Nonlinear post-Buckling of thin fgm annular spherical shells under mechanical loads and resting on elastic foundations

Volume 36 Number 4 4 2014 Vietnam Journal of Mechanics, VAST, Vol. 36, No. 4 (2014), pp. 291 – 306 NONLINEAR POST-BUCKLING OF THIN FGM ANNULAR SPHERICAL SHELLS UNDER MECHANICAL LOADS AND RESTING ON ELASTIC FOUNDATIONS Vu Thi Thuy Anh, Dao Huy Bich, Nguyen Dinh Duc∗ Vietnam National University, Hanoi, Viet Nam ∗E-mail: ducnd@vnu.edu.vn Received July 27, 2014 Abstract. This paper presents an analytical approach to investigate the nonlinear buck- ling and post-buckling of thin annular spherical shells made of functionally graded ma- terials (FGM) and subjected to mechanical load and resting on Winkler-Pasternak type elastic foundations. Material properties are graded in the thickness direction accord- ing to a simple power law distribution in terms of the volume fractions of constituents. Equilibrium and compatibility equations for annular spherical shells are derived by us- ing the classical thin shell theory in terms of the shell deflection and the stress function. Approximate analytical solutions are assumed to satisfy simply supported boundary con- ditions and Galerkin method is applied to obtain closed-form of load-deflection paths. An analysis is carried out to show the effects of material and geometrical properties and combination of loads on the stability of the annular spherical shells. Keywords: Nonlinear buckling, post-buckling, FGM annular spherical shells, mechanical loads, elastic foundations. 1. INTRODUCTION Considerable researches have focused on the thermo-elastic, dynamic and buckling analyses of functionally graded plates and shells in recent years. This is mainly due to the increased use of functionally graded materials (FGMs) as the components of structures in the advanced engineering. FGMs consisting of metal and ceramic constituents have received increasingly attention in structural applications. Smooth and continuous change in material properties enable FGMs to avoid interface problems and unexpected thermal stress concentrations. By high performance heat resistance capacity, FGMs are now chosen to use as structural components exposed to severe temperature conditions such as aircraft, aerospace structures, nuclear plants and other engineering applications. On the other hand, the static and dynamic interaction of shells with elastic foun- dation is a problem of current importance. By using the theory of elasticity and theory of shells, have many approaches to analyze the interaction between a structure and an ambient medium. One of them is approach, which most earthen soils can be appropriately 292 Vu Thi Thuy Anh, Dao Huy Bich, Nguyen Dinh Duc represented by a mathematical model from Pasternak, whereas sandy soils and liquids can be represented by Winkler’s model ([1–3]). As a result, the mechanical behavior of FGM shells, such as bending, vibration, stability, buckling, etc., have also been studied by many scientists. Notably among them are due to [4–10]. Recently, Duc et al. [11] investigated nonlinear axisymmetric response of FGM shallow spherical shells on elastic foundations, namely the Winkler-Pasternak foundations. Cheng and Batra [12] proposed a membrane analogy to derive an exact explicit eigenvalues for compression buckling, hydrothermal buckling, and vibration of FGM plates on a Winkler-Pasternak foundation based on the third-order plate theory. The free vibration, transient response, large deflection and post-buckling responses of FGM thin plates resting on Pasternak foundations were investigated by Yang and Shen [13] was using the method of differential quadrature and Galerkin procedure. Huang et al. [14] investigated Benchmark solutions for functionally graded thick plates resting onWinkler- Pasternak elastic foundations. Ying et al. [15] obtained two-dimensional elasticity solutions for functionally graded beams resting on elastic foundations. Sheng andWang [16] analyzed thermal vibration, buckling and dynamic stability of functionally graded cylindrical shells embedded in an elastic medium. Sofiyev et al. investigated the vibration analysis of simply supported FGM truncated conical shells resting on the two-parameter elastic foundation [17], and [18] studied the buckling of the homogenous and FGM truncated conical shells resting on Winkler-Pasternak type elastic foundations. Shen [19] studied post-buckling of shear deformable FGM cylindrical shells surrounded by an elastic medium. Nonlinear free vibration response, static response under uniformly distributed load, and the maximum transient response under uniformly distributed step load of orthotropic thin spherical caps on elastic foundation have been obtained by Dumir [20]. The annular spherical shell is one of the special shapes of the spherical shells. To the best of our knowledge, there has been recently a few of publications on the annular shells. Alwar and Narasimhan [21] investigated the axisymmetric nonlinear analysis of laminated orthotropic annular spherical shell, the object of this investigation is to give analytical solutions of large axisymmetric deformation of laminated orthotropic spherical shells including asymmetric laminates. Dumir et al. [22] analyzed axisymmetric dynamic buckling analysis of laminated moderately thick shallow annular spherical cap under cen- tral ring load and uniformly distributed transverse load, applied statically or dynamically as a step function load. Eslami et al. [23] studied an exact solution for thermal buckling of annular FGM plates on an elastic medium and generalized coupled thermo-elasticity of functionally graded annular disk considering the Lord-Shulman theory [24]. In this study, by using the classical thin shell theory, an approximate solution, which was proposed by Agamirov [25] and was used by Sofiyev [18] for truncated conical shells, the authors tried to apply this form to solve problems related to annular FGM spherical shells. This study is one of the first attempts about the nonlinear postbuckling analysis of thin FGM annular spherical shells under mechanical loads and resting on Winkler- Pasternak type elastic foundations. Nonlinear post-buckling of thin FGM annular spherical shells under mechanical loads and ... 293 2. GOVERNING EQUATIONS Consider an annular spherical shell is subjected to external pressure q uniformly distributed on the outer surface as shown in Fig. 1. R is the radius of curvature, r1, r0 indicate the radii and h is thickness of annular. 3 of laminated orthotropic annular spherical shell, the object of this investigation is to give analytical solutions of large axisymmetric deformation of laminated orthotropic spherical shells including asymmetric laminates. Dumir et al. [22]analyzed axisymmetric dynamic buckling analysis of laminated moderately thick shallow annular spherical cap under central ring load and uniformly distributed transverse load, applied statically or dynamically as a step function load. Eslami et al. [23] studied an exact solution for thermal buckling of annular FGM plates on an elastic mediumand generalized coupled thermo-elasticity of functionally graded annular disk considering the Lord – Shulman theory [24]. In this study, by using the classical thin shell theory, an approximate solution, which was proposed by Agamirov [25] and was used by Sofiyev [18] for truncated conical shells, the authors tried to apply this form to solve problems re ated to annular FGM spherical shells. This study is one of the first attempts about the nonlinear postbuckling analysis of thin FGM annular spherical shells under mechanical loads and resting on Winkler-Pasternak type elastic foundations. 2. Governing equations Fig. 1. Configuration of a FGM annular spherical shell Consider an annular spherical shell is subjected to external pressure q uniformly distributed on the outer surface as shown in Fig.1. R is the radius of curvature, 1 0,r r indicate the radii and h is thickness of annular. 2.1. Material properties of functionally graded shells The annular spherical shell is made of FGM, from a mixture of ceramics and metals, and is defined in coordinate system ( , ,z)  , where  and  are in the Fig. 1. Configuration of a FGM annular spherical shell 2.1. Material properties of functionally graded shells The annular spherical shell is made of FGM, from a mixture of ceramics and metals, and is defined in coordina system (ϕ, θ, z), where ϕ and θ are in the meridional and circumferential direction of the shells, respectively and z is perpendicular to the middle surface positive inwards. Suppose that the material composition of the shell varies smoothly along the thick- ness by a simple power law in terms of the volume fractions of the constituents as Vc(z) = ( 2z + h 2h )k, −h 2 ≤ z ≤ h 2 , Vm(z) = 1− Vc(z), (1) where k (volume fraction index) is a non-negative number that defines the material dis- tribution, subscripts m and c represent the metal and ceramic constituents, respectively. The modulus of elasticity E of FGM annular spherical shell can be defined as E(z) = Em + Ecm( 2z + h 2h )k, −h 2 ≤ z ≤ h 2 , (2) where the Poisson ratio ν is assumed to be constant ν(z) = const and Ecm = Ec − Em. In the present study, the classical shell theory is used to obtain the equilibrium and compatibility equations as well as expressions of buckling loads and nonlinear load- deflection curves of thin FGM annular spherical shells. For a thin annular spherical shell it is convenient to introduce a variable r, referred as the radius of parallel circle with the base of shell and defined by r = R sinϕ. Moreover, due to shallowness of the shell it is approximately assumed that cosϕ = 1, Rdϕ = dr. 294 Vu Thi Thuy Anh, Dao Huy Bich, Nguyen Dinh Duc 2.2. Winkler-Pasternak type elastic foundations The annular spherical shell is resting on the elastic foundations. For the elastic foundation, one assumes the two-parameter elastic foundation model proposed by Paster- nak [1]. The foundation medium is assumed to be linear, homogenous and isotropic. The bonding between the annular spherical shell and the foundation is perfect and frictionless. If the effects of damping and inertia force in the foundation are neglected, the foundation interface pressure may be expressed as qe = k1w − k2∆w, where ∆w = ∂2w ∂r2 + 1 r ∂w ∂r + 1 r2 ∂2w ∂θ2 , w is the deflection of the annular spherical shell, k1 is Winkler foundation modulus and k2 is the shear layer foundation stiffness of Pasternak model. 2.3. Fundamental relations and governing equations According to the classical shell theory, the strains at the middle surface and the change of curvatures and twist are related to the displacement components u, v, w in the ϕ, θ, z coordinate directions, respectively, taking into account Von Karman-Donnell nonlinear terms as [23] ε0r = ∂u ∂r − w R + 1 2 ( ∂w ∂r )2, χr = ∂2w ∂r2 , ε0θ = 1 r ∂v ∂θ + u r − w R + 1 2r2 ( ∂w ∂θ )2, χθ = 1 r ∂w ∂r + 1 r2 ∂2w ∂θ2 , γ0rθ = ∂v ∂r + 1 r ∂u ∂θ − v r + 1 r ∂w ∂r ∂w ∂θ , χrθ = 1 r ∂2w ∂r∂θ − 1 r2 ∂w ∂θ , (3) where ε0r and ε 0 θ are the normal strains, γ θ r is the shear strain at the middle surface of the spherical shell, χr, χθ, χrθ are the changes of curvatures and twist. The strains across the shell thickness at a distance z from the mid-plane are εr = ε 0 r − zχr, εθ = ε0θ − zχθ, γ0rθ = γθr − zχrθ. (4) Using Eqs. (3) and (4), the geometrical compatibility equation of an shallow spher- ical shell is written as [10] 1 r2 ∂2ε0r ∂θ2 − 1 r ∂ε0r ∂r + 1 r2 ∂ ∂r (r2 ∂ε0θ ∂r )− 1 r2 ∂2 ∂r∂θ (rγ0rθ) = − ∆w r + χ2rθ − χrχθ, (5) where ∆ = ∂2 ∂r2 + 1 r ∂ ∂r + 1 r2 ∂2 ∂θ2 is a Laplace’s operator. The stress-strain relationships are defined by the Hooke law (σr, σθ) = E(z) 1− ν2 [(εr, εθ) + ν (εθ, εr)] , σrθ = E(z) 2(1 + ν) γrθ. (6) Nonlinear post-buckling of thin FGM annular spherical shells under mechanical loads and ... 295 The force and moment resultants of an FGM spherical shell are expressed in terms of the stress components through the thickness as (Nij ,Mij) = h/2∫ −h/2 σij(1, z)dz, ij = (rr, θθ, rθ) (7) In case of (i = j = r) or (i = j = θ) for simplicity denoted Nrr = Nr, Nθθ = Nθ, Mrr = Mr,Mθθ = Mθ. By using Eqs. (4), (6), and (7) the constitutive relations can be given as (Nr, Mr) = (E1, E2) 1− ν2 ( ε0r + νε 0 θ )− (E2, E3) 1− ν2 (χr + νχθ) , (Nθ, Mθ) = (E1, E2) 1− ν2 ( ε0θ + νε 0 r )− (E2, E3) 1− ν2 (χθ + νχr) , (Nrθ, Mrθ) = (E1, E2) 2(1 + ν) γ0rθ − (E2, E3) 1 + ν χrθ. (8) From the relations one can write ε0r = 1 E1 (Nr − νNθ) + E2 E1 χr, ε 0 θ = 1 E1 (Nθ − νNr) + E2 E1 χθ, γ0rθ = 2(1 + ν) E1 Nrθ + 2E2 E1 χrθ, (9) Mr = E2 E1 Nr −D(χr + νχθ),Mθ = E2 E1 Nθ −D(χθ + νχr), Mrθ = E2 E1 Nrθ −D(1− ν)χrθ. (10) where D = E1E3 − E22 E1(1− ν2) , E1 = h/2∫ −h/2 [Ec + Ecm( 2z + h h )k]dz = ( Em + Ecm k + 1 ) h, E2 = h/2∫ −h/2 z[Ec + Ecm( 2z + h h )k]dz = h2Ecm( 1 k + 2 − 1 2k + 2 ), E3 = h/2∫ −h/2 z2[Ec + Ecm( 2z + h h )k]dz = ( Em 12 + Ecm 2(k + 1)(k + 2)(k + 3) ) h3. (11) 296 Vu Thi Thuy Anh, Dao Huy Bich, Nguyen Dinh Duc The nonlinear equilibrium equations of a perfect shallow spherical shell based on the classical shell theory [11] ∂Nr ∂r + 1 r ∂Nrθ ∂θ + Nr r − Nθ r = 0, (12) ∂Nθ r∂θ + ∂Nrθ ∂r + 2Nrθ r = 0, (13) ∂2Mr ∂r2 + 2 r ∂Mr ∂r + 2( ∂2Mrθ r∂r∂θ + 1 r2 ∂Mrθ ∂θ ) + 1 r2 ∂2Mθ ∂θ2 − 1 r ∂Mθ ∂r + 1 R (Nr +Nθ)+ + 1 r ∂ ∂r (rNr ∂w ∂r +Nrθ ∂w ∂θ ) + 1 r ∂ ∂θ (Nrθ ∂w ∂r + Nθ r ∂w ∂θ ) + q − k1w + k2∆w = 0. (14) The Eqs. (12), (13) are identically satisfied by introducing a stress function F as Nr = 1 r ∂F ∂r + 1 r2 ∂2F ∂θ2 , Nθ = ∂2F ∂r2 , Nrθ = −1 r ∂2F ∂r∂θ + 1 r2 ∂F ∂θ . (15) Substituting Eqs. (3), (9), (15) into the Eqs. (5) and substituting Eqs. (3), (10), (15) into Eq. (14) leads to 1 E1 ∆∆F = −∆w R + ( 1 r ∂2w ∂r∂θ − 1 r2 ∂w ∂θ )2 − ∂ 2w ∂r2 ( 1 r ∂w ∂r + 1 r2 ∂2w ∂θ2 ), (16) D∆∆w − ∆F R − (1 r ∂F ∂r + 1 r2 ∂2F ∂2θ ) ∂2w ∂r2 − (1 r ∂w ∂r + 1 r2 ∂2w ∂θ2 ) ∂2F ∂r2 +2( 1 r ∂2F ∂r∂θ − 1 r2 ∂F ∂θ )( 1 r ∂2w ∂r∂θ − 1 r2 ∂w ∂θ ) = q − k1w + k2∆w. (17) Regularly, the stress function F should be determined by the substitution of deflec- tion function w into compatibility equation (16) and solving resulting equation. However, such a procedure is very complicated in mathematical treatment because obtained equa- tion is a variable coefficient partial differential equation. Accordingly, integration to obtain exact stress function F (r, θ) is extremely complex. Similarly, the problem of solving the equilibrium is in the same situation. Therefore one should find a transformation to lead Eqs. (16), (17) into constant coefficient differential equations. Suppose such a transforma- tion w = w(ς), F = F0(ς)e 2ς , where r = r0eς , ς = ln r r0 . (18) Substituting Eq. (18) into Eqs. (16), (17) and establishing a lot of calculations lead to the transformed equations 1 E1 ( ∂4F0 ∂ς4 + 4 ∂3F0 ∂ς3 + 4 ∂2F0 ∂ς2 + 4 ∂3F0 ∂ς∂θ2 + 2 ∂4F0 ∂ς2∂θ2 + 4 ∂2F0 ∂θ2 + ∂4F0 ∂θ4 ) = −r 2 0 R ( ∂2w ∂ς2 + ∂2w ∂θ2 ) + 1 e4ς ( ∂2w ∂ς∂θ − ∂w ∂θ )2 + 1 e4ς ( ∂2w ∂ς2 − ∂w ∂ς )( ∂w ∂ς + ∂2w ∂θ2 ), (19) Nonlinear post-buckling of thin FGM annular spherical shells under mechanical loads and ... 297 D( ∂4w ∂ς4 − 4∂ 3w ∂ς3 + 4 ∂2w ∂ς2 − 4 ∂ 3w ∂ς∂θ2 + 2 ∂4w ∂ς2∂θ2 + 4 ∂2w ∂θ2 + ∂4w ∂θ4 ) − r 2 0e 4ς R ( ∂2F0 ∂ς2 + 4 ∂F0 ∂ς + 4F0 + ∂2F0 ∂θ2 )− (∂F0 ∂ς + 2F0 + ∂2F0 ∂θ2 )( ∂2w ∂ς2 − ∂w ∂ς )e2ς − (∂ 2F0 ∂ς2 + 2F0 + 3 ∂F0 ∂ς )( ∂w ∂ς + ∂2w ∂ς∂θ )e2ς + 2( ∂2F0 ∂ς∂θ + ∂F0 ∂θ )( ∂2w ∂ς∂θ − ∂w ∂θ )e2ς − qr40e4ς + k1wr40e4ς − k2( ∂2w ∂ς2 + ∂2w ∂θ2 )r20e 2ς = 0. (20) Eqs. (19) and (20) are the basic equations used to investigate the nonlinear buckling and postbuckling of FGM annular spherical shells resting on elastic foundations. These are nonlinear equations in terms of two dependent unknowns w and F . 3. STABILITY ANALYSIS In this section, the FGM annular spherical shell is assumed to be simply supported along the periphery and subjected to mechanical loads uniformly distributed on the outer surface and the base edges of the shell. Depending on the in-plane behavior at the edge of boundary conditions will be considered in case the edges are simply supported and immovable. For this case, the boundary conditions are u = 0, w = 0, ∂2w ∂ς2 − ∂w ∂ς = 0, Nr = N0, Nrθ = 0, with ς = 0 (i.e at r = r0), (21) where N0 is the fictitious compressive load rendering the immovable edges. The boundary conditions (21) can be satisfied when the deflection w is approxi- mately assumed as follows [18,25] w = Weς sin(β1ς) sin(nθ), β1 = mpi a , a = ln r1 r0 , (22) where W is the maximum amplitude of deflection and m,n are the numbers of half waves in meridional and circumferential direction, respectively. The form of this approximate solution was proposed by Agamirov [25] and it was used by Sofiyev [18] for truncated conical shells. Introduction of Eqs. (22) into Eq. (19) gives 1 E1 ( ∂4F0 ∂ς4 + 4 ∂3F0 ∂ς3 + 4 ∂2F0 ∂ς2 + 4 ∂3F0 ∂ς∂θ2 + 2 ∂4F0 ∂ς2∂θ2 + 4 ∂2F0 ∂θ2 + ∂4F0 ∂θ4 ) = = −r 2 0e ςW R [(1− β21 − n2) sin(β1ς) + 2β1 cos(β1ς)] sin(nθ) + W 2β21 2 (n2 − 1) cos(2β1ς) − W 2 4 (β1 − β1n− β31) sin(2β1ς) + W 2 2 β21n 2 cos(2nθ) + W 2 4 (β1 − β1n2 − β31) sin(2β1ς) cos(2nθ) + W 2 2 β21 cos(2β1ς) cos(2nθ). (23) 298 Vu Thi Thuy Anh, Dao Huy Bich, Nguyen Dinh Duc Solving this obtained equation with the boundary conditions (21) for the stress function F0 yields F0 = f1e ς sin(β1ς) sin(nθ) + f2e ς cos(β1ς) sin(nθ) + f3 cos(2β1ς) + f4 cos(2nθ) +f5 cos(2β1ς) cos(2β2θ) + f6 sin(2β1ς) cos(2nθ) + f7 sin(2β1ς) + 1/2N0r 2 0, (24) with f1 = −r 2 0E1W [A(1− β21 − n2) + 2Bβ1] (A2 +B2)R , f2 = −r 2 0E1W [B(1− β21 − n2) + 2Aβ1] (A2 +B2)R , f3 = E1l3W 2, f4 = E1l4W 2, f5 = E1l4W 2, f6 = E1l6W 2, f7 = E1l7W 2, (25) and A = 9− 22β21 + β41 + n4 − 10n2 + 2β21n2, B = 8β31 − 24β1 + 8β1n2, l3 = b6c3 + c4a6 a3b6 + a6b3 , l4 = β21 32(β22 − 1) , l5 = b5c5 + a5c6 a4b5 + a5b4 , l6 = a4c6 − b4c5 a4b5 + a5b4 , l7 = a3c4 − b3c3 a3b6 + a6b3 . (26) (the remaining constants are given in Appendix I) Applying Galerkin method with the limits of integral is given by the formula ln r1 r0∫ 0 2pi∫ 0 Φeς sin(β1ς) sin(nθ)dςdθ = 0, (27) where Φ is the left hand side of Eq. (20) after substitution Eqs. (22) and (24) in it. From Eq. (27) we obtain the following equation q − N0A5 B1r20 W − A0N0 RB1 = ( DA3 B1r40 + E1A4 B1R2 + k1A6 B1 + k2A7 B1r20 ) W + E1A2 RB1r20 W 2 + E1A1 B1r40 W 3, (28) where the constants B1, A0, A1, A2, A3, A4, A5 are given in Appendix II. Eq. (28) is used to determine the buckling loads and nonlinear equilibrium paths of FGM annular spherical shell under mechanical loads including the effect of elastic foundations. 3.1. Case N0 = 0 Eq. (28) reduces to q = ( D∗R4hA3 B1R40 + E∗1A4R 2 h B1 + K1D ∗A6 B1 + K2D ∗A7R2h B1R20 ) W ∗ + E∗1A2R 3 h B1R20 (W ∗)2 + E∗1A1R 4 h B1R40 (W ∗)3, (29) by putting E∗1 = E1 h ,E∗2 = E2 h2 ,W ∗ = W h ,Rh = h R ,R0 = r0 R ,D∗ = D h3 ,K1 = k1h 4 D ,K2 = k2h 2 D . If the FGM annular spherical shell does not rest on elastic foundations, we received q = ( D∗R4hA3 B1R40 + E∗1A4R2h B1 ) W ∗ + E∗1A2R3h B1R20 (W ∗)2 + E∗1A1R4h B1R40 (W ∗)3. (30) Nonlinear post-buckling of thin FGM annular spherical shells under mechanical loads and ... 299 Eqs. (29) and (30) may be used to find static critical buckling load and trace postbuckling load-deflection curves of FGM annular spherical shell. It is evident q(W ∗) curves originate from the coordinate origin. In addition, Eq. (29) indicates that there is no bifurcation-type buckling for pressure loaded annular spherical shell and extremum-type buckling only occurs under definite conditions. The extremum pressure buckling load of the shell can be found from Eqs. (29) and (30) using the condition dq dW ∗ = 0. 3.2. Case N0 6= 0 A FGM annular spherical shell simply supported with immovable edges under si- multaneous action of uniform external pressure q is considered. The condition expressing the immovability on the boundary edges, i.e. u = 0 on r = r0, and r = r1 is fulfilled on the average sense as pi∫ 0 r1∫ r0 ∂u ∂r rdrdθ = 0. (31) From Eqs. (3) and (9) one can obtain the following relation ∂u ∂r = ε0r + w R − 1 2 ( ∂w ∂r )2 = 1 E1 ( 1 r ∂F ∂r + 1 r2 ∂2F ∂θ2 − ν ∂ 2F ∂r2 ) + E2 E1 ∂2w ∂r2 + w R − 1 2 ( ∂w ∂r )2. (32) Using the transformation (18) into Eq. (32) yields ∂u ∂r = 1 E1r20 [ ∂F0 ∂ς + 2F0 + ∂2F0 ∂θ2 − v ( ∂2F0 ∂ς2 + 3 ∂F0 ∂ς + 2F0 )] + E2 E1r20e 2ς ( ∂2w ∂ς2 − ∂w ∂ς ) + w R − 1 2r20e 2ς ( ∂w ∂ς )2 . (33) Introduction of Eqs. (22), (24) into the Eq. (33) then substituting obtained result into Eq. (31) lead to the expression for the fictitious load N0 N0 = E1 (−1 + v) (−1 + e2a)pir20 [ E2A9 E1 + r20A10 R ] W + E1A8 (−1 + v) (−1 + e2a)pir20 W 2, (34) where the constants A8, A9, A10 are given in Appendix II. 4. RESULTS AND DISCUSSION In this section, the nonlinear response of the FGM annular spherical shell is analyzed. The shell is assumed to be simply supported along boundary edges. In characterizing the behavior of the spherical shell, deformations in which the central region of a shell moves toward the plane that contains the periphery of the shell are referred to as inward deflections (positive deflections). Deformations in the opposite direction are referred to as outward deflection (negative deflections). 300 Vu Thi Thuy Anh, Dao Huy Bich, Nguyen Dinh Duc The following properties of the FGM shell are chosen [11]: Em = 70GPa, Ec = 380GPa, Poisson’s ratio is chosen to be 0.3. The effects of material and geometric parameters on the nonlinear response of the FGM annular spherical shells under mechanical loads including the effects of elastic founda- tion are presented in Figs. 2-6. It is noted that in all figuresW/h denotes the dimensionless maximum deflection of the shell. 12 The following properties of the FGM shell are chosen[11]: 70 ; 380m cE GPa E GPa  , Poisson’s ratio is chosen to be 0.3 . The effects of material and geometric parameters on the nonlinear response of the FGM annular spherical shells under mechanical loads including the effects of elastic foundation are presented in Figs.2–6. It is noted that in all figures /W h denotes the dimensionless maximum deflection of the shell. Fig.2. shows the effects of volume fraction index (0,1,5, )k  on the nonlinear response of the FGM annular spherical shell subjected to external pressure and compressive load (mode ( , ) (1,11)m n  ). As c n be en, the load–deflection curves become lower when k increases. This is expected because the volume percentage of ceramic constituent, which has higher elasticity modulus, is dropped with increasing values of k . q(GPa) W/h k = 1 k =5 k=0 R/h=300, (m,n)=(1,11) r1=R/2; r0=R/30 k = 543210 -0.4 -0.2 0 0.2 0.4 0.6 1 2 0K K  W/h q(GPa) R/h=500 R/h=400 R/h=300 (m,n)=(1,11); k=1 r1=R/2; r0=R/30 R/h=200 543210 -0.5 0 0.5 1 1.5 1 2 0K K  Fig. 2. Effects of volume fraction index k on the load – deflection curves of the FGM annular spherical shell under external pressure. Fig. 3. Effects of curvature radius- thickness ratio on the nonlinear response of the FGM annular spherical shells under external pressure. Fig.3. depicts the effects of curvature radius - thickness ratio /R h (200, 300, 400, and 500) on the nonlinear behavior of the external pressure and compressive load of the FGM annular spherical shells (mode ( , ) (1,11)m n  ). From Fig. 3 we can conclude that when the annular spherical shells get thinner - corresponding with /R h getting bigger, the critical buckling loads will get smaller. Fig. . ffects of volume fraction index k on the load-deflection curves of the FGM annular spherical shell under external pressure 12 The following pro erties of the FGM shell are chosen[11]: 70 ; 380m cE GPa E GPa  , Poiss n’s ratio is chosen to be 0.3 . The effects of material and geometric parameters on the nonlinear response of the FGM annular spherical shells under mechanic l loads including the effects of elastic foundation are presented in Figs.2–6. It is noted that in all figures /W h denotes he dimensionless maximu deflection of the s ll. Fig.2. shows the effects of v lume fraction ndex (0,1,5, )k  on the nonlinear response of the FGM annular spherical shell subjected to external pressure and compressive load (mode ( , ) (1,11)m n  ). As can be se n, the load–deflection curves become lower when k increases. This is exp cted because th volume percentage of ceramic constituent, which has higher elasticity modul s, is dropped with increasing values of k . q(GPa) W/h k = 1 k =5 k=0 R/h=300, (m,n)=(1,11) r1=R/2; r0=R/30 k = 543210 -0.4 -0.2 0 0.2 0.4 0.6 1 2 0K  W/h q(GPa) R/h=500 R/h=400 R/h=300 (m,n)=(1,11); k=1 r1=R/2; r0=R/3 R/h=200 543210 -0.5 0 0.5 1 1.5 1 2 0K K Fig. 2. Effects of v lume fr ti n ndex k on the load – eflection curves of the FGM annular spherical shell under external pressure. Fig. 3. Effects of curvature radius- thickness ratio on the nonlinear r sponse of the FGM annular spherical shells under external pressure. Fig.3. depicts the effects of curvature radius - thickness ratio /R h (200, 3 0, 400, and 500) on the nonlinear behavior of the external pressure and compressive load of the FGM annular spherical shells (mode ( , ) (1,11)m n  ). From Fig. 3 we can con lude that when the annular spherical shells get thinner - corresponding with /R h getting bi ger, the criti al buckling loads will get smaller. Fig. 3. Effects of curvature r dius-thickness ra- tio on the nonlinear response of the FGM an- nular spherical shells under external pressure 13 Fig.4 analyzes the effects of 2 base-curvature radius ratio 1 0/r r on the nonlinear response of FGM annular spherical shells subjected to uniform external pressure. It is shown that the nonlinear response of annular spherical shells is very sensitive with change of 1 0/r r ratio characterizing the shallowness of annular spherical shell. Specifically, the enhancement of the upper buckling loads and the load carrying capacity in small range of deflection as 1 0/r r increases is followed by a very severe snap - through behaviors. In other words, in spite of possessing higher limit buckling loads, deeper spherical shells exhibit a very unstable response from the post-buckling point of view. Furthermore, in the same effects of base-curvature radius ratio 1 0/r r the load of the nonlinear response of FGM annular spherical shells is higher when the shallowness of annular spherical shell ( H ) is smaller, where H is the distance between two radii 1 0,r r , and calculated by 2 2 2 2 2 2 0 1 1 0 0 1( , ) 1 1 r H r r R r R r R R R                        I Ib IIa IIb IIIa IIIb W/h q(GPa) 543210 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 1 2 300;k 1 ( , ) (0,0) R h K K    Fig.4. Effects of radius of base-curvature radius ratio 1 0/r r on the nonlinear response of the FGM annular spherical shells. Figure 5 examines the dependence of the nonlinear response of FGM annular spherical shells on the mode ( , )m n . It is easily recognized that with 1m  , the more increased the value of n , the higher increasing of the value of extreme point, corresponding to the higher load capacity of the shells. Note that, when m is even or Fig. 4. Effects of radius of base-curvature radius ratio r1/r0 on the nonlinear response of the FGM annular spherical shells Fig. 2 shows the effects of volume fraction index k(0, 1, 5,+∞) on the nonlinear response of the FGM annular spherical shell subjected to external pre sure and compressive load (mode (m,n) = (1, 11)). As can be seen, the load-deflection curves become lower when Nonlinear post-buckling of thin FGM annular spherical shells under mechanical loads and ... 301 k increases. This is expected because the volume percentage of ceramic constituent, which has higher elasticity modulus, is dropped with increasing values of k. Fig. 3 depicts the effects of curvature radius-thickness ratio R/h (200, 300, 400, and 500) on the nonlinear behavior of the external pressure and compressive load of the FGM annular spherical shells (mode (m,n) = (1, 11)). From Fig. 3 we can conclude that when the annular spherical shells get thinner-corresponding with R/h getting bigger, the critical buckling loads will get smaller. Fig. 4 analyzes the effects of 2 base-curvature radius ratio r1/r0 on the nonlinear response of FGM annular spherical shells subjected to uniform external pressure. It is shown that the nonlinear response of annular spherical shells is very sensitive with change of r1/r0 ratio characterizing the shallowness of annular spherical shell. Specifically, the enhancement of the upper buckling loads and the load carrying capacity in small range of deflection as r1/r0 increases is followed by a very severe snap-through behaviors. In other words, in spite of possessing higher limit buckling loads, deeper spherical shells exhibit a very unstable response from the post-buckling point of view. Furthermore, in the same effects of base-curvature radius ratio r1/r0 the load of the nonlinear response of FGM annular spherical shells is higher when the shallowness of annular spherical shell (H) is smaller, where H is the distance between two radii r1, r0, and calculated by H(r1, r0) = √ R2 − r20 − √ R2 − r21 = R [√ 1− (r0 R )2 −√1− (r1 R )2] . 14 3m  , the graphic consists of symmetric curves through the origin of the coordinate system and the extreme point does not exist in the load-deflection curves. q(Gpa) W/h (m,n)=(1,9 ) (m,n)=(1,11) (m,n)=(3,3) (m,n)=(1,7) (m,n)=(2,3) (m,n)=(4,3) (m,n)=(2,5) 543210 -0.1 0 0.1 0.2 1 0 1 2 300;k 1 ; 2 30 ( , ) (0,0) R h R Rr r K K      q(Gpa) W/h R/h=300, (m,n)=(1,11) r1=R/2; r0=R/30 3210 0.4 0.3 0.2 0.1 0 -0.1 -0.2 4 1 2 3 4 1 2 1 2 1 2 1 2 1: ( , ) (0.0) 2 : ( , ) (100.0) 3: ( , ) (100.10) 4 : ( , ) (50.20) K K K K K K K K     Fig. 5. Effects of mode ( , )m n on the nonlinear response of the FGM annular spherical shells. Fig. 6. Effects of the elastic foundations 1 2( , )K K on the nonlinear response of the FGM annular spherical shells. Effects of the elastic foundations 1 2( , )K K on the nonlinear response of FGM annular spherical shells are shown in Fig 6. Obviously, elastic foundations played positive role on nonlinear static response of the FGM annular spherical shell: the large 1K and 2K coefficients are, the larger loading capacity of the shells is. It is clear that the elastic foundations can enhance the mechanical loading capacity for the FGM annular spherical shells, and the effect of Pasternak foundation 2K on critical uniform external pressure is larger than the Winkler foundation 1K . 5. Concluding remarks Due to practical importance of FGM annular spherical shells and the lack of investigations on stability of these structures, the present paper aims to propose an analytical approach to study the problem of nonlinear buckling and postbuckling of FGM thin annular spherical shells on elastic foundations. Based on the classical thin shell theory, the equilibrium and compatibility equations are derived in terms of the Fig. 5. Effects of mode (m,n) on the nonlinear response of the FGM annular spherical shells 3m  , the graphic consists of sy etri h the origin of the c ordinate sy tem and the xtreme point does not i -deflection curves. q(Gpa) W/h (m,n)=(1,9 ) (m,n)=(1, ) (m,n)=(3, ) (m,n)=(1,7) (m,n)=(2,3) (m,n)=(4,3) (m,n)=(2,5) 43210 -0.1 0 0.1 0.2 1 0 1 2 30 ;k 1 ; 2 30 ( , ) (0,0) R h R Rr r K K      ( pa) W/h R/h=300, (m,n)=( ,11) r1=R/2; r0=R/30 321 4 1 2 3 4 1 2 1 2 1 2 1 2 : ( , ) (0.0) : ( , ) (1 0.0) : ( , ) (1 0.10) : ( , ) (50.20) K K K K     Fig. 5. Effects of mode ( , )m n on the no linear esponse of the FGM annul r spherical shells. . ffects of the elastic foundations ) the nonlinear response of the annular spherical shells. Effects of the elastic foundatio s t e nonlinear response of FGM annular spherical shells are shown in i sly, elastic foundations played positive role on onlinear static respons nular spherical shell: the large 1K and 2K coefficients are, the larger loa i f the she ls is. It is clear that the elastic foundations can enhance the e i capacity for the FGM a nular spherical shells, and the ef ect of Paster 2 on critical uniform external pressure is larger than the Winkler fou 5. Concluding remarks Due to practical importance of erical shells and the lack of investigations on stability of these str t r sent paper aims to propose an an lytical ap roach to study the probl r buckling and postbuckling of FGM thin an ular spherical shel s on l ti s. Based on the classical thin shell theory, the equil brium and co ti ti s are derived in terms of the Fig. 6. Effects of the elastic foundations (K1,K2) on the nonlinear response of the FGM annular spherical shells Fig. 5 examines t dependence of the nonlinear respo se of FGM annular spherical shells on the mode (m,n). It is easily recognized that with m = 1, the more increased the value of n, the higher increasing of the value of extreme point, corresponding to the higher load capacity of t e shells. Note that, when m is even or m ≥ 3, the graphic consists of symmetric curves through the origin of the coordinate system and the extreme point does not exist in the load-deflection curves. 302 Vu Thi Thuy Anh, Dao Huy Bich, Nguyen Dinh Duc Effects of the elastic foundations (K1,K2) on the nonlinear response of FGM annular spherical shells are shown in Fig. 6. Obviously, elastic foundations played positive role on nonlinear static response of the FGM annular spherical shell: the large K1 and K2 coefficients are, the larger loading capacity of the shells is. It is clear that the elastic foundations can enhance the mechanical loading capacity for the FGM annular spherical shells, and the effect of Pasternak foundation K2 on critical uniform external pressure is larger than the Winkler foundation K1. 5. CONCLUDING REMARKS Due to practical importance of FGM annular spherical shells and the lack of inves- tigations on stability of these structures, the present paper aims to propose an analytical approach to study the problem of nonlinear buckling and postbuckling of FGM thin an- nular spherical shells on elastic foundations. Based on the classical thin shell theory, the equilibrium and compatibility equations are derived in terms of the shell deflection and the stress function. This system of equations has been transformed into another system of more simple equations, so the appropriate formulas for FGM annular spherical shells are found as a special case. The results show the effects of the material composition, volume fraction of constituent materials, Winkler and Pasternak type elastic foundations on the nonlinear response of FGM annular spherical shells are very appreciable. ACKNOWLEDGMENT This paper was supported by the Grant “Nonlinear analysis on stability and dynam- ics of functionally graded shells with special shapes” code QG.14.02 of Vietnam National University, Hanoi. The authors are grateful for this support. REFERENCES [1] P. L. Pasternak. On a new method of analysis of an elastic foundation by means of two foun- dation constants. State Publisher on Construction and Architecture, Moscow, USSR, (1954). (in Russian). [2] A. D. Kerr. Elastic and viscoelastic foundation models. Journal of Applied Mechanics, 31, (3), (1964), pp. 491–498. [3] M. I. Gorbunov-Possadov, T. A. Malikova, and V. I. Solomin. Design of structures on elastic foundation. State Publisher on construction and architecture, Moscow, USSR, 3rd edition, (1984). (in Russian). [4] V. A. Bajenov. The bending of the cylindrical shells in an elastic medium. Kiev, High school, (1975). (in Russian). [5] Y. Nath and R. K. Jain. Orthotropic annular shells on elastic foundations. Journal of Engi- neering Mechanics, 111, (10), (1985), pp. 1242–1256. [6] R. Yang, H. Kameda, and S. Takada. Shell model fem analysis of buried pipelines under seismic loading. Bulletin of the Disaster Prevention Research Institute, 38, (3), (1988), pp. 115–146. [7] D. N. Paliwal, R. K. Pandey, and T. Nath. Free vibrations of circular cylindrical shell on Winkler and Pasternak foundations. International Journal of Pressure Vessels and Piping, 69, (1), (1996), pp. 79–89. Nonlinear post-buckling of thin FGM annular spherical shells under mechanical loads and ... 303 [8] D. N. Paliwal and R. K. Pandey. The free vibration of a cylindrical shell on an elastic foun- dation. Journal of Vibration and Acoustics, 120, (1), (1998), pp. 63–71. [9] M. Amabili and G. Dalpiaz. Free vibrations of cylindrical shells with non-axisymmetric mass distribution on elastic bed. Meccanica, 32, (1), (1997), pp. 71–84. [10] D. H. Bich, D. V. Dung, and L. K. Hoa. Nonlinear static and dynamic buckling analysis of functionally graded shallow spherical shells including temperature effects. Composite Struc- tures, 94, (9), (2012), pp. 2952–2960. [11] N. D. Duc, V. T. T. Anh, and P. H. Cong. Nonlinear axisymmetric response of FGM shal- low spherical shells on elastic foundations under uniform external pressure and temperature. European Journal of Mechanics-A/Solids, 45, (2014), pp. 80–89. [12] Z. Q. Cheng and R. C. Batra. Exact correspondence between eigenvalues of membranes and functionally graded simply supported polygonal plates. Journal of Sound and Vibration, 229, (4), (2000), pp. 879–895. [13] J. Yang and H. S. Shen. Dynamic response of initially stressed functionally graded rectangular thin plates. Composite Structures, 54, (4), (2001), pp. 497–508. [14] Z. Y. Huang, C. F. Lu¨, and W. Q. Chen. Benchmark solutions for functionally graded thick plates resting on Winkler-Pasternak elastic foundations. Composite Structures, 85, (2), (2008), pp. 95–104. [15] J. Ying, C. Lu¨, and W. Q. Chen. Two-dimensional elasticity solutions for functionally graded beams resting on elastic foundations. Composite Structures, 84, (3), (2008), pp. 209–219. [16] X. Wang and G. G. Sheng. Thermal vibration, buckling and dynamic stability of functionally graded cylindrical shells embedded in an elastic medium. Journal of Reinforced Plastics and Composites, 27, (2), (2007), pp. 117–134. [17] A. Sofiyev, E. Schnack, and A. Korkmaz. The vibration analysis of simply supported FGM truncated conical shells resting on two-parameter elastic foundations. In IRF’2009 3rd Int. Conf. Integrity, Reliability and Failure, Porto, Portugal, (July, 2009). pp. 277–288. [18] A. H. Sofiyev. The buckling of FGM truncated conical shells subjected to axial compressive load and resting on Winkler-Pasternak foundations. International Journal of Pressure Vessels and Piping, 87, (12), (2010), pp. 753–761. [19] H. S. Shen. Postbuckling of shear deformable FGM cylindrical shells surrounded by an elastic medium. International Journal of Mechanical Sciences, 51, (5), (2009), pp. 372–383. [20] P. C. Dumir. Nonlinear axisymmetric response of orthotropic thin spherical caps on elastic foundations. International Journal of Mechanical Sciences, 27, (11), (1985), pp. 751–760. [21] R. S. Alwar and M. C. Narasimhan. Axisymmetric non-linear analysis of laminated or- thotropic annular spherical shells. International Journal of Non-linear Mechanics, 27, (4), (1992), pp. 611–622. [22] P. C. Dumir, G. P. Dube, and A. Mallick. Axisymmetric buckling of laminated thick annu- lar spherical cap. Communications in Nonlinear Science and Numerical Simulation, 10, (2), (2005), pp. 191–204. [23] Y. Kiani and M. R. Eslami. An exact solution for thermal buckling of annular FGM plates on an elastic medium. Composites Part B: Engineering, 45, (1), (2013), pp. 101–110. [24] A. Bagri and M. Eslami. Generalized coupled thermoelasticity of functionally graded annular disk considering the Lord-Shulman theory. Composite Structures, 83, (2), (2008), pp. 168–179. [25] V. L. Agamirov. Problem of nonlinear shells theory. Science Edition, Moscow, (1990). (in Russian). 304 Vu Thi Thuy Anh, Dao Huy Bich, Nguyen Dinh Duc APPENDIX I a3 = 16(β 4 1 − β21), b3 = 32β31 , a4 = 16(β 4 1 − β21 + 2β21n− n2 + n4), b4 = 32(β31 + β1n2), a5 = 32(β 3 1 + β1n 2), b5 = 16(β1 − β21 + 2β21n2 − n2 + n4), a6 = 32β 3 1 , b6 = 16(β 4 1 − β21), c3 = 0.5(β 2 1n 2 − β21), c5 = 0.5β21 , c4 = 0.25(β1n 2 − β1 + β31), c6 = −c4. APPENDIX II C1 = a4, C2 = b5, B1 = −2mpia(−1 + e5a(−1)m) n(25a2 +m2pi2) , A0 = m2pi3 ( 1− e6a) (9a2 +m2pi2) , A1 = h11 + h12 (mpi a t3 + t4 − 2t4n2 ) + h13 (mpi a t6 + t5 ) + h14 (mpi a t5 + t6 ) + + h15 (mpi a t4 − t3 − 2t3n2 ) + h16 ( 2m2pi2 a2 t3 + 3mpi a t4 − t3 ) + + h17 (−2m2pi2 a2 t4 + 3mpi a t3 + t4 ) + h18 ( 2m2pi2 a2 t5 + 3mpi a t6 − t5 ) + + h19 (−2m2pi2 a2 t6 + 3mpi a t5 + t6 ) + h110 ( t3 − 2mpi a t4 ) + h111 ( t4 + 2mpi a t3 ) , A2 = h21 ( −6t1 + m 2pi2 a2 t1 + 5mpi a t2 ) + h22 ( 6t2 − m 2pi2 a2 t2 + 5mpi a t1 ) + + h23 ( −t3 + 2mpi a t4 + m2pi2 a2 t3 + n 2t3 ) + h24 ( t3 + 2mpi a t3 − m 2pi2 a2 t4 + n 2t4 ) + + h25 ( m2pi2 a2 t5 + 2mpi a t6 − t5 ) + h26 (−m2pi2 a2 t6 + 2mpi a t5 + t6 ) + h27 + + h28 ( 2nt1 − mnpi a t2 ) + h29 ( 2nt2 + mnpi a t1 ) + h210 ( −3t1 + mpi a t2 + n 2t1 ) + + h211 ( 3t2 + mpi a t1 − n2t2 ) , A3 = 1 8 m2pi3 ( e2a − 1) (a4 + 2m2pi2a2 +m4pi4 − 2n2a4 + 2m2n2pi2a2 + n4a4) a4(a2 +m2pi2) , A4 = −m2pi3 (e6a − 1) (−9t1a2 + 6t2mpia+ t1m2pi2 + t1n2a2) 24a2(9a2 +m2pi2) + + mpi2 ( e6a − 1) (−9t2a2 − 6t1mpia+ t2m2pi2 + t2n2a2) 8a(9a2 +m2pi2) , Nonlinear post-buckling of thin FGM annular spherical shells under mechanical loads and ... 305 A5 = −8m3pi3 (−3m2pi2 − 7a2 + 3e5am2pi2(−1)m + 7e5a(−1)ma2) 3an(625a4 + 250m2pi2a2 + 9m4pi4) , A6 = m2pi3 ( e6a − 1) 24 (9a2 +m2pi2) , A7 = m2pi3 (−m2pi2 + 3a2e4a − 3a2 + e4am2pi2) 16a2 (4a2 +m2pi2) , A8 = m2pi2 16a2 − ( t6a 2 + t5mpia− vt6a2 + 2vt6m2pi2 − 3vt5mpia ) a2 +m2pi2 + + mpi ( t5a 2 − t6mpia− vt5a2 + 2vt5m2pi2 + 3vt6mpia ) a (a2 +m2pi2) A9 = −pim (−1 + ea(−1)m) an , A10 = apim (−1 + e3a(−1)m) n (9a2 +m2pi2) − mpi (−1 + e3a(−1)m) (−t2mpia+ 3t1a2 − n2t1a2 + vm2pi2t1 + 5vt2mpia− 6vt1a2) an (9a2 +m2pi2) + 3 (−1 + e3a(−1)m) (mpit1a+ 3t2a2 − n2t2a2 − 5vmpit1a+ vt2m2pi2 − 6vt2a2) n (9a2 +m2pi2) , h11 = m3pi4 ( e4a − 1) [4a2pim (3n2 − 1)+ (2n2 − 1) (m3pi3 − 2a3)] 512a4(n2 − 1)(4a2 +m2pi2) + + pi4m(e4a − 1) (a3 − pim3) 256a2(n2 − 1)(a2 +m2pi2) , h12 = −pi2m (e4a − 1) (2pim− a) 32 (a2 +m2pi2) + pi5m4 ( e4a − 1) (pi2m2 − 2a2) 32a2(4a4 + 5a2m2pi2 +m4pi4) , h13 = pi3m2 ( e4a − 1) (2pim− a) 16a (a2 +m2pi2) + 9m7npi10(−1 + e4a)2 16a(4a4 + 5a2m2pi2 +m4pi4) h14 = −2h12, h15 = −1 2 h13, h16 = −3api4m3(−1 + e2a) 8 (a4 + 5a2m2pi2 + 4m4pi4) + pi3m2(−1 + e2a) 4 (a2 + 4m2pi2) , h17 = pi2m ( e2a − 1) [2pi3m3 − a2pim+ a3 + am2pi2] 8 (a2 + 4m2pi2) (a2 +m2pi2) , h18 = pi3m2(−1 + e2a) (3apim− 2a2 − 2pi2m2) 4 (a2 + 4m2pi2) (a2 +m2pi2) , h19 = pi2m ( e2a − 1) [−4pi3m3 + 2a2pim+ a3 + am2pi2] 8 (a2 + 4m2pi2) (a2 +m2pi2) , h110 = −pi4nm3(−1 + e4a) 16a(a2 +m2pi2) , h111 = npi3m2(−1 + e4a) 16(a2 +m2pi2) , 306 Vu Thi Thuy Anh, Dao Huy Bich, Nguyen Dinh Duc h21 = 8am2pi2 (a−mpi) (−1 + (−1)me3a) 9n (9a4 + 10a2m2pi2 +m4pi4) , h22 = 4ampi (−1 + (−1)me3a) (−2ampi +m2pi2 + 3a2) 9n (9a4 + 10a2m2pi2 +m4pi4) , h23 = 160a2m2pi2(−1 + (−1)me5a) 3n(625a4 + 250a2m2pi2 + 9m4pi4) , h24 = −8apim (−1 + (−1)me5a) (25a2 − 3m2pi2) 3n(625a4 + 250a2m2pi2 + 9m4pi4) , h25 = 160a2m2pi2 3n(625a4 + 250a2m2pi2 + 9m4pi4) , h26 = −h24, h27 = ampi3 (−1 + (−1)me5a) 12n(25a2 +m2pi2) , h28 = −40pi3m3a (−1 + (−1)me5a) 3(625a4 + 250a2m2pi2 + 9m4pi4) , h29 = 4m2pi2 (−1 + (−1)me5a) (3m2pi2 + 25a2) 3(625a4 + 250a2m2pi2 + 9m4pi4) , h210 = −8m2pi2 (−1 + (−1)me5a) (−5a3 + 10a2mpi + 3m3pi3) 3an(625a4 + 250a2m2pi2 + 9m4pi4) , h211 = 4pim (−1 + (−1)me5a) (50a2mpi − 25a3 − 3am2pi2 + 16m3pi3) 3n(625a4 + 250a2m2pi2 + 9m4pi4) , t1 = { A(1− β21 − n2) + 2Bβ1 } (A2 +B2) , t2 = { B(1− β21 − n2) + 2Aβ1 } (A2 +B2) , t3 = { C1(−β1n2 + β1 − β31)− 64(β31 + β1n2)β21 } 4 [ C1C2 + (32β31 + 32β1n 2) 2 ] , t4 = { C2β 2 1 + 16(β 3 1 + β1n 2)(−β1n2 + β1 − β31) } 2 [ C1C2 + (32β31 + 32β1n 2) 2 ] , t5 = 4 { (β41 − β21)(β1n2 − β1 + β31)− 4β31(β21n2 − β21) } (16β41 − 16β21)2 + 1024β61 , t6 = 8 { (β41 − β21)(β21n2 − β21) + (β1n2 − β1 + β31)β31 } (16β41 − 16β21)2 + 1024β61 . VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY VIETNAM JOURNAL OF MECHANICS VOLUME 36, N. 4, 2014 CONTENTS Pages 1. N. T. Khiem, P. T. Hang, Spectral analysis of multiple cracked beam subjected to moving load. 245 2. Dao Van Dung, Vu Hoai Nam, An analytical approach to analyze nonlin- ear dynamic response of eccentrically stiffened functionally graded circular cylindrical shells subjected to time dependent axial compression and external pressure. Part 2: Numerical results and discussion. 255 3. Lieu B. Nguyen, Chien H. Thai, Ngon T. Dang, H. Nguyen-Xuan, Tran- sient analysis of laminated composite plates using NURBS-based isogeometric analysis. 267 4. Tran Xuan Bo, Pham Tat Thang, Do Thanh Cong, Ngo Sy Loc, Experimental investigation of friction behavior in pre-sliding regime for pneumatic cylinder 283 5. Vu Thi Thuy Anh, Dao Huy Bich, Nguyen Dinh Duc, Nonlinear post-buckling of thin FGM annular spherical shells under mechanical loads and resting on elastic foundations. 291 6. N. D. Anh, N. N. Linh, A weighted dual criterion for stochastic equivalent linearization method using piecewise linear functions. 307