 12 Dao Huy Bich, Nguyen Xuan Nguyen, Hoang Van Tung To illustrate the proposed approach, we consider a ceramic-metal functionally graded cylindrical shell that consists of aluminum and alumina with the following properties Em = 70 GPa , αm = 23× 10 −6 ◦C−1 , Km = 204 W/mK Ec = 380 GPa , αc = 7.4× 10 −6 ◦C−1 , Kc = 10.4 W/mK (34) whereas Poisson’s ratio is chosen to be 0.3. Table 2. Critical buckling compressive loads Pxcr (in GPa) for FGM cylindrical shells, R/h = 100. k L/R 1.0 2.0 3.0 6.0 0 2.262(1,7)e 2.229(1,5) 2.262(3,7) 2.079(1,3) 0.5 1.554 1.545 1.554 1.445 1.0 1.230 1.228 1.230 1.151 5.0 0.736 0.723 0.736 0.674 e The numbers in brackets indicate the buckling mode (m, n) Tab. 2 considers the effects of volume fraction index k and L/R ratio on critical buckling loads Pxcr of FGM cylindrical shells under axial compression. As expected, the critical values of buckling loads are decreased when k increases due to drop in the volume percentage of ceramic constituent. It is also seen that critical loads are not always decreased when L/R increases. Fig. 2 gives the effects of k on the postbuckling behavior of FGM cylindrical shells under axial compression. As can be seen, both buckling compressive loads and postbuckling load carrying capacity of cylindrical shells are reduced when k is increased. However, the increase in buckling loads and postbuckling strength is paid by a more severe snap-through phenomenon, i.e. a bigger difference between bifurcation and lower buckling loads and curves become more unstable. Fig. 3 shows the effects of L/R ratio on the postbuckling of FGM cylindrical shells under axial compression. Although there is not much change of bifurcation point loads, buckling modes and postbuckling curves are considerably varied due to the variation of L/R ratio. Specifically, both number of waves in the circumferential direction and post- buckling bearing capability of shells are reduced when L/R is enhanced. In addition, the increase in L/R is accompanied by an unstable postbuckling behavior, i.e. a more severe snap-through response. Figs. 4 and 5 illustrate the effects of buckling mode and pre-existent axial com- pressive load on the nonlinear response of FGM cylindrical shells subjected to uniform external pressure. As can be observed, for a specific buckling mode nonlinear equilibrium paths become lower and the intensity of snap-through is enhanced for higher values of Postbuckling of functionally graded cylindrical shells based on improved Donnell equations 13 0 0.5 1 1.5 20 1 2 3 4 5 6 W/h P x (GPa) k = 0 k = 1 k = 5 L/R = 2.0, R/h = 100, (m,n) = (1,5) 0 0.5 1 1.5 20 1 2 3 4 5 6 W/h P x (GPa) k = 0 k = 1 k = 5 L/R = 2.0, R/h = 100, (m,n) = (1,5) Fig. 2. Effects of k on the postbuckling be- havior of FGM cylindrical shells under axial compression Fig. 3. Effects of L/R ratio on the postbuck- ling behavior of FGM cylindrical shells un- der axial compression pre-existent axial compressive load. Furthermore, the cylindrical shells carry better exter- nal pressure and the nonlinear response to be more benign as the number of waves in the circumferential direction increases. 0 1 2 3 40 0.5 1 1.5 2 2.5 3 x 10 −3 W/h q (GPa) 1 2 3 L/R = 2.0, R/h = 100, k = 1.0, (m,n) = (1,3) 1: P x = 0 2: P x = 0.5 GPa 3: P x = 1.0 GPa 0 0.5 1 1.5 2 2.5−2 0 2 4 6 8 10 12 x 10 −3 W/h q (GPa) 1 2 3 L/R = 2.0, R/h = 100, k = 1.0, (m,n) = (1,5) 1: P x = 0 2: P x = 0.5 GPa 3: P x = 1.0 GPa Fig. 4. Effects of pre-existent compressive load on the nonlinear response of FGM cylindrical shells under external pressure Fig. 5. Counterpart of Fig. 4 for case of n = 5 Figs. 6 and 7 depict the effects of environment temperature and through the thick- ness temperature gradient on the nonlinear response of FGM cylindrical shells under uni- form external pressure in the presence of pre-existent axial compressive load. As mentioned above, due to thermal loading conditions, FGM cylindrical shells experience a bifurcation type buckling behavior. The increase in thermal loads is followed by both higher bifurca- tion point pressure and more severe snap-through behavior. It is interesting to note that all pressure-deflection curves go across a point for various values of temperature differ- ence ∆T . This behavior trend of FGM shells is similar to the nonlinear response of FGM 14 Dao Huy Bich, Nguyen Xuan Nguyen, Hoang Van Tung cylindrical panels subjected to simultaneous action of external pressure and thermal loads presented in [21]. 0 1 2 3 4−0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 W/h q (GPa) L/R = 2.0, R/h = 100,k = 1.0, 1: ∆T = 0 2: ∆T = 200oC 3: ∆T = 400oC 1 1 2 2 3 3 P x = 1.0 GPa, (m,n) = (1,5) 0 1 2 3 4−0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 W/h q (GPa) L/R = 2.0, R/h = 100, k = 1.0 P x = 1.0 GPa, (m,n) = (1,5), T m = 27oC 1: T c = 27oC 2: T c = 400oC 3: T c = 800oC 3 3 2 2 1 1 Fig. 6. Effects of the environment temper- ature on the nonlinear response of FGM cylindrical shells under external pressure Fig. 7. Effects of the temperature gradient on the nonlinear response of FGM cylindri- cal shells under external pressure 6. CONCLUDING REMARKS This paper presents an analytical approach to investigate buckling and postbuckling behaviors of FGM circular cylindrical shells subjected to axial compressive load, uniform external pressure accounting for the effects of temperature conditions. Equilibrium equa- tions are established within the framework of improved Donnell shell theory taking into account the nonshallowness of cylindrical shell and geometrical nonlinearity. One-term approximate solution satisfying simply supported boundary conditions is assumed and ex- plicit expressions of buckling loads and postbuckling load-deflection curves are determined by using Galerkin method. The study shows that buckling loads and postbuckling behavior of FGM cylindrical shells are greatly influenced by material and geometrical parameters and temperature conditions. The results also reveal that buckling mode and pre-existent axial compressive load have significant effects on the nonlinear response of the shells. The improved theory should be used to predict the nonlinear behavior of nonshallow cylindrical shells. ACKNOWLEDGEMENT This paper was supported by National Foundation for Science and Technology De- velopment of Vietnam - NAFOSTED. The authors are grateful for this support. REFERENCES [1] D.O. Brush, B.O. Almroth, Buckling of Bars, Plates and Shells, McGraw-Hill, New York, 1975. [2] V. Birman, C.W. Bert, Dynamic stability of reinforced composite cylindrical shells in thermal fields,J. Sound and Vibration 142(2) (1990) 183-190. Postbuckling of functionally graded cylindrical shells based on improved Donnell equations 15 [3] V. Birman, C. W. 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R. Eslami, On thermal dynamic buckling analysis of imperfect laminated cylindrical shells, ZAMM 80(3) (2000) 171-182. [21] N.D. Duc, H. V. Tung, Nonlinear response of pressure-loaded functionally graded cylindrical panels with temperature effects, Compos. Struct. 92 (2010) 1664-1672. Received January 08, 2012 VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY VIETNAM JOURNAL OF MECHANICS VOLUME 35, N. 1, 2013 CONTENTS Pages 1. Dao Huy Bich, Nguyen Xuan Nguyen, Hoang Van Tung, Postbuckling of functionally graded cylindrical shells based on improved Donnell equations. 1 2. Bui Thi Hien, Tran Ich Thinh, Nguyen Manh Cuong, Numerical analysis of free vibration of cross-ply thick laminated composite cylindrical shells by continuous element method. 17 3. Tran Ich Thinh, Bui Van Binh, Tran Minh Tu, Static and dynamic analyses of stiffended folded laminate composite plate. 31 4. Nguyen Dinh Kien, Trinh Thanh Huong, Le Thi Ha, A co-rotational beam element for geometrically nonlinear analysis of plane frames. 51 5. Nguyen Chien Thang, Qian Xudong, Ton That Hoang Lan, Fatigue perfor- mance of tubular X-joints: Numberical investigation. 67 6. Hoang H. Truong, Chien H. Thai, H. Nguyen-Xuan, Isogeometric analysis of two–dimensional piezoelectric structures. 79 7. Pham Chi Vinh, Do Xuan Tung, Explicit homogenized equations of the piezo- electricity theory in a two-dimensional domain with a very rough interface of comb-type. 93