DSpace at VNU: Nonlinear postbuckling of an eccentrically stiffened thin FGM plate resting on elastic foundations in thermal environments

DSpace at VNU: Nonlinear postbuckling of an eccentrically stiffened thin FGM plate resting on elastic foundations in the...

Thin-Walled Structures 75 (2014) 103–112 Contents lists available at ScienceDirect Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws Nonlinear postbuckling of an eccentrically stiffened thin FGM plate resting on elastic foundations in thermal environments Nguyen Dinh Duc n, Pham Hong Cong Vietnam National University, Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Viet Nam art ic l e i nf o a b s t r a c t Article history: Received 29 May 2013 Received in revised form September 2013 Accepted 17 October 2013 Available online December 2013 This paper first time presents an analytical investigation on the nonlinear postbuckling of imperfect eccentrically stiffened thin FGM plates under temperature and resting on elastic foundation using a simple power-law distribution (P-FGM) Both of the FGM plate and stiffeners are deformed under thermal loads The formulations are based on the classical plate theory taking into account geometrical nonlinearity, initial geometrical imperfection, temperature-dependent properties and the Lekhnitsky smeared stiffeners technique with Pasternak type elastic foundation By applying Galerkin method and using stress function, effects of material and geometrical properties, temperature, elastic foundation and eccentrically stiffeners on the buckling and postbuckling loading capacity of the eccentrically stiffened FGM plate in thermal environments are analyzed and discussed Some results were compared with the one of the other authors & 2013 Elsevier Ltd All rights reserved Keywords: Nonlinear postbuckling Eccentrically stiffened P-FGM plates Classical plate theory Elastic foundation Thermal environments Introduction The FGM plates and shells, as other composite structures, usually reinforced by stiffening member to provide the benefit of added load-carrying static and dynamic capability with a relatively small additional weight penalty Thus study on static and dynamic problems of reinforced FGM plates and shells with geometrical nonlinearity are of significant practical interest Up to date, the investigation on static and dynamic of eccentrically stiffened FGM structures has received comparatively little attention Recently, Bich et al studied nonlinear postbuckling and dynamic of eccentrically stiffened functionally graded shallow shells [1,2], buckling and postbuckling of an eccentrically stiffened functionally graded cylindrical panels [3] Dung et al considered nonlinear stability of eccentrically stiffened functionally graded imperfect plates resting on elastic foundation [4] Duc investigated nonlinear dynamic response of imperfect eccentrically stiffened doubly curved FGM shallow shells on elastic foundations [5] Notice that in all the publication mentioned above [1–5], the eccentrically stiffened FGM plates and shells are considered without temperatures There has been no publication on the FGM plates and shells reinforced by eccentrically stiffeners in thermal environment The most difficult part in this type of problem is to calculate the thermal mechanism of FGM plates and shells as well as eccentrically stiffeners under thermal loads n Corresponding author Tel.: ỵ 84 37547978; fax: ỵ84 37547724 E-mail address: ducnd@vnu.edu.vn (N Dinh Duc) 0263-8231/$ - see front matter & 2013 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.tws.2013.10.015 In this paper, our investigation is the first proposal for an imperfection eccentrically stiffened FGM plate in thermal environments and resting on elastic foundation in which we studied the nonlinear postbuckling using a simple power-law distribution (P-FGM) We consider the eccentrically stiffened thin FGM plate in thermal environments with temperature independent material property, i.e the Young's modulus E, thermal expansion coefficient a, the mass density ρ, the thermal conduction K and even Poisson ratio v are independent to the temperature Those vary in the thickness direction z as well as temperature T in the two variables function of z and T The investigation under those assumptions for FGM plates is a very challenging work Moreover, the presence of the eccentrically stiffeners makes it more difficult to solve Here, we have solved this problems taking into account all above assumptions The formulations are based on the classical plate theory taking into account geometrical nonlinearity, initial geometrical imperfection, temperature-dependent properties and the Lekhnitsky smeared stiffeners technique with Pasternak type elastic foundation Using Galerkin method and stress function, the effects of geometrical and material properties, temperature, elastic foundation and eccentrically stiffeners on the nonlinear response of the P-FGM plate in thermal environments are analyzed and discussed Eccentrically stiffened FGM plate on elastic foundations Consider a ceramic–metal eccentrically stiffened FGM plate of length a, width b and thickness h resting on an elastic foundation 104 N Dinh Duc, P Hong Cong / Thin-Walled Structures 75 (2014) 103–112 Eccentrically stiffened thin FGM plate under temperatures Fig Geometry and coordinate system of an eccentrically stiffened FGM plate on elastic foundation A coordinate system ðx; y; zÞ is established in which ðx; yÞ plane on the middle surface of the plate and z is thickness direction ð À h=2 r z r h=2Þ as shown in Fig The volume fractions of constituents are assumed to vary through the thickness according to the following power law distribution (P-FGM):  V c zị ẳ N 2z ỵ h ; V m zị ẳ V c ðzÞ 2h ð1Þ where N is volume fraction index (0 r N o 1) The material properties of P-FGM shells have been assumed to be temperaturedependent and graded in the thickness direction z: ẵEz; Tị; z; Tị; z; Tị; Kz; Tị ẳ ẵEm Tị; m Tị; m Tị; K m Tị  N 2z ỵ h vz; Tị ỵ ½Ecm ðTÞ; ρcm ðTÞ; αcm ðTÞ; K cm ðTފ 2h  ẳ m Tị ỵ cm Tị N 2z ỵ h 2h 2ị cm Tị ẳ c Tị m Tị; cm Tị ẳ c Tị m Tị; ssh xy ẳ Ez; Tị ẵx ; y ị ỵ y ; x ị ỵ ịTzị1; 1ị z; Tị 8ị Ez; Tị 21 ỵ z; Tịị xy st sst x ; sy ị ẳ E x ; y ị cm Tị ẳ αc ðTÞ Àαm ðTÞ; ð3Þ It is evident from Eqs (2) and (3) that the upper surface of the plate z ẳ h=2ị is ceramic -rich, while the lower surface z ẳ h=2ị is metal-rich A material property P r , such as the elastic modulus E, Poisson ratio ν, the mass density ρ, the thermal expansion coefficient α and coefficient of thermal conduction K can be expressed as a nonlinear function of temperature [68]: 4ị in which T ẳ T ỵ Tzị and T ẳ 300 K (room temperature); P0, P À 1, P1, P2 and P3 are coefficients characterizing of the constituent materials In short, we will use T-D (temperature dependent) for the cases in which the material properties depend on temperature Otherwise, we use T-ID for temperature independent cases The material properties for the later one have been determined by (4) at room temperature, i.e T ¼ 300 K The load–displacement relationship of the elastic foundation is assumed as following [9–12]: qe ¼ k1 w À k2 ∇2 w In which u; v are the displacement components along the x; y directions, respectively Interestingly, comparing to the other [1–5], we have assumed that the eccentrically outside stiffeners also depend on temperature Hooke law for an FGM plate with temperature-dependent properties is defined as [13,14] where ΔT is temperature rise from stress free initial state, and more generally, T ẳ Tzị; Ez; Tị; vz; Tị are the FGM plate's elastic moduli which is determined by (2) For stiffeners in thermal environments with temperaturedependent properties, we have proposed its form adapted from [2] as the follows: Ecm Tị ẳ Ec Tị Em Tị; P r ẳ P P T ỵ þ P T þ P T þ P T Þ where ε0x and ε0y are the normal strains, γ 0xy is the shear strain at the middle surface of the plate, and kx ; ky ; kxy are the curvatures In the framework of the classical plate theory, the strains at the middle surface and the curvatures are related to the displacement components u; v; w in the coordinates as [13,14] 0 1 1 εx À wxx kx ux ỵ w2x =2 B C B C B C B B y C ẳ @ vy ỵ wy =2 A; @ ky A ¼ @ À wyy C 7ị A @ A wxy kxy uy ỵ vx þ w x w y γ 0xy sh ðssh x ; sy ị ẳ where K cm Tị ẳ K c ðTÞ À K m ðTÞ; N Z 0; N Z In the present study, the classical plate theory is used to obtain the equilibrium and compatibility equations as well as expressions of buckling loads and postbuckling equilibrium paths of FGM plates The strains across the plate thickness at a distance z from the mid-plane are [13,14] 0 1 εx εx kx B C B εy C B ε0 C B C ð6Þ @ A ẳ @ y A ỵ z@ ky A xy 2kxy xy 5ị where ẳ =x2 ỵ =y2 , w is the deection of the FGM plate, k1 and k2 are Winkler foundation stiffness and shear layer stiffness of Pasternak foundation, respectively E0 α0 ðTÞΔðTÞð1; 1Þ À 2ν0 ðTÞ ð9Þ Here, E0 ¼ E0 ðTÞ; ν0 ¼ ν0 ðTÞ; α0 ¼ α0 ðTÞ are the Young's modulus, Poisson ratio and thermal expansion coefficient of the stiffeners, respectively The FGM plate reinforced by eccentrically longitudinal and transversal stiffeners is shown in Fig E0 is elasticity modulus in the axial direction of the corresponding stiffener which is assumed identical for both types of longitudinal and transversal stiffeners In order to provide continuity between the plate and stiffeners, suppose that stiffeners are made of full metal E0 ẳ Em ị if putting them at the metal-rich side of the plate, and conversely full ceramic stiffeners E0 ẳ Ec ị at the ceramic-rich side of the plate (this assumption first time was proposed by Bich in [1] and has been used in [1–5]) The contribution of stiffeners can be accounted for using the Lekhnitsky smeared stiffeners technique [1–5] In order to investigate the FGM plates with stiffeners in the thermal environment, we have not only taken into account the materials moduli with temperature-dependent properties but also we have assumed that all elastic moduli of FGM plates and stiffener are temperature dependence and they are deformed in the presence of temperature Hence, the geometric parameters, the plate's shape and stiffeners are varied through the deforming process due to the temperature change We have assumed that the thermal stress of stiffeners is subtle which distributes uniformly through the whole plate structure, therefore, we can ignore it Lekhnitsky smeared stiffeners technique can be adapted from N Dinh Duc, P Hong Cong / Thin-Walled Structures 75 (2014) 103–112 [1–5] for eccentrically stiffened FGM plate under temperatures as the follows: ! ET0 AT1 N x ẳ I 10 ỵ T x ỵ I 20 0y ỵ I 11 ỵ C T1 ịkx ỵI 21 ky ỵ s1 ! E T AT N y ẳ I 20 0x ỵ I 10 ỵ T 0y ỵ I 21 kx ỵ I 11 ỵ C T2 ịky ỵ s2 N xy ẳ I 30 0xy ỵ 2I 31 kxy ! ET0 I T1 Mx ¼ I 12 þ T kx þ I 22 ky þ Φ2 s1 ! ET0 I T2 T 0 M y ¼ I 21 x ỵ I 11 ỵ C ịy þI 22 kx þ I 12 þ T ky þ Φ2 s2 ð10Þ The relation (10) is our most important finding, where I ij ði ¼ 1; 2; 3; j ¼ 0; 1; 2Þ: Z h=2 EðzÞ j z dz I 1j ẳ zị2 h=2 Z h=2 Ezịzị j I 2j ẳ z dz h=2 zị Z h=2 Ezị zj dz ẳ I 1j I 2j ị I 3j ẳ h=2 2ẵ1 ỵ zị Z h=2 Ezịzị Tzị1; zịdz ; ị ẳ h=2 zị ỵ AT1 zT1 ị2 ; I T2 ẳ 12 E0 AT1 zT1 T E0 AT2 zT2 C T1 ¼ ; C2 ¼ sT1 sT2 I T1 ¼ T zT1 ¼ T T 12 ỵAT2 zT2 ị2 T 11ị T d1 ẳ d1 ỵ m Tzịị; d2 ẳ d2 ỵ m Tzịị; T T ẳ s1 ỵ m TðzÞÞ; ð12Þ where the coupling parameters C ; C are negative for outside stiffeners and positive for inside one; s1 ; s2 are the spacing of the longitudinal and transversal stiffeners; I ; I are the second moments of cross-section areas; z1 ; z2 are the eccentricities of stiffeners with respect to the middle surface of plate; and the width and thickness of longitudinal and transversal stiffeners are denoted by d1 ; h1 and d2 ; h2 respectively A1 ; A2 are the crosssection areas of stiffeners Although the stiffeners are deformed by temperature, we, however, have assumed that the stiffeners keep its rectangular shape of the cross section Therefore, it is straightforward to calculate AT1 ; AT2 The nonlinear equilibrium equations of a perfect plate based on the classical plate theory are given by [13,14] N x;x ỵ N xy;y ẳ 13aị N xy;x ỵ N y;y ẳ 13bị M x;xx þ2M xy;xy þ M y;yy þ N x wxx þ 2Nxy wxy ỵ N y wyy k1 w ỵk2 w ẳ 13cị 0x 0y ẳ A22 N x A12 N y ỵ B11 wxx ỵB12 wyy A22 A12 ị1 ẳ A11 N y A12 N x ỵ B21 wxx ỵ B22 wyy A11 A12 ị1 16ị B21 f xxxx ỵ B12 f yyyy ỵ B11 ỵ B22 2B66 ịf xxyy D11 wxxxx D22 wyyyy D12 ỵ D21 þ 4D66 Þwxxyy þ þ N x wxx þ 2Nxy wxy ỵ N y w;yy k1 w ỵ k2 w ẳ where D11 ẳ I 12 ỵ ET0 I T1 B11 I 11 ỵ C T1 ị I 21 B21 sT1 D22 ẳ I 12 ỵ ET0 I T2 B22 I 11 ỵ C T2 Þ À I 21 B12 sT2 ð17Þ f ðx; yÞ is stress function defined by Nx ¼ f yy ; N y ¼ f xx ; N xy ¼ À f xy ð18Þ For an imperfect FGM plate, Eq (16) are modified into form as [12,15–17] À D11 wxxxx À D22 wyyyy D12 ỵ D21 ỵ 4D66 ịwxxyy zT2 ẳ z2 ỵ m Tzịị; sT2 ẳ s2 þ αm TðzÞÞ calculated from Eq (10) Nx;x þ Nxy;y ẳ Nxy;x ỵ N y;y ẳ B21 f xxxx ỵ B12 f yyyy ỵ B11 ỵ B22 2B66 ịf xxyy h1 ẳ h1 ỵ m Tzịị; h2 ẳ h2 ỵ m Tzịị; ẳ z1 þ αm TðzÞÞ; ð15Þ Substituting once again Eq (14) into the expression of M ij in (10), then M ij into the Eq (13c) leads to D21 ¼ I 22 B21 I 11 ỵ C T2 ị I 21 B11 D66 ¼ I 32 À I 31 B66 After the thermal deformation process, the geometric shapes of stiffeners which can be determined as follows: zT1 sT1 ! ET0 AT1 ET0 AT2 I 20 I 10 ỵ T A11 ẳ I 10 ỵ T ị; A22 ẳ ; A12 ¼ ; A66 ¼ Δ Δ I 30 Δ s1 s2 ! ! ET0 AT1 ET0 AT2 Δ ẳ I 10 ỵ T I 10 ỵ T I 220 s1 s2 D12 ¼ I 22 À B12 I 11 ỵ C T1 ị I 21 B22 h1 þ h h þh ; zT2 ¼ 2 T where B21 ¼ A11 I 21 ÀA12 ðI 11 ỵ C T1 ị; B22 ẳ A11 I 11 ỵ C T2 ị A12 I 21 I 31 B66 ẳ I 30 M xy ẳ I 31 0xy ỵ 2I 32 kxy T T d2 ðh2 Þ3 ð14Þ B11 ẳ A22 I 11 ỵ C T1 ị A12 I 21 ; B12 ¼ A22 I 21 ÀA12 ðI 11 þ C T2 Þ ðI 11 þ C T1 Þε0x þ I 21 ε0y þ T T d1 ðh1 Þ3 0xy ẳ A66 N xy ỵ 2B66 wxy 105 þ f yy ðwxx þ wnxx Þ À 2f xy wxy ỵ wnxy ị ỵ f xx wyy ỵ wnyy ị k1 w ỵ k2 w ẳ ð19Þ n in which w ðx; yÞ is a known function representing initial small imperfection of the plate The geometrical compatibility equation for an imperfect plate is written as [12,15–17]: 0x;yy ỵ 0y;xx 0xy;xy ẳ w2xy wxx wyy ỵ2wxy wnxy wxx wnyy wyy wnxx 20ị From the constitutive relation (15) in conjunction with Eq (19) one can write ε0x ¼ A22 f yy À A12 f xx ỵ B11 wxx ỵ B12 wyy A22 A12 ị1 0y ẳ A11 f xx A12 f yy ỵ B21 wxx ỵ B22 wyy A11 A12 ị1 0xy ẳ A66 f xy ỵ 2B66 wxy ð21Þ Setting Eq (21) into Eq (20) gives the compatibility equation of an imperfect FGM plate as A11 f xxxx ỵ A22 f yyyy ỵ A66 2A12 ịf xxyy þ B21 wxxxx þ B12 wyyyy þ ðB11 þ B22 2B66 ịwxxyy w2xy wxx wyy ỵ 2wxy wnxy wxx wnyy wyy wnxx ị ẳ 22ị Eqs (19) and (22) are nonlinear equations in terms of variables w and f and used to investigate the stability of FGM plate on elastic foundations subjected to mechanical, thermal and thermomechanical loads 106 N Dinh Duc, P Hong Cong / Thin-Walled Structures 75 (2014) 103–112 In the present study, the edges of FGM plates are assumed to be simply supported Depending on the in-plane restraint at the edges, three cases of boundary conditions [15–17], labeled as Case 1, and 3, may be considered Case Four edges are simply supported and freely movable (FM) The associated boundary conditions are w ¼ Nxy ¼ M x ¼ 0; N x ¼ Nx0 at x ¼ 0; a w ¼ Nxy ¼ M y ¼ 0; N y ¼ N y0 at y ẳ 0; b 23ị Case Four edges are simply supported and immovable (IM) The associated boundary conditions are w ¼ u ¼ M x ¼ 0; N x ¼ N x0 at x ¼ 0; a w ¼ v ¼ M y ¼ 0; N y ¼ Ny0 at y ¼ 0; b ð24Þ Nonlinear buckling and postbuckling analysis 4.1 Mechanical buckling and postbuckling analysis Consider a simply supported FGM plate with all movable edges which is rested on elastic foundations and subjected to in-plane edge compressive loads F x ; F y uniformly distributed on edges x ¼ 0; a and y ¼ 0; b respectively In this case, buckling force resultants are given Nx0 ¼ À F x h; N y0 ¼ À F y h ð29Þ and Eq (28) leads to F x ¼ Àb1 W À b1 WðW þ 2μÞ W W þμ þb1 W þμ Case The edges are simply supported Uniaxial edge loads are applied in the direction of the x-coordinate The edges x ¼ 0; a are considered freely movable, the remaining two edges being unloaded and immovable For this case, the boundary conditions are defined as w ¼ Nxy ¼ M x ¼ 0; N x ¼ Nx0 at x ¼ 0; a w ¼ v ¼ M y ¼ 0; N y ẳ Ny0 at y ẳ 0; b 25ị To solve two Eqs (19) and (22) for unknowns w and f, and with the consideration of the boundary conditions (23)–(25), we assume the following approximate solutions [15–17] ð26aÞ 1 f ẳ A1 cos 2m x ỵ A2 cos 2n y ỵ A3 sin m x sin n y ỵ N x0 y2 ỵ N y0 x2 2 26bị where λm ¼ mπ=a, δn ¼ nπ=b, W is amplitude of the deflection and μ is parameter of imperfection; m; n are odd natural numbers The coefficients Ai ði ¼ 1=3Þ are determined by substitution of Eqs (26a and 26b) into Eq (22) as A1 ẳ B21 4m ỵ B12 4n ỵ B11 ỵ B22 2B66 ị2m 2n W A11 4m ỵ A22 4n ỵ A66 2A12 ị2m 2n where the coefficients b1 ; b1 ; b1 ; b1 are described in detail in Appendix B For a perfect FGM plate, Eq (30) reduces to an equation from which buckling compressive load may be obtained as F xb ¼ À b1 A simply supported FGM plate with all immovable edges is considered The in-plane condition on immovability at all edges, i e u ¼ at x ¼ 0; a and v ¼ at y ¼ 0; b, is fulfilled in an average sense as [6–8,15–17]: Z bZ a Z aZ b ∂u ∂v dx dy ¼ 0; dy dx ¼ ð31Þ ∂x 0 0 ∂y From Eqs (6) and (14) one can obtain the following expressions in which Eq (18) and imperfection have been included ∂u ¼ A22 f yy À A12 f xx ỵB11 wxx ỵ B12 wyy A22 À A12 ÞΦ1 À w2x À wx wnx ∂x ∂v ¼ A11 f ;xx À A12 f yy þB22 wyy þ B21 wxx À ðA11 À A12 ÞΦ1 À w2y À wy wny ∂y ð32Þ Substitution of Eqs (26a), (26b) into Eq (32) and then the result into Eq (31) give fictitious edge compressive loads as δ2n 2m W ỵ 2hịW; A2 ẳ W ỵ 2hịW 32A22 2n 32A11 m A3 ẳ 30ị 4.2 Thermal buckling and postbuckling analysis where Nx0 ; N y0 are pre-buckling force resultants in directions x and y, respectively w; wn ị ẳ W; hị sin m x sin n y ỵb1 WW ỵ 2ị 27ị Subsequently, substitution of Eqs (26a), (26b) into Eq (19) and applying the Galerkin procedure for the resulting equation yield > ẵB21 4m ỵ B12 4n ỵ B11 ỵ B22 À 2B66 Þλ2m δ2n Š2 > = ab< À A11 ỵ A22 ỵ A66 2A12 ị2 n m m n W > > 4: À D11 4m D22 4n D12 ỵ D21 ỵ 4D66 ị2m 2n k1 2m ỵ 2n ịk2 ; " # 8m n B21 4m ỵ B12 4n ỵ B11 þ B22 À2B66 Þλ2m δ2n À WðW þ μhÞ þ A11 4m ỵ A22 4n ỵ A66 2A12 ị2m 2n   2m n B21 B12 ab WW ỵ 2hị N x0 2m ỵ Ny0 2n ịW ỵ hị ỵ A11 A22 ! ab 4m 4n ỵ WW ỵ hịW ỵ2hị ẳ 28ị 64 A22 A11 Eq (28) is used to determine nonlinear buckling and postbuckling response of rectangular eccentrically stiffened FGM plates in thermal environments It is not so difficult to realize that Eq (28) is more complicated than the equation written in [4] without the temperature and ν ¼ const An interested characteristics of Eq (28) is temperature dependent, which are displayed in the B21 ; B12; ; B11 ; B22 ; D11 ; D22 ; A11 ; A22 ; ::: coefficients as shown in Eqs (15) and (17) ðA11 2m ỵ A12 2n ịWW ỵ 2hị 8A11 A22 A212 ị B12 A11 ỵ B22 A12 4n 4 ỵ B þ B δ þ ðB þ B À 2B Þλ δ 2 ðA11 A22 ÀA212 Þ 21A m ỵ12A n ỵ11A 222A ị662 2m n mb A A A Nx0 ẳ ỵ 11 22 Wỵ 12 11 m 22 n 66 12 m n 4m B11 A11 ỵ B21 A12 W na2 A11 A22 A212 A12 2m ỵ A22 2n ịWW ỵ2hị 8A11 A22 A212 ị B21 A22 ỵB11 A12 À 4m 2 4 þ B λ þ B δ þ ðB þ B À 2B Þλ δ 21 12 11 22 66 na A A À A2 ðA11 A22 À A12 ị A m ỵ A n ỵ A 2A ị2 2m n Ny0 ẳ ỵ 11 22 Wỵ 12 11 m 4n B12 A12 ỵ B22 A22 mb A11 A22 À A212 22 n 66 12 m n ð33Þ W Inserting Eq (33) into Eq (28) gives the following expression for the thermal parameter: ! W WW ỵ 2ị 34ị b2 b2 WW ỵ 2ị ẳ h b2 W ỵ b2 W ỵ W ỵ where the coefficients b2 ; b2 ; b2 ; b2 are described in detail in Appendix B N Dinh Duc, P Hong Cong / Thin-Walled Structures 75 (2014) 103–112 107 By using Eq (11), the thermal parameter Φ1 can be expressed in terms of ΔT: Employing these relations in Eq (28) yields Φ1 ¼ LhΔT F x ¼ b3 W ỵ b3 35ị where the coefcient L are described in detail in Appendix A Although ΔTis included in the expression for L due to the temperature dependence of material properties T ẳ T ỵ Tị, one may formally express ΔTfrom Eqs (34) and (35) as follows ! 1 W WW ỵ2ị b W ỵ b2 36ị T ẳ b2 b2 WW þ 2μÞ L W þμ W þμ þ W W ỵ ị WW ỵ2ị ỵb3 W ỵị ỵ b3 WW ỵ2ị A11 A12 ịLn2 T 38ị A11 m2 B2a ỵA12 n2 =A11 ịị where the coefficients b3 ; b3 ; b3 ; b3 are described in detail in Appendix B Eq (38) is a crucial equation to investigate the nonlinear response of eccentrically stiffened FGM plate under both of thermal and mechanical loads Eq (36) is the analytical form to determine the non-linear relation between the bending deflection and temperature for both of the perfect and imperfect plates under the thermal loads (for perfect plate, μ ¼ 0) Using Eq (36), we have derived the temperature change, ΔT b ¼ ðb2 =LÞ, which sets them into the buckling state under the condition W ¼ In case of T-D, the two hand sides of Eq (36) are temperature dependence which makes it very difficult to solve Fortunately, we have applied a numerical technique using the iterative algorithm to determine the buckling loads as well as to determine the deflection–load relations in the postbuckling period of the FGM plate More details, given the material parameter N, the geometrical parameter ðb=a; b=hÞ and the value of W=h, we can use these to determine ΔT in (36) as the follows: we choose an initial step for ΔT on the right hand side in Eq (36) with ΔT ¼ (since T ¼ T ¼ 300 K, the initial room temperature) In the next iterative step, we replace the known value of ΔT found in the previous step to determine the right hand side of Eq (36) ΔT This iterative procedure will stop at the kth step if ΔT k satisfies the condition ΔT ÀΔT k jr ε Here, ΔT is a desired solution for the temperature and ε is a tolerance used in the iterative steps Numerical results and discussion Here, several numerical examples will be presented for perfect and imperfect simply supported midplane-symmetric FGM plates The typical values of the coefficients of the materials mentioned in (4) are listed in Table Unlike the other works, we here assume that all coefficients depend on both of the thickness z and temperature T Technically, it is much more difficult to capture and solve the fundamental set of equations In this paper, we have contribute significantly to this transformation process The parameters for the stiffeners are z1 ¼ 0:0225mị; z2 ẳ 0:0225mị; s1 ẳ 0:2mị; s2 ẳ 0:2mị h1 ẳ 0:03mị; h2 ẳ 0:03mị; d1 ẳ 0:004mị; d2 ¼ 0:004ðmÞ 4.3 Thermo-mechanical postbuckling analysis Let us consider a simply supported eccentrically stiffened FGM plate, with movable edges x ¼ 0; a and immovable ones y ¼ 0; b (Case 3), subjected to the simultaneous action of a thermal field and an in-plane compressive load F x distributed uniformly along the edges x ¼ 0; a From the first of Eq (29) and the second of Eqs (31) and (32), we have N x0 ¼ ÀF x hN y0 2 m B21 m ỵ B12 n ỵ B11 ỵ B22 2B66 ịm n n A11 4m ỵ A22 4n ỵ A66 2A12 ị2m 2n ẳ ab þ B21 λ4m þ B12 δ4n þ ðB11 þ B22 2B66 ị2m 2n A12 n ỵB n m 22 A11 m ỵ B21 A11 n A11 m A þ A δ4 þ ðA À 2A Þλ2 δ2 11 m 22 n 66 12 m n A12 ðA11 À A12 ị Wỵ N x0 ỵ ỵ n W W ỵ 2hị A11 A11 8A11 Fig Postbuckling of FGM plates under mechanical loads (1,2: stiffened FGM plate; 3, 4: un-stiffened FGM plate) ð37Þ Table Material properties of the constituent materials of the considered FGM plates [6,8,16] Material Property P0 PÀ1 P1 P2 P3 Si3N4 (Ceramic) E(Pa) ρ (kg/m3) 348.43e9 2370 5.8723e-6 0 À 3.70e-4 9.095e-4 2.160e-7 0 À 8.946e-11 0 13.723 0.24 0 0 0 0 201.04e9 8166 12.330e-6 0 3.079e-4 8.086e-4 -6.534e-7 0 0 15.379 0.3177 0 0 0 0 α ðK À Þ k ðW=mKÞ ν SUS304 (Metal) E (Pa) ρ (kg/m3) α ðK À Þ k ðW=mKÞ ν 108 N Dinh Duc, P Hong Cong / Thin-Walled Structures 75 (2014) 103–112 Fig compares nonlinear postbuckling response of eccentrically stiffened FGM plate and un-stiffened FGM plate under mechanical loads It is clear that the stiffeners can enhance the thermal loading capacity for the imperfect and perfect FGM plates The similar conclusion has been reported for nonlinear static and dynamic analysis of eccentrically stiffened FGM plates and shells [1–5] Fig presents postbuckling of the eccentrically stiffened FGM plate under thermal loads (T-D) with Poisson ratio ẳ const and ẳ zị It is clear that there is no significant difference in case of ẳ const and ẳ zị In our calculation, the equations under the condition ẳ zị are much more complicated Hence, for a sake of simplicity, the others often choose the condition of ν ¼ const In particular case of a FGM plate without stiffeners with the conditions: A1 ¼ A2 ¼ 0and I ¼ I ¼ 0, we have compared the numerical results of un-stiffened FGM plate with Dung's results [18] (without temperatures, ẳ zị) and with Duc's results [15] ( ẳ const,N ¼ 0) under only mechanical loads Those have been presented in Figs and which show the good agreement between our findings for postbuckling of FGM plates and the others Figs and present the positive influence of elastic foundations on the postbuckling of eccentrically stiffened FGM plates under uniaxial compressive load (all FM edges) and uniform temperature (all IM edges) The effect of Pasternak foundation K on the critical compressive loads and the thermal resistance of Fig Effect of Poisson's ratio on postbuckling of FGM plate Fig Comparing the force–bending curve of FGM plate without stiffeners with Tung [15] (ν ¼ const) Fig Comparing the force–bending curve of FGM plate without stiffeners with Dung [18] ( ẳ zị) Fig Effects of elastic foundation on the postbuckling of eccentrically stiffened FGM plate under compression (all FM edges) N Dinh Duc, P Hong Cong / Thin-Walled Structures 75 (2014) 103–112 Fig Effects of elastic foundation on the postbuckling of eccentrically stiffened FGM plate under thermal loads (T-D; all IM edges) Fig Effects of temperatures on the postbuckling of eccentrically stiffened FGM plate under compression (all IM edges) FGM is larger than the Winkler foundation K This conclusion has been also reported in others publications [9–12,17] Figs and present effects of thermo-mechanical loads on nonlinear response of FGM plates in thermal environment Fig investigated effects of temperatures on postbuckling of eccentrically stiffened FGM plate under compressive loads with all IM edges and T-D properties These show us that the decrease in temperature reduces the mechanical loading ability of the perfect plate as well as the imperfect plate Fig also studies the effects of the compressive loads on postbuckling of eccentrically stiffened FGM plate in thermal environment with all FM edges and T-D properties Obviously, both of the presence of the compressive and thermal loads reduce significantly the loading capacity of the FGM plates Figs and 10show effects of boundary conditions (two edges FM x ¼ 0; a and two edges IM y ¼ 0; b) on the postbuckling of eccentrically stiffened FGM plates under compressive loads The 109 Fig Effects of compressive loads on the postbuckling of eccentrically stiffened FGM plate in thermal environment (all FM edges) Fig 10 Effects of boundary conditions (FM and IM) on the postbuckling of eccentrically stiffened FGM plates under compressive loads curves in case of FM have been plotted using Eq (30) with the loading ratio β ¼ Whereas, the curves in case of IM have been plotted using Eq (38) with ΔT ¼ This has illustrated that the boundary conditions have a significant effects on the buckling and postbuckling of FGM plates Although the perfect FGM plate have only been buckling in case of the large loads, the loading ability of the imperfect FGM plates in the postbuckling period as well as under the boundary condition of the edge y ¼ 0; b is much better than those of perfect one Fig 11 presents effects of volume fraction index on the postbuckling of eccentrically stiffened FGM plates under thermal loads (all IM edges) These postbuckling curves show the loading ability of FGM get worse with the increase of N; N Figs 12 and 13 show effects of imperfection on postbuckling response of eccentrically stiffened FGM plate under mechanical and thermal loads In postbuckling period, those suggest us that 110 N Dinh Duc, P Hong Cong / Thin-Walled Structures 75 (2014) 103–112 Fig 11 Effects of volume fraction index on the postbuckling of eccentrically stiffened FGM plates under thermal loads (all IM edges) Fig 13 Effect of imperfection on postbuckling of eccentrically stiffened FGM plate under thermal loads (all IM edges) Fig 12 Effect of imperfection on postbuckling of eccentrically stiffened FGM plate under compressive edge loads (all FM edges) Fig 14 Comparison of postbuckling curves for un-stiffened FGM plates under uniaxial compression with the results of Duc [12] the imperfect properties have affected actively on the loading ability in the limit of large enough W=h In other words, the loading ability increases with μ This has been reported in the other papers [4,5;15–17] To validate the accuracy of the proposed approach, the obtained numerical results for un-stiffened FGM plate and ν ¼ const under uniaxial compression are compared with those in [12] Computations have been carried out for the following material and the geometrical parameters of FGM plate: Em ¼ 70 GPa; Ec ¼ 380 GPa; ν ¼ 0:3; and a=b ¼ 1; b=h ¼ 40; N ¼ 1; m ¼ n ¼ 1; β ¼ It is seen that these results (in Figs 14 and 15) are in good agreement to those one of Duc [12] We have also compared our findings with Dung [4] for stiffened FGM plate under uniaxial compression Computations have been carried out with the following material and the geometrical parameters of the FGM plate: a ¼ b ¼ 1:50 m; h ¼ 0:008 m and stiffeners with z1 ẳ z2 ẳ 0:019mị; s1 ẳ 0:15mị; s2 ẳ 0:15mị,h1 ẳ 0:03mị; h2 ẳ 0:03mị; d1 ẳ 0:003mị; d2 ẳ 0:003mị It is seen that obtained results in Fig 16 are in good agreement to those one of Dung [4] Conclusion This paper first time presents an analytical investigation on the nonlinear postbuckling for imperfect eccentrically stiffened N Dinh Duc, P Hong Cong / Thin-Walled Structures 75 (2014) 103–112 111 Acknowledgment This work was supported by the National Foundation for Science and Technology Development of Viet Nam – NAFOSTED under Grant number 107.02-2013.06 Appendix A "Z E m m du ỵ N1 vm Àvcm u # Z αcm Ecm u2N du þ N1 À vm À vcm u L¼ Z m Ecm ỵcm Em ịuN du À vm À vcm uN1 Appendix B b1 Fig 15 Comparison of postbuckling curves for un-stiffened FGM plates under uniform temperature rise with the results of Duc [12] À Á 32mn B2a Ba m4 B21 ỵ n4 B12 ỵ B11 ỵ B22 2B66 Ba m2 n2   ¼ À B2h B4 m4 A ỵ n4 A ỵ A 2A B m2 n 11 a 22 66 12 a m2 B2a ỵ n2 Ba m4 B21 ỵ n4 B12 þ B11 þ B22 À 2B66 B2a m2 n2 À B4a m4 A11 ỵ A22 n4 þ A66 À 2A12 B2a m2 n2 π2 À Á  b1 ¼ D B4 m4 D n4 D ỵ D ỵ 4D B2 m2 n2  22 12 21 66 a 11 a 7B2 B2 m2 ỵ n2   h a K B4a D11 D11 K B2a 2 À π4 À π2 Ba m ỵ n b1 ẳ b1 ẳ 8mnB2a B21 3B2h ỵ A11 m4 B4a 16B2h A22 ỵ B12 ! A22 m2 B2a ỵ n2 n4 ! A11 m2 B2a ỵ n2 Ba m4 B21 ỵ n4 B12 þ B11 þ B22 À 2B66 B2a m2 n2 π À Á À À A11 B4a m4 þ A22 n4 þ A66 À 2A12 B2a m2 n2 7 b2 ¼ D B4 m4 D n4 D ỵD þ 4D ÁB2 m2 n2 π 22 12 21 66 a 11 a   2 2 2 À Ba K D11 À Ba K π D11 Ba m þ n  Fig 16 Comparison of present postbuckling curves for stiffened FGM plates under uniaxial compression with the results of Dung [4] B2a m2 ỵ n2  B2h π 2 b2 ¼ B2h 32mnBa 6 4 B2a m2 ỵ n2 3Bh B4a m4 B21 ỵ n4 B12 ỵ B11 ỵ B22 2B66 m2 n2 B2a A11 B4a m4 ỵ A22 n4 ỵ A66 2A12 thin FGM plates using a simple power-law distribution (P-FGM) under temperatures Both of FGM plate and stiffeners are deformed under thermal loads The formulations are based on the classical plate theory taking into account geometrical nonlinearity, initial geometrical imperfection, temperature-dependent properties and the Lekhnitsky smeared stiffeners technique with Pasternak type elastic foundation Using Galerkin method and stress function, effects of material and geometrical properties, elastic foundation and eccentrically outside stiffeners on the buckling and postbuckling loading capacity of the imperfect eccentrically stiffened P-FGM plate in thermal environments are analyzed and discussed Some results were compared with the ones of the other authors B12 A11 ỵ A12 B22 À B C A22 A11 À A12 C 4nmB2a B BÀ C À Á Á À B B4 m4 B21 ỵ n4 B12 ỵ B11 ỵ B22 2B66 m2 n2 B2 C Bh @ a a A A11 B4a m4 ỵ A22 n4 ỵ A66 2A12 m2 n2 B2a   B11 A11 ỵ A12 B21 4m3 B4 4mnB2 a a  À  À nB4h B4h A22 A11 À A212 2À Á A22 B21 ỵ A12 B11 A22 A11 À A212 À À Á Á7 6 B4 m4 B21 ỵ n4 B12 ỵ B11 þ B22 À 2B66 m2 n2 B2 a a A11 Ba m4 ỵ A22 n4 ỵ A66 2A12 m2 n2 Ba m2 n2 B2a Á 112 N Dinh Duc, P Hong Cong / Thin-Walled Structures 75 (2014) 103–112   A12 B12 ỵ A22 B22 n3   mB4h A22 A11 À A212 ! 8mnB2a B21 B12  ỵ b2 ẳ  B2a m2 ỵ n2 B2h A11 A22 π2 b2 ¼  B2a m2 ỵ n2 b3 ẳ  n2 8B2h A22 A11 À A212   3 m2 B2a 2 B A ỵ n A m ỵ 11 12 a 8Bh A22 A11 À A212   2 A12 m Ba ỵ A22 n m2 B2a þ 4n2 Ba A12 n 4 Ba þ B12 n4 ỵ B11 ỵ B22 2B66 ịm2 n2 B2a A11 m4 B4a ỵ A22 n4 ỵ A66 2A12 Þm2 n2 B2a  À B21 m ! A11  m2 B2a ỵ A12 n A11  B m4 B4a ỵ B12 n4 ỵ B11 ỵ B22 À 2B66 Þm2 n2 B2a A12 n À Banm 21 A11 m4 B4a ỵ A22 n4 ỵ A66 2A12 Þm2 n2 B2a A11 Ba m Â4 ỵ B22 n ỵ Ba B21 m ỵ  B2h  A11 Ba m b3 A11 n   B2h m2 B2a ỵ A12 n A11  Ã2 > > B21 m4 B4a ỵ B12 n4 ỵ B11 ỵ B22 2B66 Þm2 n2 B2a > > > > À > > > > A11 m4 Ba ỵ A22 n4 þ ðA66 À 2A12 Þm2 n2 Ba = < 4 2  À D11 m Ba D22 n D12 ỵ D21 ỵ 4D66 ịm n Ba > > > >   > > > > K D11 B4a B2a > > ; : m2 B2a ỵ n2 K Dπ11 b3 ¼ À ¼ 8mnB2a 3B2h b3 ẳ 16   m2 B2a ỵ A12 n A11 2 m2 B2a ỵ n A12 A 11   B21 A11 m4 B4a B2h A22 ỵ þ B12 ! A22 n4 A11 ! þ n4 π   8A11 B2h m2 B2a ỵ n2 A12 A11 ¼ hA11 ; A12 ¼ hA12 ; A22 ¼ hA22 ; A66 ¼ hA66 B11 ¼ D11 h ; D22 ¼ D22 h ; D12 ¼ D12 h ; D21 ¼ D21 h ; D66 ¼ D66 h Fy k1 a4 k2 a2 W b b ; K2 ¼ ; W ¼ ; Ba ¼ ; Bh ¼ ; β ¼ h a h D11 D11 Fx References  32mnB2a 3B2h K1 ¼  B4a m4 ỵn 16B2h A22 A11 6 4ỵ D11 ẳ B11 B12 B21 B22 B66 ; B12 ¼ ; B21 ¼ ; B22 ¼ ; B66 ¼ h h h h h A11 [1] Bich DH, Nam VH, Phuong NT Nonlinear postbuckling of eccentrically stiffened functionally graded plates and shallow shells Vietnam J Mech 2011;33(3):131–47 [2] Bich DH, Dung DV, Nam VH Nonlinear dynamic analysis of eccentrically stiffened imperfect functionally graded double curved thin shallow shells J Compos Struct 2013;96:384–95 [3] Bich DH, Dung DV, Nam VH Nonlinear dynamic analysis of eccentrically stiffened functionally graded cylindrical panels J Compos Struct 2012;94: 2465–73 [4] Dung DV, Thiem HT On the nonlinear stability of eccentrically stiffened functionally graded imperfect plates resting on elastic foundation In: Proceedings of the 2nd international conference on engineering mechanics and automation (ICEMA2), Hanoi; 2012, p 216–225 [5] Duc ND Nonlinear dynamic response of imperfect eccentrically stiffened FGM double curved shallow shells on elastic foundation J Compos Struct 2013;99:88–96 [6] Liew KM, Jang J, Kitipornchai S Postbuckling of piezoelectric FGM plates subject to thermo-electro-mechanical loading Int J Solids Struct 2003;40: 3869–92 [7] Samsam Shariat BA, Eslami MR Buckling of thick functionally graded plates under mechanical and thermal loads J Compos Struct 2007:433–9 [8] Shen HS Thermal postbuckling behavior of shear deformable FGM plates with temperature-dependent properties Int J Mech Sci 2007;49:466–78 [9] Huang ZY, Lu CF, Chen WQ Benchmark solutions for functionally graded thick plates resting on Winkler–Pasternak elastic foundations J Compos Struct 2008;85:95–104 [10] Zenkour AM Hygro-thermo-mechanical effects on FGM plates resting on elastic foundations J Compos Struct 2010;93:234–8 [11] Shen HS, Wang ZX Nonlinear bending of FGM plates subjected to combined loading and resting on elastic foundations J Compos Struct 2010;92:2517–24 [12] Duc ND, Nam D, Tung HV Effects of elastic foundation on nonlinear stability of FGM plates under compressive and thermal loads In: Proceedings of Xth national conference on mechanics of deformed solid, Thai Nguyen, Vietnam; 2010, p.191–197 [13] Brush DD, Almroth BO Buckling of bars, plates and shells Mc Graw-Hill; 1975 [14] Reddy JN Mechanics of laminated composite plates and shells: theory and analysis Boca Raton: CRC Press; 2004 [15] Tung HV, Duc ND Nonlinear analysis of stability for functionally graded plates under mechanical and thermal loads J Compos Struct 2010;92:1184–91 [16] Duc ND, Tung HV Mechanical and thermal post-buckling of shear-deformable FGM plates with temperature-dependent properties J Mech Compos Mater 2010;46(5):461–76 [17] Duc ND, Tung HV Mechanical and thermal postbuckling of higher order shear deformable functionally graded plates on elastic foundations J Compos Struct 2011;93:2874–81 [18] D.V Dung, N.T Nga, Nonlinear stability analysis of imperfect functionally graded plates with the Poisson's ratio v ¼v(z) subjected to mechancal and thermal loads In: Proiceedings of Xth national conference on mechanics of deformed solid, Thai Nguyen, Vietnam; 2010, p.142–154  ... of thermo-mechanical loads on nonlinear response of FGM plates in thermal environment Fig investigated effects of temperatures on postbuckling of eccentrically stiffened FGM plate under compressive... (19) and (22) are nonlinear equations in terms of variables w and f and used to investigate the stability of FGM plate on elastic foundations subjected to mechanical, thermal and thermomechanical... plates resting on elastic foundation In: Proceedings of the 2nd international conference on engineering mechanics and automation (ICEMA2), Hanoi; 2012, p 216–225 [5] Duc ND Nonlinear dynamic response