DSpace at VNU: Nonlinear postbuckling of imperfect eccentrically stiffened P-FGM double curved thin shallow shells on el...
Composite Structures 106 (2013) 590–600 Contents lists available at SciVerse ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct Nonlinear postbuckling of imperfect eccentrically stiffened P-FGM double curved thin shallow shells on elastic foundations in thermal environments Nguyen Dinh Duc ⇑, Tran Quoc Quan Vietnam National University, Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Viet Nam a r t i c l e i n f o Article history: Available online 14 July 2013 Keywords: Nonlinear postbuckling Eccentrically stiffened P-FGM double curved thin shallow shells Imperfection Elastic foundation Thermal environments a b s t r a c t This paper first time presents an analytical investigation on the nonlinear postbuckling for imperfect eccentrically stiffened FGM double curved thin shallow shells on elastic foundation using a simple power-law distribution (P-FGM) in thermal environments The formulations are based on the classical shell 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, explicit relations of thermal load–deflection curves for simply supported curved eccentrically stiffened FGM shells are determined Effects of material and geometrical properties, temperature, elastic foundation and eccentrically stiffeners on the buckling and postbuckling loading capacity of the imperfect eccentrically stiffened P-FGM double curved shallow shells in thermal environments are analyzed and discussed Ó 2013 Elsevier Ltd All rights reserved Introduction Functionally Graded Materials (FGMs), which are microscopically composites and made from mixture of metal and ceramic constituents, have received considerable attention in recent years due to their high performance heat resistance capacity and excellent characteristics in comparison with conventional composites By continuously and gradually varying the volume fraction of constituent materials through a specific direction, FGMs are capable of withstanding ultrahigh temperature environments and extremely large thermal gradients Therefore, these novel materials are chosen to use in temperature shielding structure components of aircraft, aerospace vehicles, nuclear plants and engineering structures in various industries As a result, buckling and postbuckling behaviors of FGM plate and shell structures under different types of loading are attractive to many researchers in the world Regarding to the static buckling and postbuckling of FGM shells, Shen has studied postbuckling of FGM cylindrical panels under axial compression [1], external pressure [2] by using higher order shell theory in conjunction with boundary layer theory of shell buckling Shen and Liew also investigated the postbuckling of FGM cylindrical panels with piezoelectric actuators in thermal ⇑ Corresponding author Tel.: +84 37547978; fax: +84 37547424 E-mail address: ducnd@vnu.edu.vn (N.D Duc) 0263-8223/$ - see front matter Ó 2013 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.compstruct.2013.07.010 environments [3] FGM cylindrical shells in thermal environment under various loading types has been treated by similar methods and different shell theories [4–7] For Shen’s works, an semianalytical approach is used to expand deflection and stress functions in form of power functions of small parameters, and then an iteration is adopted to determine buckling loads and postbuckling curves Huang and Han [8,9] investigated the nonlinear buckling of FGM cylindrical shells under axial and external pressure by a semi-analytical approach Duc and Tung presented analytical investigations on the nonlinear response of imperfect FGM cylindrical shells under axial compression [10] and thermal loads [11] Geometrically nonlinear analysis of functionally graded shells is considered by Zhao and Liew [12] Sohn and Kim investigated structural stability of FGM shells subjected to aero-thermal loads [13] Sofiyev studied the stability of compositionally graded ceramic–metal cylindrical shells under periodic axial impulsive loading [14] In the recent years, there have been several works for the more complicated FGM shells for example: spherical, conical and double curved FGM shallow shells Shahsiah et al [15] studied the linear buckling of shallow FGM spherical shells under two types of thermal loads Naj et al [16] used an analytical method and adjacent equilibrium criterion with assumption on small deflection to determine critical buckling loads of FGM truncated conical shells subjected to mechanical and thermal loadings The stability of FGM truncated conical shells under compression, external pressure, impulsive and thermal loads also treated in works by Sofiyev using 591 N.D Duc, T.Q Quan / Composite Structures 106 (2013) 590–600 an analytical method [17–19] Bich and Tung used an analytical approach to investigate the nonlinear buckling of FGM shallow spherical shell under uniform external pressure including temperature effects [20] and conical panels under mechanical loads [21] Duc and Quan [30] studied nonlinear stability of double curved shallow FGM panel The components of structures widely used in aircraft, reusable space transportation vehicles and civil engineering are usually supported by an elastic foundation Therefore, it is necessary to include effects of elastic foundation for a better understanding of the buckling behavior and loading carrying capacity of plates and shells Librescu and his co-workers have investigated the postbuckling behavior of flat and curved laminated composite shells resting on Winkler elastic foundations [22,23] In spite of practical importance and increasing use of FGM structures, investigations on the effects of elastic media on the response of FGM plates and shells are comparatively scarce Bending behavior of FGM plates on Pasternak type foundations has been studied by Huang et al [24] and Zenkour [25] using analytical methods, Shen and Wang [26] making use of asymptotic perturbation technique Shen et al [27,28] investigated the postbuckling behavior of FGM cylindrical shells subjected to axial compressive loads and internal pressure and surrounded by an elastic medium of tensionless elastic foundation of the Pasternak type Duc et al used the third order shear deformation theory plate for studying nonlinear postbuckling of FGM plates [29] and classical theory of shells for studying nonlinear postbuckling of double curved shallow FGM panel on elastic foundation [30] Recently, Duc extend his investigations for nonlinear dynamic of the FGM shells on elastic foundation [38] However, in practice, the FGM structure are usually exposed to high-temperature environments, where significant changes in material properties are unavoidable Therefore, the temperature dependence of their properties should be considered for an accurate and reliable prediction of deformation behavior of the composites In [31,32] Shen studied thermal postbuckling behavior of functionally graded cylindrical shells and panels with temperature-dependent properties Also Shen investigated postbuckling analysis of axially loaded piezolaminated cylindrical panels with temperature-dependent properties [33] Yang et al studied thermo-mechanical postbuckling of FGM cylindrical panels with temperature-dependent properties [34] It is evident from the literature that investigations considering the temperature dependence of materials properties for FGM shells are few in number Notice that in all the publication mentioned above [31–34], all authors use the displacement functions and volume fraction follows a simple power law (P-FGM) Recently, Duc and Quan investigated nonlinear buckling and postbuckling for FGM double curved shallow FGM panel on elastic foundation in thermal environments (without temperature-dependent properties) using stress function [30] In fact, 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 However, up to date, the investigation on static and dynamic of eccentrically stiffened FGM structures has received comparatively little attention Bich et al studied (without temperatures) static and dynamic analysis for eccentrically stiffened FGM shallow shells [35,36] and dynamic analysis for eccentrically stiffened functionally graded cylindrical panels [37] Duc investigated nonlinear dynamic response of imperfect eccentrically stiffened doubly curved FGM shallow shells on elastic foundations [38] To the best of our knowledge, there has been recently 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 Our paper is the first proposal for an imperfection eccentrically stiffened FGM double curved shallow shells on elastic foundation in which we investigate the nonlinear postbuckling using a simple power-law distribution (P-FGM) under thermal environments Here, we have considered the FGM shell under temperature independent material property, i.e the Young’s modulus E, thermal expansion coefficient a, the mass density q, the thermal conduction K and even Poisson ratio m 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 shells is 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 shell theory taking into account geometrical nonlinearity, initial geometrical imperfection, temperature-dependent properties and the Lekhnitsky smeared stiffeners technique with Pasternak type elastic foundation Explicit expressions of buckling loads and postbuckling loads–deflection curves for simply supported double curved shallow thin FGM shells are determined by Galerkin method and using stress function The effects of geometrical and material properties, temperature, elastic foundation and eccentrically stiffeners on the nonlinear response of the P-FGM shallow shells in thermal environments are analyzed and discussed Eccentrically stiffened FGM double curved shallow shells on elastic foundations Consider a ceramic–metal eccentrically stiffened FGM double curved shallow shell of radii of curvature Rx, Ry length of edges a, b and uniform thickness h resting on an elastic foundation (Fig 1) For FGM shell, the volume fractions of constituents are assumed to vary through the thickness according to the following power law distribution (P-FGM): V m zị ẳ N 2z ỵ h ; V c zị ẳ V m zị 2h 1ị where N is volume fraction index (0 N < 1) Effective properties Preff of FGM shell are determined by linear rule of mixture as Preff z; Tị ẳ Prm TịV m zị ỵ Prc TịV c zị 2ị where Pr denotes a temperature dependent material property, and subscripts m and c stand for the metal and ceramic constituents, respectively Specific expressions of material coefficients are obtained by substituting Eq (1) into Eq (2) as z h b a y x Ry Rx Fig Geometry and coordinate system of an eccentrically stiffened double curved FGM shell on elastic foundation 592 N.D Duc, T.Q Quan / Composite Structures 106 (2013) 590–600 ẵEz; Tị; v z; Tị; qz; Tị; az; Tị; Kz; Tị ẳ ẵEc Tị; mc Tị; qc Tị; ac Tị; K c Tị ỵ ẵEmc Tị; mmc Tị; qmc Tị; amc Tị; K mc Tị 3ị rsh xy ẳ Emc Tị ẳ Em Tị Ec Tị; mmc Tị ¼ mm ðTÞ À mc ðTÞ; qmc ðTÞ ¼ qm Tị qc Tị; amc Tị ẳ am Tị ac Tị;K mc Tị ẳ K m Tị K c ðTÞ ð4Þ The values with subscripts m and c belong to metal and ceramic respectively It is evident from Eqs (3) and (4) that the upper surface of the shell (z = Àh/2) is ceramic-rich, while the lower surface (z = h/2) is metal-rich, and the percentage of ceramic constituent in the shell is enhanced when N increases A material property Pr, such as the elastic modulus E, Poisson ratio m, the mass density q, the thermal expansion coefficient a and coefficient of thermal conduction K can be expressed as a nonlinear function of temperature [31–34]: Pr ¼ P0 P1 T ỵ ỵ P T þ P2 T þ P3 T Þ ð5Þ in which T = T0 + DT(z) and T0 = 300 K (room temperature); P0, PÀ1, P1, P2 and P3 are coefficients characterizing of the constituent materials The shell–foundation interaction is represented by Pasternak model as qe ¼ k1 w À k2 r2 w 2 ð6Þ 2 where r = @ /@x + @ /oy , w is the deflection of the shell, k1 is Winkler foundation modulus and k2 is the shear layer foundation stiffness of Pasternak model Theoretical formulation In this study, the classical shell theory, the Lekhnitsky smeared stiffeners technique are used to establish governing equations and determine the nonlinear response of FGM double curved thin shallow shells and took into account for FGM shells as well as stiffeners which are both deformed by temperature According to the classical thin shell theory the strain at the middle surface and curvatures are related to the displacement components u, v, w in the x, y, z coordinate as: 0 1 ex ex kx B e C B e0 C B C @ y A ẳ @ y A ỵ z@ ky A cxy 2kxy c0xy Eðz; TÞ c 2ð1 þ mðz; TÞÞ xy where DT is temperature rise from stress free initial state, and more generally, DT = DT(z); E(z, T), v(z, T) are the FGM shell’s elastic moduli which is determined by (3) For stiffeners in thermal environments with temperaturedependent properties, we have proposed its form adapted from [37] as the follows: rstx ; rsty ¼ E0 ex ; ey ị Nx ẳ 1 e0x u;x w=Rx ỵ w2;x =2 Àwx;x kx B e0 C B C B C B C @ y A ¼ @ v ;y À w=Ry þ w2;y =2 A; @ ky A ¼ @ Àwy;y A w;xy kxy cxy u;y ỵ v ;x ỵ w;x w;y I10 ỵ In which u, v are the displacement components along the x, y directions, respectively Interestingly, comparing to the other [35–38], we have assumed that the eccentrically outside stiffeners depend on temperature Hooke law for an FGM shell with temperature-dependent properties is defined as ET0 AT1 sT1 ! 10ị e0x ỵ I20 e0y ỵ I11 þ C T1 kx þ I21 ky þ U1 Ny ẳ I20 e0x ỵ I10 ỵ ET0 AT2 sT2 ! e0y ỵ I21 kx ỵ I11 ỵ C T2 ky ỵ U1 Nxy ẳ I30 c0xy ỵ 2I31 kxy ! ET I T M x ẳ I11 ỵ e ỵ e ỵ I12 ỵ 0T kx ỵ I22 ky ỵ U2 s1 ! ET IT M x ẳ I21 e0x ỵ I11 ỵ C T2 e0y ỵ I22 kx ỵ I12 ỵ 0T ky þ U2 s2 ð8Þ E0 a0 ðTÞDðTÞð1; 1Þ À 2m0 ðTÞ Here, E0 = E0(T); m0 = m0(T), a0 = a0(T) are the Young’s modulus, Poisson ratio and thermal expansion coefficient of the stiffeners, respectively The shell 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 shell and stiffeners, suppose that stiffeners are made of full metal (E0 = Em) if putting them at the metal-rich side of the shell, and conversely full ceramic stiffeners (E0 = Ec) at the ceramic-rich side of the shell (this assumption has been used in [35–38]) The shallow shell is assumed to have a relative small rise as compared with its span The contribution of stiffeners can be accounted for using the Lekhnitsky smeared stiffeners technique [35–38] In order to investigate the FGM shells 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 shells and stiffener are temperature dependence and they are deformed in the presence of temperature Hence, the geometric parameters, the shell’s shape and stiffeners are varied through the deforming process due to the temperature change However, we have assumed that the thermal stress of stiffeners is subtle which distributes uniformly through the whole shell structure Therefore, we can ignore it and Lekhnitsky smeared stiffeners technique can be adapted from [35–38] as the follows: ð7Þ where e0x , e0x and c0xy are normal and shear strain at the middle surface of the shell, and kx, ky, kxy are the curvatures and Rx, Ry are radii of curvatures, 1/Rx, 1/Ry are principal curvatures of the shell The nonlinear strain–displacement relationship based upon the von Karman theory for moderately large deflection and small strain are [39]: Ez; Tị ẵex ; ey ị ỵ mey ; ex ị ỵ mịaDTzị1; 1ị m2 z; Tị 9ị N 2z ỵ h 2h where sh rsh ¼ x ; ry C T1 x 11ị I21 0y M xy ẳ I31 c0xy þ 2I32 kxy The relation (11) is our most important finding, where Iij (i = 1, 2, 3; j = 0, 1, 2): 593 N.D Duc, T.Q Quan / Composite Structures 106 (2013) 590–600 Z I1j ¼ h=2 Àh=2 Z I2j ẳ Ezị mzị Ezịmzị h=2 mzị2 h=2 h=2 Z h=2 h=2 ẳ A11 ẳ zj dz 12ị Ezị zj dz ẳ I1j I2j ị 2ẵ1 ỵ mzị U1 ; U2 ị ẳ À IT1 zj dz h=2 Z I3j ¼ ! 3 T T d1 h1 12 ỵ Ezịazị DTzị1; zÞdz À mðzÞ À Á2 AT1 zT1 ; IT2 C T1 ¼ E0 AT1 zT1 T E0 AT2 zT2 ; C2 ẳ sT1 sT2 zT1 ẳ h1 ỵ h T h2 ỵ h ; z2 ẳ 2 T T T T ¼ 3 T T d2 h2 12 ỵ AT2 zT2 I20 ; A66 ¼ I30 D ! ! ET0 AT1 ET0 AT2 D ẳ I10 ỵ T I10 ỵ T I220 s1 s2 B11 ẳ A22 I11 ỵ C T1 A12 I21 B22 ẳ A11 I11 ỵ C T2 À A12 I21 B12 ¼ A22 I21 A12 I11 ỵ C T2 B21 ẳ A11 I21 A12 I11 ỵ C T1 A12 ¼ B66 ¼ T E T AT E T AT I10 ỵ T ; A22 ẳ I10 ỵ T D D s1 s2 ! I31 I30 Substituting once again Eq (15) into the expression of Mij in (11), then Mij into the Eq (14c) leads to: T AT1 ¼ d1 sT1 ; AT2 ¼ d2 sT2 Nx;x ỵ Nxy;y ẳ Nxy;x ỵ N y;y ¼ Where the geometric shapes of stiffeners after the thermal deformation process in Eq (12) can be determined as the follows: T d2 ẳ d2 ỵ am Tzịị; h1 ẳ h1 ỵ am Tzịị; T h2 ẳ h2 ỵ am Tzịị; zT1 ẳ z1 ỵ am Tzịị; zT2 ẳ z2 ỵ am Tzịị; sT1 ẳ s1 ỵ am Tzịị; T d1 d1 ẳ d1 ỵ am Tzịị; T ẳ d1 ỵ am Tzịị 13ị 14aị Nxy;x ỵ Ny;y ẳ 14bị Nx Ny ỵ ỵ Nx w;xx ỵ 2Nxy w;xy þ Ny w;yy Rx Ry þ q À k1 w ỵ k2 r2 w ẳ 14cị e ẳ A22 Nx A12 Ny ỵ B11 w;xx ỵ B12 w;yy A22 A12 ịU1 c0xy ẳ A66 Nxy ỵ 2B66 w;xy D22 D12 D21 18ị D66 ẳ I32 I31 B66 f(x, y) is stress function defined by Nx ¼ f;yy ; Ny ¼ f;xx ; Nxy ¼ Àf;xy ð19Þ For an imperfect FGM curved shell, Eq (17) are modied into form as B21 f;xxxx ỵ B12 f;yyyy ỵ B11 ỵ B22 2B66 ịf;xxyy D11 w;xxxx D22 w;yyyy D12 ỵ D21 ỵ 4D66 ịw;xxyy ỵ f;yy w;xx ỵ w;xx 2f ;xy w;xy ỵ w;xy ỵ f;xx w;yy ỵ w;yy f;yy f;xx ỵ ỵ q k w ỵ k r2 w ẳ Rx Ry 20ị in which w⁄(x, y) is a known function representing initial small imperfection of the shell The geometrical compatibility equation for an imperfect double curved shallow shell is written as [30,38]: e0x;yy ỵ e0y;xx c0xy;xy ẳ w2;xy w;xx w;yy þ 2w;xy wÃ;xy À w;xx wÃ;yy x e ¼ A11 Ny A12 Nx ỵ B21 w;xx ỵ B22 w;yy À ðA11 À A12 ÞU1 ET0 IT1 B11 I11 ỵ C T1 I21 B21 T s1 ET I T ẳ I12 ỵ 0T B22 I11 ỵ C T2 I21 B12 s2 ẳ I22 B12 I11 ỵ C T1 À I21 B22 ¼ I22 À B21 I11 ỵ C T2 I21 B11 D11 ẳ I12 þ þ Calculated from Eq (11) y B21 f;xxxx þ B12 f;yyyy þ ðB11 þ B22 À 2B66 Þf;xxyy D11 w;xxxx D22 w;yyyy where Nx;x ỵ Nxy;y ẳ M x;xx ỵ 2M xy;xy ỵ M y;yy þ ð17Þ À ðD12 þ D21 þ 4D66 Þw;xxyy þ Nx w;xx ỵ 2Nxy w;xy ỵ Ny w;yy Nx Ny þ þ þ q À k1 w þ k2 r2 w ¼ Rx Ry T Where the coupling parameters C1, C2 are negative for outside stiffeners and positive for inside one; s1, s2 are the spacing of the longitudinal and transversal stiffeners; I1, I2 are the second moments of cross-section areas; z1, z2 are the: eccentricities of stiffeners with respect to the middle surface of shell; and the width and thickness of longitudinal and transversal stiffeners are denoted by d1, h1 and d2, h2 respectively A1, A2 are the cross-section 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 FGM double curved shallow shell based on the classical shell theory are [30,38]: where ð16Þ ð15Þ À w;yy wÃ;xx À w;yy w;xx À : Rx Ry ð21Þ From the constitutive relations (15) in conjunction with Eq (19) one can write 594 N.D Duc, T.Q Quan / Composite Structures 106 (2013) 590–600 e0x ¼ A22 f;yy À A12 f;xx þ B11 w;xx þ B12 w;yy À ðA22 À A12 ịU1 e0y ẳ A11 f;xx A12 f;yy ỵ B21 w;xx ỵ B22 w;yy A11 A12 ịU1 c0xy ẳ A66 f;xy ỵ 2B66 w;xy 22ị Setting Eq (22) into Eq (21) gives the compatibility equation of an imperfect FGM double curved shell as A11 f;xxxx ỵ A22 f;yyyy þ ðA66 À 2A12 Þf;xxyy þ B21 w;xxxx þ B12 w;yyyy ỵ B11 ỵ B22 2B66 ịw;xxyy w;yy w;xx ẳ0 w2;xy w;xx w;yy ỵ 2w;xy wÃ;xy À w;xx wÃ;yy À w;yy wÃ;xx À À Rx Ry ð23Þ Eqs (20) and (23) are nonlinear equations in terms of variables w and f and used to investigate the stability of FGM double curved shells on elastic foundations subjected to mechanical, thermal and thermo-mechanical loads In the present study, the edges of curved shells are assumed to be simply supported Depending on the in-plane restraint at the edges, three cases of boundary conditions [26–30], labeled as Cases 1, and may be considered Case Four edges of the shell are simply supported and freely movable (FM) The associated boundary conditions are w ¼ Nxy ¼ M x ¼ 0; Nx ¼ Nx0 at x ¼ 0; a w ¼ Nxy ¼ M y ¼ 0; Ny ¼ Ny0 at y ¼ 0; b: ð24Þ Case Four edges of the shell are simply supported and immovable (IM) In this case, boundary conditions are w ¼ u ¼ Mx ¼ 0; Nx ¼ Nx0 at x ¼ 0; a w ¼ v ¼ M y ¼ 0; Ny ẳ Ny0 at y ẳ 0; b: 25ị Case All edges are simply supported Two edges x = 0, a are freely movable, whereas the remaining two edges y = 0, b are immovable For this case, the boundary conditions are defined as w ¼ Nxy ¼ M x ¼ 0; w ¼ v ¼ M y ¼ 0; Nx ¼ Nx0 at x ¼ 0; a Ny ¼ Ny0 at y ¼ 0; b ð26Þ where Nx0, Ny0 are in-plane compressive loads at movable edges (i.e Case and the first of Case 3) or are fictitious compressive edge loads at immovable edges (i.e Case and the second of Case 3) The approximate solutions of w and f satisfying boundary conditions (24)–(26) are assumed to be [29,30,38]: w; w ị ẳ W; lhị sin km x sin dn y 27aị f ẳ A1 cos 2km x ỵ A2 cos 2dn y ỵ A3 sin km x sin dn y ỵ Nx0 y2 2 þ Ny0 x ð27bÞ km = mp/a, dn = np/b W is amplitude of the deflection and l is imperfection parameter The coefficients Ai (i = Ä 3) are determined by substitution of Eqs (27a, 27b) into Eq (23) as A1 ẳ d2n 32A11 k2m WW ỵ 2lhị; A2 ẳ k2m 32A22 d2n WW ỵ 2lhị; ! d2n k2m A3 ẳ ỵ W A11 k4m ỵ A22 d4n ỵ A66 2A12 ịk2m d2n Rx Ry B21 k4 ỵ B12 d4n ỵ B11 ỵ B22 À 2B66 Þk2m d2n W À À m A11 km ỵ A22 d4n ỵ A66 2A12 Þk2m d2n ð28Þ Subsequently, substitution of Eqs (27a) and (27b) into Eq (20) and applying the Galerkin procedure for the resulting equation yield 4 2 d2n k2m ẵB21 km ỵB12 dn ỵB11 ỵB22 À2B66 Þkm dn > > > > > > Rx ỵ Ry A11 k4m ỵA22 d4n ỵA66 2A12 Þk2m d2n > > ½ > > > > > > > > > > > > 4 2 > > B k ỵB d ỵB ỵB 2B ịk d ẵ 21 m 12 n 11 22 66 m n > > > >À = < 4 2 A11 km ỵA22 dn ỵA66 2A12 ịkm dn ẵ mnp W 4km dn > > > > > > d2n k2m > > > > Rx ỵ Ry > > ẵA11 k4m ỵA22 d4n ỵA66 2A12 ịk2m d2n > > > > > > > > > > > > > > 4 2 2 : ÀD11 km À D22 dn À D12 ỵ D21 ỵ 4D66 ịkm dn k2 km þ dn À k1 ; 2 3 dn ỵ kRmy A11 k4m ỵA22 d4n ỵA66 2A12 ịk2m d2n ị Rx 7WW ỵ lhị ỵ 8km3 dn B k4 ỵB d4 ỵB ỵB À2B Þk2 d2 ð 21 m 12 n 11 22 66 m n ị A k4 ỵA d4 þðA À2A Þk2 d2 ð 11 m 22 n 66 12 m n Þ h i k ỵ AB12 km dn WW ỵ 2lhị þ 12k1m dn A22mRx þ A11dnRy À 13 AB21 11 22 ab 64 ỵ km4dn k4m þ Ad11n A22 N x0 k2m þ Ny0 d2n ðW þ lhÞ WðW þ lhÞðW þ 2lhị ab N Nx0 ỵ Ry0y Rx ỵ km4qdn ẳ 29ị where m, n are odd numbers This is basic equation governing the nonlinear response of eccentrically stiffened FGM double curved shallow shells under mechanical, thermal and thermo-mechanical loading conditions In what follows, some thermal loading conditions will be considered It is not so difficult to realize that the Eq (29) is more complicated than the equation written in [35–38] without the temperature A simply supported FGM curved shell on elastic foundations with all immovable edges is considered The shell is subjected to uniform external pressure q and simultaneously exposed to temperature environments or subjected to through the thickness temperature gradient The in-plane condition on immovability at all edges, i.e u = at x = 0, a and v = at y = 0, b, is fulfilled in an average sense as [27–34]: Z b Z a @u dxdy ¼ 0; @x Z a Z b @v dydx ¼ 0: @y ð30Þ From Eqs (7) and (15) one can obtain the following expressions in which Eq (19) and imperfection have been included @u ẳ A22 f;yy A12 f;xx ỵ B11 w;xx ỵ B12 w;yy @x w A22 A12 ịU1 w2;x w;x w;x ỵ Rx @v ẳ A11 f;xx A12 fyy ỵ B22 w;yy ỵ B21 w;xx A11 A12 ịU1 @y w w2;y w;y w;y ỵ : Ry ð31Þ Substitution of Eqs (27a) and (27b) into Eq (31) and then the result into Eq (30) give fictitious edge compressive loads as n Nx0 ¼ U1 þ ðA A ÀA2 Þ À mn1p2 AR11x þ AR12y 11 22 12 2 3 2 33 dn > ỵ km = 4 2 6 A11 km ỵA22 dn ỵA66 2A12 ịkm dn ị Rx Ry 77 ỵ42A12 A11 ỵ A12 B21 ỵ A11 B11 55 nam2 W B21 k4m ỵB12 d4n ỵB11 ỵB22 2B66 ịk2m d2n ị > ; A k4 ỵA d4 ỵA 2A ịk2 d2 ð 11 m 22 n 66 12 m n Þ 2 33 2 d k n m A11 k4m ỵA22 d4n þðA66 À2A12 Þk2m d2n Þ Rx þ Ry o 6 77 ỵ4 A12 ỵ A11 A22 ỵ A12 B22 ỵ A11 B12 55mbn W B21 k4m ỵB12 d4n ỵB11 ỵB22 2B66 ịk2m d2n ị A k4 ỵA d4 ỵA 2A ịk2 d2 11 m 22 n 66 12 m n Þ À ỵ A A1 A2 A11 k2m ỵ A12 d2n WW ỵ 2lhị 11 22 12 ị 32ị 595 N.D Duc, T.Q Quan / Composite Structures 106 (2013) 590–600 n elevated and nonlinear steady temperature conduction is governed by one-dimensional Fourier equation À mn1p2 AR12x þ AR22y ðA11 A22 ÀA212 Þ 2 3 2 33 dn > ỵ kRmy > A11 k4m ỵA22 d4n ỵA66 2A12 ịk2m d2n ị Rx ð 6 77 = ỵ A12 B11 ỵ A22 B21 77 m2 W þ6 A þ A A 11 22 12 4 B k4 ỵB d4 ỵB ỵB 2B ịk2 d2 55 na > ð 21 Þ > 12 11 22 66 ; A mk4 ỵA n d4 ỵA 2A Þk2 dm2 n ð 11 m 22 n 66 12 m n Þ 2 3 2 33 dn > ỵ km > 4 2 6 A11 km ỵA22 dn ỵA66 2A12 ịkm dn Þ Rx Ry 77 n = 7 A ỵ A B ỵ A B W þ6 2A 22 22 12 12 55 12 22 B k4 ỵB d4 ỵB ỵB 2B ịk2 d2 mb2 > ð 21 Þ > 12 11 22 66 ; A mk4 ỵA n d4 ỵA 2A Þk2 dm2 n ð 11 m 22 n 66 12 m n ị ỵ A A A2 A12 k2m ỵ A22 d2n WW ỵ 2lhị 11 22 12 ị N y0 ẳ U1 ỵ ! d dT ẳ 0; Kzị dz dz Tz ẳ h=2ị ¼ T c ; Tðz ¼ h=2Þ ¼ T m : ð37Þ Using K(z) in Eq (3), the solution of Eq (37) may be found in terms of polynomial series, and the first eight terms of this series gives the following approximation [29–31]: j P NK mc =K c Þ r 5jẳ0 r jNỵ1 Tzị ẳ T m ỵ DT À DT P ðÀK mc =K c Þj ð33Þ j¼0 Specific expressions of parameter U1 in two cases of thermal loading will be determined Obviously, in the presence of temperature, the expressions for eccentrically stiffened FGM double curved shallow shells are more complicated than the results in the absence of temperature 38ị jNỵ1 where r = (2z + h)/2h and DT = Tc À Tm is defined as the temperature change between two surfaces of the FGM shallow shell Introduction of Eq (38) into Eq (12) gives the thermal parameter as U1 ẳ L HịhDT Numerical results and discussion ð39Þ where P5 ðÀK mc =K c Þj jNỵ1 h Ec ac jNỵ2 amc ỵEmc ac Emc amc þ Ecðjþ1ÞNþ2 þ ðjþ2ÞNþ2 i Here, several numerical examples will be presented for perfect and imperfect simply supported midplane-symmetric FGM shells The silicon nitride and stainless steel are regarded as constituents of the FGM shells The typical values of the coefficients of the materials mentioned in (5) are listed in Table Subsequently, setting Eq (34) into Eq (32) and then the result into Eq (29) give 4.1 Uniform temperature rise q ẳ b1 W ỵ b2 WW ỵ lị þ b3 WðW þ 2lÞ þ b4 WðW þ lÞ j¼0 H¼ P5 j¼0 1 ðÀK mc =K c ịj jNỵ1 40ị : 1 W ỵ 2lị þ b5 LDT The FGM curved shell is exposed to temperature environments uniformly raised from stress free initial state Ti to final value Tf and temperature difference DT = Tf À Ti is considered to be independent from thickness variable The thermal parameter is obtained from Eq (12) as U1 ¼ LhDT where b1 ¼ mnp2 B4a D11 K 16B4h 34ị ỵ Emc amc P ỵ Ec amc N ỵ 1ịmmc m2mc Z 1 ds ỵ Emc ac ị ỵ N mc À mmc À mc À mmc s À mnp4 B2a D11 K 2 m B ỵ n a 16B4h m np6 B4a D11 þ mn5 p6 D22 þ m3 n3 p6 B2a ðD12 þ D21 þ 4D66 Þ 16B4h mnp4 Ba ðn2 Rax þ m2 Ba Rby Þ 8B3h h B21 m4 B4a þ ðB11 þ B22 À 2B66 Þm2 n2 B2a þ B12 n4 h i A11 m4 B4a ỵ A66 2A12 ịm2 n2 B2a ỵ A22 n4 35ị and P ẳ mmc Ec amc ỵ ac Emc ị Emc amc ỵ Emc amc ac ỵ where Lẳ 41ị 36ị ỵ h mnBa n2 Rax ỵ m2 Ba Rby ị i iỵ m5 np6 B4a 16B2h A11 m4 B4a ỵ A66 2A12 ịm2 n2 B2a ỵ A22 n4 h i2 B21 m4 B4a ỵ B11 ỵ B22 2B66 ịm2 n2 B2a ỵ B12 n4 h i A11 m4 B4a ỵ A66 2A12 ịm2 n2 B2a ỵ A22 n4 4.2 Through the thickness temperature gradient The metal-rich surface temperature Tm is maintained at stress free initial value while ceramic-rich surface temperature Tc is 16B4h Table Material properties of the constituent materials of the considered FGM shells [29–34] Material Property P0 PÀ1 P1 P2 P3 Si3N4 (Ceramic) E (Pa) q (kg/m3) a (KÀ1) k (W/mK) 348.43e9 2370 5.8723eÀ6 13.723 0.24 0 0 À3.70eÀ4 9.095eÀ4 0 2.160eÀ7 0 0 À8.946eÀ11 0 0 201.04e9 8166 12.330eÀ6 15.379 0.3177 0 0 3.079eÀ4 8.086eÀ4 0 À6.534eÀ7 0 0 0 0 m SUS304 (Metal) E (Pa) q (kg/m3) a (KÀ1) k (W/mK) m 596 N.D Duc, T.Q Quan / Composite Structures 106 (2013) 590–600 b2 ¼ À 3B3h h 2m2 n2 p4 B3a n2 Rax ỵ m2 Ba Rby A11 m4 B4a ỵ A66 h 2A12 ịm2 n2 B2a ỵ A22 n4 i 2m2 n2 p4 B2a B21 m4 B4a ỵ B11 ỵ B22 2B66 ịm2 n2 B2a ỵ B12 n4 h i ỵ 3B4h A11 m4 B4a ỵ A66 2A12 ịm2 n2 B2a þ A22 n4 i equations In this paper, we have contribute significantly to this transformation process The numerical results have been calculated with DT = c onst The parameters for the stiffeners are: z1 = 0.0225 (m), z2 = 0.0225 (m), s1 = 0.4 (m), s2 = 0.4 (m) h1 ¼ 0:003ðmÞ; h2 ¼ 0:003ðmÞ; d1 ¼ 0:004ðmÞ; d2 ¼ 0:004mị 42ị b3 ẳ b4 ẳ p A11 m2 B3a Rax ỵ A22 n2 Rby mnp6 48A11 A22 B3h m4 B4a ỵ n4 ! þ m2 n2 p4 B2a B21 12B4h A11 þ B12 ! Fig illustrates the nonlinear postbuckling of pressure–deflection curves relation for eccentrically reinforced spherical shallow A22 A11 256B4h A22 2 mnp m Ba ỵ n2 1 W ỵ lị Ba Rax ỵ Rby Þ b5 ¼ Bh 16B2h where k1 a4 k2 a2 ; K2 ¼ D11 D11 A11 ¼ hA11 ; A22 ¼ hA22 ; A12 ¼ hA12 ; A66 ¼ hA66 B11 B22 B12 B21 B66 B11 ¼ ; B22 ¼ ; B12 ẳ ; B21 ẳ ; B66 ẳ 43ị h h h h h D11 D22 D12 D21 D66 D11 ¼ ; D22 ¼ ; D12 ¼ ; D21 ¼ ; D66 ¼ h h h h h K1 ¼ Eq (41) expresses explicit relation of pressure–deflection curves for eccentrically stiffened FGM curved shells rested on elastic foundations and under combined action of uniformly raised temperature field and uniform external pressure A similar expression for eccentrically stiffened FGM curved shells simultaneously subjected to uniform external pressure and temperature gradient across the thickness may be obtained as Eq (38), provided L is replaced by (L–H) 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 Fig Effects of mechanical loads on the pressure–deflection relation of eccentrically stiffened cylindrical FGM shallow panel (without temperatures) Fig Effects of thermo-mechanical loads on the pressure–deflection relation of spherical FGM shallow shell (without stiffeners) Fig Effects of N index and imperfection on the pressure–deflection curves of reinforced spherical FGM panel N.D Duc, T.Q Quan / Composite Structures 106 (2013) 590–600 shell FGM (1/Rx = 1/Ry = 0.5) in the thermal environment and comparing with Duc [30] (without stiffeners) It is clear that the stiffeners can enhance the thermal loading capacity for the spherical FGM shallow shells Fig describes a nonlinear postbuckling of pressure–deflection curves relation for eccentrically reinforced FGM cylindrical panel (1/Rx = 1/Ry = 0.5) under mechanical loads and comparing with Bich et al [35] (in the presence of stiffeners without temperature) There is no significant difference between our finding (with v = v(z)) and Bich’s report (v = const) Fig shows us the effect of the volume fraction index N and the imperfection l on the nonlinear response for eccentrically Fig Effects of temperature on the nonlinear response of reinforced spherical FGM panel Fig Effects of elastic foundation on the nonlinear response of reinforced spherical FGM shallow shell 597 stiffened FGM spherical shallow shell Obviously, the thermomechanical loading capacity of FGM shells increase as N decreases It is consistent with P-FGM which has been reported in [29–31] and the imperfect plates has a better thermo-mechanical loading capacity than the perfect plates [29,38] Fig illustrates the effect of temperature on a nonlinear postbuckling response for eccentrically stiffened FGM spherical shallow shells in thermal environment with DT = (K), 300 (K), 500 (K), 1000 (K) It also shows us that the temperature is able to reduce the loading capacity of the shells And, it is therefore consistent with the well-known results of the temperature effects on the nonlinear postbuckling response of FGM structures [1,2,11,29–31] Fig Effects of initial imperfect on the postbuckling curves of reinforced spherical FGM shallow shell under uniform temperature rise Fig Comparison of postbuckling behaviors of reinforced and unreinforced spherical FGM shallow shells 598 N.D Duc, T.Q Quan / Composite Structures 106 (2013) 590–600 Fig investigates the effects of elastic foundations on the nonlinear response for eccentrically stiffened spherical shallow shell in thermal environment The curve is a case without an elastic foundation We have seen that the elastic foundation (increasing K1, K2) has enhanced the loading capacity of the shell And, K2 in Pasternak’s model has a stronger effect than K1 in Winkler’s model This is consistent with the conclusions in [22–30] Fig shows us the effect of the imperfection of the initial shape on the non-linear response of eccentrically stiffened spherical shell corresponding to the parameters l = À0.5, À0.2, 0, 0.2, 0.5 Indeed, the imperfection affects very complicated on the postbuckling behaviors of the shells, and imperfect eccentrically stiffened Fig Effects of Poisson’s ratio on the pressure–deflection curves of reinforced spherical FGM shallow shells Fig 10 Effects of ratio b/a on the pressure–deflection curves of reinforced spherical FGM panels spherical shell with the positive coefficient l which seems to be able getting a better loading capacity in the postbuckling period with the large bending In Fig 8, it is obvious that the effects of stiffeners on the nonlinear postbuckling of spherical FGM shallow shells under thermo-mechanical loads We can see that the loading capacity of FGM shell increases in the presence of stiffeners, particularly in the parts of the shell which start the buckling process This is very important in engineering applications Fig illustrates the effects of Poisson ratio on nonlinear response of reinforced spherical FGM shallow shells in the thermal environment Similar to the shell without stiffeners, there is no big difference between m = const and m = m(z) cases [30] However, Fig 11 Effects of ratio b/h on the pressure–deflection curves of reinforced spherical FGM panels Fig 12 Effects of ratio a/Rx on the pressure–deflection curves of imperfect eccentrically stiffened P-FGM double curved shallow shells on elastic foundations in thermal environments N.D Duc, T.Q Quan / Composite Structures 106 (2013) 590–600 599 References Fig 13 Effects of ratio b/Ry on the pressure–deflection curves of imperfect eccentrically stiffened P-FGM double curved shallow shells on elastic foundations in thermal environments the critical loads with m = m(z) gives us the smaller results than these with m = const Fig 10 shows us the effects of a ratio b/a on nonlinear response of the shells under the same temperature condition In the first period, the shell with b/a = 0.75 has the best loading capacity However, in the limit of large bending, the shell with b/a = 1.5 has the best loading capacity, even in the postbuckling period Similarly, Figs 11–13 show us the effects of the geometric parameters b/h and the curvatures of the shell a/Rx, b/Ry on nonlinear response of imperfect eccentrically stiffened P-FGM double curved shallow shells with elastic foundations in the thermal environment Concluding remarks This paper presents an analytical investigation on the nonlinear postbuckling for imperfect eccentrically stiffened double curved thin shallow FGM shells using a simple power-law distribution (P-FGM) in thermal environments Our most important finding is the systematical investigation of the thin FGM shell reinforced by stiffeners in the thermal environment Both of FGM and stiffeners are deformed under mechanical, thermal and thermo-mechanical loads The formulations are based on the classical shell 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, explicit relations of thermal load–deflection curves for simply supported curved eccentrically stiffened FGM shells are determined 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 double curved shallow shells in thermal environments are analyzed and discussed Some results were compared with the ones of the other authors Acknowledgment This work was supported by Grant in Mechanics of the National Foundation for Science and Technology Development of VietnamNAFOSTED The authors are grateful for this support [1] Shen HS Postbuckling analysis of axially loaded functionally graded cylindrical panels in thermal environments Int J Solids Struct 2002;39: 5991–6010 [2] Shen HS, Leung AYT Postbuckling of pressure-loaded functionally graded cylindrical panels in thermal environments J Eng Mech ASCE 2003;129: 414–25 [3] Shen HS, Liew KM Postbuckling of axially loaded functionally graded cylindrical panels with piezoelectric actuators in thermal environments J Eng Mech ASCE 2004;130:982–95 [4] Shen HS Postbuckling of axially loaded FGM hybrid cylindrical shells in thermal environments J Compos Sci Technol 2005;65:1675–90 [5] Shen HS, Noda N Postbuckling of FGM cylindrical shells under combined axial and radial mechanical loads in thermal environments Int J Solids Struct 2005;42:4641–62 [6] Shen HS Postbuckling analysis of axially loaded piezolaminated cylindrical panels with temperature-dependent properties J Compos Struct 2007;79: 390–403 [7] Shen HS, Noda N Postbuckling of pressure-loaded FGM hybrid cylindrical shells in thermal environments J Compos Struct 2007;77:546–60 [8] Huang H, Han Q Nonlinear elastic buckling and postbuckling of axially compressed functionally graded cylindrical shells Int J Mech Sci 2009;51:500–7 [9] Huang H, Han Q Nonlinear buckling and postbuckling of heated functionally graded cylindrical shells under combined axial compression and radial pressure Int J Nonlinear Mech 2009;44:209–18 [10] Duc ND, Tung HV Nonlinear analysis of stability for functionally graded cylindrical shells under axial compression J Comput Mater Sci 2010;49: 313–6 [11] Duc ND, Tung HV Nonlinear response of pressure-loaded functionally graded cylindrical shells with temperature effects J Compos Struct 2010;92: 1664–72 [12] Zhao X, Liew KM Geometrically nonlinear analysis of functionally graded shells Int J Mech Sci 2009;51:131–44 [13] Sohn KJ, Kim JH Structural stability of functionally graded shells subjected to aerothermal loads J Compos Struct 2008;82:317–25 [14] Sofiyev AH The stability of compositionally graded ceramic-metal cylindrical shells under aperiodic axial impulsive loading J Compos Struct 2005;69: 247–57 [15] Shahsiah R, Eslami MR, Naj R Thermal instability of functionally graded shallow spherical shell J Therm Stress 2006;29:771–90 [16] Naj R, Boroujerdy MS, Eslami MR Thermal and mechanical instability of functionally graded truncated conical shells J Thin Wall Struct 2008;46: 65–78 [17] Sofiyev AH The stability of functionally graded truncated conical shells subjected to aperiodic impulsive loading Int J Solids Struct 2004;41: 3411–24 [18] Sofiyev AH Thermoelastic stability of functionally graded truncated conical shells J Compos Struct 2007;77:56–65 [19] Sofiyev AH, Kuruoglu N, Turkmen M Buckling of FGM hybrid truncated conical shells subjected to hydrostatic pressure J Thin Wall Struct 2009;47: 61–72 [20] Bich DH, Tung HV Nonlinear axisymetric response of FGM shallow spherical shell under uniform external pressure including temperature effects Int J Nonlinear Mech 2011;46:1195–204 [21] Bich DH, Phuong NT, Tung HV Buckling of functionally graded conical panels under mechanical loads J Compos Struct 2012;94:1379–84 [22] Librescu L, Lin W Postbuckling and vibration of shear deformable flat and curved shells on a non-linear elastic foundation Int J Non-Linear Mech 1997;32(2):211–225 [23] Lin W, Librescu L Thermomechanical postbuckling of geometrically imperfect shear-deformable flat and curved shells on a nonlinear foundation Int J Eng Sci 1998;36(2):189–206 [24] 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 [25] Zenkour AM Hygro-thermo-mechanical effects on FGM plates resting on elastic foundations J Compos Struct 2010;93:234–8 [26] 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 [27] Shen HS Postbuckling of shear deformable FGM cylindrical shells surrounded by an elastic medium Int J Mech Sci 2009;51:372–83 [28] Shen HS, Yang J, Kitipornchai S Postbuckling of internal pressure loaded FGM cylindrical shells surrounded by an elastic medium Euro J Mech A/Solids 2010;29:448–60 [29] Duc ND, Tung HV Mechanical and thermal posbuckling of higher order shear deformable fuctionally graded plates on elastic foundations J Compos Struct 2011;93:2874–81 [30] Duc ND, Quan TQ Nonlinear stability analysis of double curved shallow FGM panel on elastic foundation in thermal environments J Mech Compos Mater 2012;48(4):435–48 [31] Shen HS Thermal postbuckling behavior of functionally graded cylindrical shells with temperature-dependent properties Int J Solids Struct 2004;41: 1961–74 600 N.D Duc, T.Q Quan / Composite Structures 106 (2013) 590–600 [32] Shen HS Postbuckling analysis of axially loaded piezolaminated cylindrical panels with temperature-dependent properties J Compos Struct 2007;79: 390–403 [33] Shen HS Postbuckling analysis of axially loaded piezolaminated cylindrical panels with temperature-dependent properties J Compos Struct 2007;79: 390–403 [34] Yang J, Liew KM, Wu YF, Kitipornchai S Thermo-mechanical postbuckling of FGM cylindrical panels with temperature-dependent properties Int J Solids Struct 2006;43:307–24 [35] Bich DH, Nam VH, Phuong NT Nonlinear postbuckling of eccentrically stiffened functionally graded plates and shallow shells Vietnam J Mech VAST 2011;33(3):131–47 [36] 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 [37] Bich DH, Dung DV, Nam VH Nonlinear dynamic analysis of eccentrically stiffened functionally graded cylindrical panels J Compos Struct 2012;94:2465–73 [38] Duc ND Nonlinear dynamic response of imperfect eccentrically stiffened FGM double curved shallow shells on elastic foundation J Compos Struct 2013;99: 88–96 [39] Brush DD, Almroth BO Buckling of bars, plates and shells Mc Graw-Hill; 1975 ... curvatures of the shell a/Rx, b/Ry on nonlinear response of imperfect eccentrically stiffened P-FGM double curved shallow shells with elastic foundations in the thermal environment Concluding... analytical investigation on the nonlinear postbuckling for imperfect eccentrically stiffened double curved thin shallow FGM shells using a simple power-law distribution (P-FGM) in thermal environments. .. have investigated the postbuckling behavior of at and curved laminated composite shells resting on Winkler elastic foundations [22,23] In spite of practical importance and increasing use of FGM