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Buckling Analysis of Eccentrically Stiffened Functionally Graded Toroidal Shell Segments under Mechanical Load Downloaded from ascelibrary.org by National Taiwan University on 06/28/15 Copyright ASCE For personal use only; all rights reserved Bich Huy Dao 1; Ninh Gia Dinh 2; and Thinh Ich Tran Abstract: An analytical approach is presented to investigate the linear buckling of eccentrically stiffened functionally graded thin toroidal shell segments subjected to axial compression, lateral pressure, and hydrostatic pressure On the basis of classical thin shell theory, the smeared stiffener technique and the adjacent equilibrium criterion, the governing equations of buckling of eccentrically stiffened functionally graded toroidal shell segments are derived The functionally graded toroidal shell segments with simply supported edges are reinforced by a ring and stringer stiffener system on an external surface The resulting equations in the case of compressive and pressure loads are solved directly The obtained results show the effects of stiffeners and input factors on the buckling behavior of these structures In this paper, the results are also compared with the solutions published in the literature for the specific cases of a toroidal shell DOI: 10.1061/(ASCE)EM 1943-7889.0000964 © 2015 American Society of Civil Engineers Author keywords: Functionally graded material; Toroidal shell segments; Stiffeners; Critical buckling loads; Axial compression; Lateral pressure; Hydrostatic pressure Introduction Functionally graded materials (FGMs) are composite materials that have mechanical properties varying continuously from one surface of a structure to another Nowadays, the application of FGMs is becoming so widespread that many problems related to new structures as well as the static and dynamic behaviors of structures are being noticed Many studies on the buckling analysis of FGMs have been reported in the literature Expressions for the buckling load and postbuckling equilibrium path of an axially compressed thin homogeneous cylindrical shell have been obtained by Karman and Tsien (1941) Shen (2002) investigated the postbuckling analysis of axially loaded functionally graded cylindrical shells in thermal environments using classical shell theory with a von Kármán–Donnell type of kinematic nonlinearity Shen and Leung (2003) investigated the postbuckling of a pressure-loaded FGM cylindrical panel subjected to lateral pressure in a thermal environment based on Reddy’s higher-order shear deformation shell theory with a von Karman–Donnell type of kinematic nonlinearity The postbuckling of shear deformation FGM cylindrical shells surrounded by a Pasternak foundation with two kinds of micromechanics models, the Voigt model and the Mori–Tanaka model, was researched by Shen (2013) The governing equations were based on a higherorder shear deformation shell theory, and the material properties of FGMs were assumed to be temperature dependent Professor, Vietnam National Univ., No 144 Xuan Thuy St., Cau Giay District, Hanoi 10000, Vietnam Lecturer, Hanoi Univ of Science and Technology, No Dai Co Viet St., Hai Ba Trung District, Hanoi 10000, Vietnam (corresponding author) E-mail: ninhdinhgia@gmail.com; ninh.dinhgia@hust.edu.vn Professor, Hanoi Univ of Science and Technology, No Dai Co Viet St., Hai Ba Trung District, Hanoi 10000, Vietnam Note This manuscript was submitted on December 23, 2014; approved on April 6, 2015; published online on May 26, 2015 Discussion period open until October 26, 2015; separate discussions must be submitted for individual papers This paper is part of the Journal of Engineering Mechanics, © ASCE, ISSN 0733-9399/04015054(10)/$25.00 © ASCE The nonlinear dynamic buckling of functionally graded cylindrical shells subjected to a time-dependent axial load was researched by Huang and Han (2010) These authors, whose work was mentioned previously (Huang and Han 2008), researched the buckling of imperfect functionally graded cylindrical shells under axial compression Sofiyev (2010) analyzed the buckling of functionally graded material circular truncated conical and cylindrical shells subjected to combined axial extension loads and hydrostatic pressure and resting on a Pasternak-type elastic foundation The buckling of a simply supported three-layer circular cylindrical shell under axial compressive load was studied by Li and Batra (2006) A postbuckling analysis was performed by Shen and Noda (2007) for a functionally graded cylindrical shell with piezoelectric actuators subjected to lateral or hydrostatic pressure combined with electric loads in thermal environments Bich et al (2013) investigated the nonlinear static and dynamic buckling of imperfect eccentrically stiffened functionally graded thin circular cylindrical shells subjected to axial compression based on classical thin shell theory with geometrical nonlinearity in a von Karman–Donnell sense Moreover, Bich et al (2012) researched the linear problem of functionally graded conical panels buckling under mechanical loads using classical thin shell theory in which the Galerkin method was applied to obtain closed-form relations of bifurcation-type buckling loads Dung and Hoa (2013) researched nonlinear torsional buckling and postbuckling of eccentrically stiffened functionally graded thin circular cylindrical shells A linear buckling analysis of eccentrically stiffened functionally graded circular cylindrical thin shells under mechanical load was conducted by Phuong and Bich (2013) Stein (1964) was the first to recognize the importance of nonlinear prebuckling deformations on the buckling load of perfect cylindrical shells Weaver (2000) pointed out that the load-bearing efficiency of cylindrical shells derived from both the properties of the material of which they are made and from the shape itself Toroidal shells have been proposed for use in such applications as fusion reactor vessels, satellite support structures, underwater toroidal pressure hulls, rocket fuel tanks, and diver oxygen tanks Nowadays FGMs consisting of metal and ceramic components have received considerable attention in structural applications The smooth and continuous changes in material properties enable 04015054-1 J Eng Mech J Eng Mech Downloaded from ascelibrary.org by National Taiwan University on 06/28/15 Copyright ASCE For personal use only; all rights reserved FGMs to avoid interface problems and unexpected thermal stress concentrations Some of the aforementioned structural components may be made of FGMs Thus, buckling analysis has become an important part of the design process for such shells Stein and McElman (1965) researched the buckling of homogeneous toroidal shell segments In a NASA technical note, McElman (1967) investigated eccentrically stiffened shallow shells of double curvature Furthermore, Hutchinson (1967) presented a study on the initial postbuckling behavior of homogeneous toroidal shell segments The present paper is concerned with studies on the linear buckling analysis of eccentrically stiffened functionally graded toroidal shell segments under axial compression, lateral pressure, and hydrostatic pressure In this study, shells are assumed to be perfect, and the analysis does not consider geometric or material imperfections The fundamental equations for the buckling analysis of stiffened FGM toroidal shell segments based on classical shell theory using the smeared stiffener technique and adjacent equilibrium criterion are derived Governing Equations Functionally Graded Material (FGM) FGMs are microscopically inhomogeneous materials whose properties vary smoothly and continuously from one surface of a material to another These materials are made from a mixture of ceramic and metal or a combination of different materials Such a mixture of ceramic and metal with a continuously varying volume fraction can be manufactured In particular, FGM thin-walled structures with ceramic in the inner surface and metal in the outer surface are widely used in practice It is assumed that the elasticity modulus E changes in the thickness direction z, while the Poisson ratio ν is constant Denote by V m and V c the volume fractions of metal and ceramic phases, respectively These parameters are related by V m ỵ V c ¼ 1, in which V c is expressed as V c zị ẳ ẵ2z ỵ hị=2 hịk , where h is the thickness of a thin-walled structure, k is the volume-fraction exponent (k ≥ 0), and z is the thickness coordinate and varies from −h=2 to h=2 The subscripts m and c refer to the metal and ceramic constituents, respectively According to the aforementioned law, the Young’s modulus can be expressed in the form 2z ỵ h k Ezị ẳ Em V m ỵ Ec V c ẳ Em ỵ Ec Em ị 1ị 2h Fig Configuration and coordinate system of eccentrically stiffened toroidal shell segments: (a and b) convex shell; (c and d) concave shell and the Poisson ratio ν is assumed to be constant Constitutive Relations and Governing Equations Consider a functionally graded toroidal shell segment of thickness h and length L formed by the rotation of a plane circular arc of radius R around an axis in the plane of the curve as shown in Fig The geometry and coordinate system of a stiffened FGM toroidal shell segment are described in Fig For the middle surface of the toroidal shell, from the figure we have Fig Geometry and coordinate system of stiffened FGM toroidal shell segments: (a) stringer stiffeners; (b) ring stiffeners r ¼ a − Rð1 − sin φÞ The radius of arc R is positive with the convex toroidal shell segment and negative with the concave one Suppose the FGM toroidal shell segment is reinforced by closely spaced stringer and ring stiffeners To provide continuity between the shell and the stiffeners and to make it easier to manufacture them, homogeneous stiffeners can be used Because the pure ceramic stiffeners are brittle, metal stiffeners are used and arranged on the metal-rich side of the shell Applying the law indicated by Eq (1), the outer where a is the equator radius and φ is the angle between the axis of revolution and the normal to the shell surface For a sufficiently shallow toroidal shell in the region of the equator of the torus, the angle φ is approximately equal to π=2; thus, sin φ ≈ 1, cos φ ≈ 0, and r ¼ a (Stein and McElman 1965) The form of the governing equation is simplified by setting © ASCE dx1 ¼ Rdφ; 04015054-2 J Eng Mech dx2 ¼ adθ J Eng Mech surface of the shell is metal-rich, so the external metal stiffeners are arranged on this side In this paper, classical shell theory and the Lekhnitsky smeared stiffener technique are used to obtain the equilibrium and compatibility equations, as well as expressions for the buckling loads of stiffened FGM toroidal shell segments The strains across the shell thickness at a distance z from the midsurface are Downloaded from ascelibrary.org by National Taiwan University on 06/28/15 Copyright ASCE For personal use only; all rights reserved ε1 ¼ ε01 − zχ1 ; ε2 ¼ ε02 − zχ2 ; u w ặ ; x1 R ẳ 2w ; x21 02 ẳ ẳ v w ỵ ; x2 a ∂2w ; ∂x22 χ12 ¼ γ 012 ¼ ∂u v ỵ ; x2 x1 2w x1 x2 3ị From Eqs (3), the strains must satisfy the deformation compatibility equation ∂ ε01 ∂ ε02 ∂ γ 012 2w 2w ỵ ẳặ ỵ 2 x2 ∂x1 ∂x1 ∂x2 R∂x2 a∂x21 ð4Þ The constitutive stress–strain equations determined by Hooke’s law for the shell material are omitted here for brevity The contribution of the stiffeners can be calculated using the Lekhnitsky smeared stiffener technique Integrating the stress–strain equations and their moments through the thickness of the shell, the expressions for the force and moment resultants of an eccentrically stiffened FGM toroidal shell are obtained: E A N ẳ A11 ỵ m 01 ỵ A12 02 B11 ỵ C1 ị1 B12 s1 E A N ¼ A12 ε01 þ A22 þ m ε02 − B12 χ1 − B22 ỵ C2 ị2 s2 N 12 ẳ A66 012 − 2B66 χ12 ð5Þ E I M ẳ B11 ỵ C1 ị01 ỵ B12 02 D11 ỵ m 1 D12 s1 E I M ẳ B12 01 ỵ B22 þ C2 Þε02 − D12 χ1 − D22 þ m χ2 s2 M 12 ¼ B66 γ 012 − 2D66 χ12 where Aij ; Bij ; Dij ði; j ¼ 1; 2; 6Þ are the extensional, coupling, and bending stiffnesses of a shell without stiffeners, E1 E :ν A11 ¼ A22 ¼ ; A12 ¼ ; − ν2 1−ν E2 E :ν B11 ¼ B22 ¼ ; B12 ¼ 2 ; − ν2 1−ν E3 E :ν ; D12 ¼ ; D11 ¼ D22 ¼ − ν2 1−ν with © ASCE C1 ẳ E1 ; A66 ẳ 21 ỵ ị E2 ; B66 ẳ 21 ỵ ị E3 D66 ẳ 21 ỵ ị 7ị Em A1 z1 ; s1 C2 ¼ − Em A2 z2 s2 ð9Þ In Eqs (5), (6) and (9), the spacing of the stringer and ring striffeners are denoted by s1 and s2 , respectively The quantities A1 and A2 are the cross-sectional areas of the stiffeners, and I , I , z1 , and z2 are the second moments of the cross-sectional areas and eccentricities of the stiffeners with respect to the middle surface of the shell, respectively The minus sign for C1 and C2 depends on the external stiffeners Remark: Conversely, if the inner side of the stiffened FGM shell is metal-rich, then all calculated expressions can be used, but Ec and Em must be replaced each other in Eqs (8) and the plus sign is taken in Eqs (9) The eccentrically stiffened FGM shell is subjected to mechanical loads in two cases: Case 1: An axial compression load p uniformly distributed on the two end edges of the shell and a lateral pressure q uniform distributed on the surface; and Case 2: Hydrostatic pressure load The nonlinear equilibrium equations of a toroidal shell based on classical shell theory are given by (Brush and Almroth 1975) ∂N ∂N 12 þ ¼ 0; ∂x1 ∂x2 ∂N 12 ∂N þ ¼ 0; ∂x1 ∂x2 ∂ M1 ∂ M 12 M 2w 2w ỵ2 ỵ ỵ N ỵ 2N 12 2 x1 x2 x1 x2 x1 x1 x2 ỵ N2 2w N1 N2 ỵqẳ0 R a x22 10ị The stability equations of eccentrically stiffened functionally graded shells may be established by the adjacent equilibrium criterion It is assumed that the equilibrium state of an eccentrically stiffened functionally graded shell under applied load is represented by the displacement components u0 , v0 , and w0 The state of adjacent equilibrium differs that of stable equilibirum by increments u1 , v1 , w1 , and the total displacement component of a neighboring configuration are u ẳ u0 ỵ u1 ; 6ị 8ị and 12 ẳ 012 2z12 2ị where ε01 and ε02 = normal strains; γ 012 = shear strain at middle surface of shell; and χij = the curvatures According to classical shell theory, the strains at the middle surface and the curvatures are related to the displacement components u, v, and w in the x1 , x2 , and z coordinate directions as (Brush and Almroth 1975) ε01 ¼ E − Em ðEc − Em ịkh2 h; E2 ẳ Em ỵ c kỵ1 2k ỵ 1ịk ỵ 2ị Em 1 E3 ẳ ỵ Ec Em ị ỵ h3 12 k ỵ k ỵ 4k ỵ E1 ẳ v ẳ v0 ỵ v1 ; w ẳ w0 ỵ w1 11ị Similarly, the force and moment resultants of a neighboring state are represented by N ¼ N 01 ỵ N 11 ; N ẳ N 02 ỵ N 12 ; M ẳ M01 ỵ M 11 ; M ẳ M 02 ỵ M 12 ; N 12 ẳ N 012 ỵ N 112 ; M 12 ẳ M012 ỵ M 112 12ị where terms with subscripts correspond to the u0 , v0 , and w0 displacements and those with subscript represent the increments of force and moment resultants that are linear in u1 , v1 , and w1 Subsequently, inserting Eqs (11) and (12) into Eq (10) and subtracting from the resulting equations the term relating to the stable equilibrium state, neglecting the nonlinear term in u1 , v1 , and w1 or their counterparts in the form of, for example, 04015054-3 J Eng Mech J Eng Mech N 11 , N 12 , N 112 , : : : and prebuckling the rotations yields the following stability equations: ∂N 11 N 112 ỵ ẳ0 x1 x2 A11 N 112 N 12 ỵ ẳ0 x1 x2 Downloaded from ascelibrary.org by National Taiwan University on 06/28/15 Copyright ASCE For personal use only; all rights reserved ∂ w1 ∂ w1 ỵ N 02 ẳ0 x1 x2 x22 13ị D11 Considering the first two of Eqs (13), a stress function may be defined as N 11 ∂2F ¼ 2; ∂x2 N 12 ∂2F ¼ 2; ∂x1 N 112 ∂2F ¼− ∂x1 ∂x2 ∂ w1 ∂ w1 ∂ w1 ỵ B12 2 R x22 x1 x2 x42 ỵ B11 ỵ B22 2B66 ị M 11 ∂ M112 ∂ M 12 N 11 N 12 w1 ỵ N 01 ỵ2 ỵ 2 x1 x2 R a x1 x2 x21 ỵ 2N 012 4F 4F 4F w1 ỵ A66 2A12 ị 2 ỵ A22 ỵ B21 x41 x1 x1 x2 x2 Ã Ã Ã Ã ε01 ẳ A22 N A12 N ỵ B11 ỵ B12 17ị w1 4w w1 4F ỵ D12 ỵ D21 ỵ 4D66 ị 2 ỵ D22 B21 4 x1 x1 x1 x2 x2 B11 ỵ B22 2BÃ66 Þ − N 01 ð14Þ The reverse relations are obtained from Eqs (5) for increments ∂ w1 ¼0 a ∂x21 ∂4F ∂4F ∂2F ∂2F − B12 ặ ỵ 2 x1 x2 x2 R ∂x22 a ∂x21 ∂ w1 ∂ w1 ∂ w1 − 2N 012 − N 02 ¼0 ∂x1 ∂x2 ∂x1 ∂x22 ð18Þ Eqs (17) and (18) are the basic equations used to investigate the stability of eccentrically stiffened functionally graded toroidal shell segments They are linear equations in terms of two dependent unknowns, w1 and F Ã Ã Ã Ã ε01 ¼ A11 N A12 N ỵ B21 þ B22 χ2 Ã Ã γ 01 12 ẳ A66 N 12 ỵ 2B66 12 15ị where E A E A AÃ22 ẳ A11 ỵ ; A22 ỵ ; Δ Δ s1 s2 A 12 Ẫ12 ¼ ; Ẫ66 ¼ A66 Δ E A E A ẳ A11 ỵ : A22 þ − A212 s1 s2 Buckling Analysis of Eccentrically Stiffened Functionally Graded Toroidal Shell Segments Subjected to Axial Compressive Load and Lateral Pressure In this paper, an eccentrically stiffened FGM shell is free and is simply supported at two end edges and subjected to mechanical loads in two cases In Case 1, an axial compression load p is uniformly distributed on the two end edges of the shell and a lateral pressure q is uniformly distributed on the external surface In Case 2, there is a hydrostatic pressure load By solving the membrane form of the equilibrium equations, prebuckling force resultants are determined as (Stein and McElman 1965) A11 ẳ B11 ẳ A22 B11 ỵ C1 ị A12 B12 ; B22 ẳ A11 B22 ỵ C2 ị − Ẫ12 B12 Case 1∶ N 01 ¼ −ph; BÃ12 ẳ A22 B12 A12 B22 ỵ C2 ị; B21 ẳ A11 :B12 A12 B11 ỵ C1 ị; B66 ¼ B66 A66 Case 2∶ N 01 N 02 ¼ −qa; a ¼ −q ; N 02 N 012 ¼ a ¼ −qa 1∓ ; 2R N 012 ẳ 20ị Substituting Eq (15) into Eq (6) for increments yields The boundary conditions considered in the present study are written for increments as follows: M 11 ẳ B11 N 11 ỵ B21 N 12 D11 χ11 − DÃ12 χ12 ; M 12 ¼ BÃ12 N 11 ỵ B22 N 12 D21 11 D22 χ12 ; M 112 ¼ BÃ66 N 112 − 2DÃ66 χ112 DÃ22 w1 ¼ XX m DÃ12 ¼ D12 − B11 ỵ C1 ịB12 B12 B22 D66 ẳ D66 − B66 BÃ66 The substitution of Eqs (15) into the compatibility Eqs (4) and substituting Eqs (16) into the third of Eqs (13), taking into account Eqs (3) and (14), yields a system of equations â ASCE N 11 ẳ 0; M ¼ 0; at x1 ¼ 0; L ð21Þ Conditions (21) can be satisfied if the buckling mode shape is represented by E I ẳ D11 ỵ B11 ỵ C1 ịB11 B12 B21 s1 E0 I ẳ D22 ỵ B12 B21 B22 ỵ C2 ịB22 s2 D21 ẳ D12 B12 B11 B22 ỵ C2 ịB21 w1 ẳ 0; 16ị where DÃ11 ð19Þ W mn sin n mπx1 nx sin L a ð22Þ where W mn is a maximum deflection; m, n are the numbers of half waves in axial and circumferential direction respectively Substituting Eqs (22) into Eq (17) and solving the obtained equation for unknown F leads to F¼ XX m n fmn sin mπx1 nx sin L a ð23Þ where 04015054-4 J Eng Mech J Eng Mech Downloaded from ascelibrary.org by National Taiwan University on 06/28/15 Copyright ASCE For personal use only; all rights reserved f mn ẳ ẵB21 m4 ỵ B11 ỵ B22 2:B66 ịm2 n2 2 ỵ B12 n4 λ4 − am2 π2 λ2 ∓a2 n2 λ4 =R W mn A11 m4 ỵ A66 2A12 ịm2 n2 2 ỵ A22 n4 Case 1: Axial Compression Load p Uniformly Distributed on Two End Edges of Shell and Lateral Pressure q Uniformly Distributed on External Surface Inserting expressions (22), (23) and (19) into Eqs (18) leads to X X B2 ỵ phm2 qan2 ịL2 :W mn Dỵ A m n mx1 nx sin ẳ 25ị ì sin L a where denote A ẳ A11 m4 ỵ A66 2A12 ịm2 n2 2 ỵ A22 n4 ; ẳ L=a; B ẳ B21 m4 ỵ B11 ỵ B22 2:B66 ịm2 n2 2 ỵ B12 n4 − am2 π2 λ2 − a2 n2 λ4 =R; D¼ D11 m4 ỵ D12 ỵ D21 ỵ D22 n4 λ4 Case 2: Hydrostatic Pressure Load The substitution of expressions (22), (23) and (20) into Eqs (18) leads to X X B2 a 2 a 2 n L2 :W mn D ỵ þ −q: m π − q:a 1∓ 2R A m n mx1 nx2 sin ẳ 0; 34ị ì sin L a A ẳ A11 m4 ỵ A66 2A12 ịm2 n2 2 ỵ A22 n4 ; 26ị B2 Dỵ phm2 L2 ẳ A ð27Þ From Eq (27) the compressive buckling load can be obtained B2 pẳ 28ị D ỵ A hm2 L2 Introducing parameters ỵ B11 ỵ B22 ¼ L=a; − 2:BÃ66 Þm2 n2 π2 λ2 D ¼ D11 m4 ỵ D12 ỵ D21 ỵ 4D66 ịm2 n2 2 ỵ D22 n4 Eq (34) is satisfied by all x1 and x2 , so that B2 a 2 a 2 Dỵ ỵ q m qa n L2 ẳ A 2R 35ị From Eq (35) the buckling hydrostatic pressure load can be obtained: B2 qẳ Dỵ 36ị a 2 a A m ỵ a n2 λ2 L2 2R or B B¯ ¼ ; h A¯ ¼ Ah The critical axial compression load of eccentrically stiffened FGM toroidal shell is determined by condition pcr ¼ minpvs.ðm; nÞ If the toroidal shell is subjected to only lateral pressure (p ¼ 0), Eq (26) becomes B2 Dỵ qan2 L2 ẳ A The pressure buckling load can be determined B2 q¼ 2 Dỵ A an L B Dỵ qẳ A a m2 a L ỵ n2 λ 2 h 2R h ð29Þ into Eq (28), the compressive buckling load can be obtained ỵB 30ị pẳ 2 D m L=hị2 A â ASCE B21 m4 ỵ B12 n4 ỵ am2 2 a2 n2 =R; Now investigate the linear buckling of reinforced FGM toroidal shell segments subjected to only axial compression (q ¼ 0), The Eq (26) becomes or ð33Þ The critical lateral pressure of eccentrically stiffened FGM toroidal shell is determined by condition qcr ẳ minqvs.m; nị Bẳ B2 ỵ phm2 qan2 ịL2 ẳ A ẳ D; D h3 ỵB D a=hịn2 L=hị2 A where ỵ 4D66 ịm2 n2 2 Eq (25) satisfies for all x1 , x2 so that Dỵ qẳ ð24Þ ð37Þ The critical hydrostatic pressure load of an eccentrically stiffened FGM toroidal shell is determined by the condition qcr ¼ minqvs.ðm; nÞ Numerical Result and Discussions Validation Studies ð31Þ ð32Þ Cylindrical shells are a specific case of toroidal shell segments (when R ỵ) The buckling of a simply supported FGM cylindrical shell without stiffeners under axial compression is considered To demonstrate the formulation presented here, calculations of a FGM cylindrical shell under mechanical loads are compared with the results of Huang and Han (2010) Numerical results are given for cylindrical shells made of zirconia (ZrO2 ) and titanium (Ti-6Al-4 V) The elasticity moduli of zirconia and titanium at the 04015054-5 J Eng Mech J Eng Mech Table Comparison of Present Critical Load (MPa) with Theoretical Results Reported by Huang and Han (T o ¼ 300 K, L=a ¼ 2) kða=h ¼ 500Þ Reference 0.2 Downloaded from ascelibrary.org by National Taiwan University on 06/28/15 Copyright ASCE For personal use only; all rights reserved a 5.0 400 600 800 189.262 (2, 11) 164.352 (2, 11) 144.471 (2, 11) 236.578 (5, 15) 157.984 (3, 14) 118.849 (2, 12) 189.324 (2, 11) 0.033 164.386 (2, 11) 0.021 144.504 (2, 11) 0.023 236.464 (5, 15) −0.048 158.022 (3, 14) 0.024 118.898 (2, 12) 0.041 a Huang and Han (2010) ðσscr ¼ σdcr = cr ị Present Difference (%) a=hk ẳ 0.2ị 1.0 Numbers in parentheses indicate buckling mode (m, n) initial temperature T o ¼ 300 K are taken to be 168.08 GPa and 105.69 GPa, respectively The Poisson ratio is chosen to be 0.3 As can be seen in Table 1, good agreement is obtained in this comparison study It can be shown that for simply supported FGM cylindrical shells with stiffeners under axial compression, the compressive buckling load determined by Eq (30) (when R ỵ) in the text coincides completely with Eq (32), deduced by Bich et al (2013), and Eq (26) of Phuong and Bich (2013) For shells under lateral pressure the present buckling load determined by Eq (33) coincides identically with Eq (27) of Phuong and Bich (2013) This fact shows the accuracy of the present approach Secondly, the results in this paper are compared with those in the monograph of Brush and Almroth (1975) for a stiffened homogeneous cylindrical shell under axial compression, as shown in Table The comparison shows that good results are achieved Another comparison is established for homogeneous isotropic toroidal shell segments under lateral pressure, based on the formula of a critical buckling load given by Hutchinson (1967) and the formula in the present paper The results presented in Table show the good agreement of the present formulation Results of Buckling Analysis of FGM Toroidal Shell Segments To illustrate the present formulation, the FGM toroidal shell segments are made of aluminum Em ẳ ì 1010 N=m2 and alumina Ec ẳ 38 × 1010 N=m2 The Poisson ratio ν is chosen to be 0.3 for Table Critical Buckling Load per Unit Length p cr ẳ pcr h106 N=mị of Externally Stiffened Homogeneous Cylindrical Shells under Axial Compression versus a=h Ratio Ratio a=h 100 200 500 Present Brush and Almroth (1975) Difference (%) 3.9529 (9, 3) 2.1410 (9, 4) 1.2764 (6, 6) 3.9551 (9, 2) 2.1369 (9, 4) 1.2897 (6, 6) 0.06 0.28 1.04 Note: 50 rings, 50 stringers; L ¼ m; a ¼ 0.5 m; E ¼ ì 1010 N=m2 ị; v ẳ 0.3; d1 ẳ d2 ¼ 0.0025 m; h1 ¼ h ¼ 0.01 m simplicity The height of the stiffener is equal to 0.005 m, and its width is 0.002 m The Young’s moduli of the external stringer stiffeners and external ring stiffeners is E0 ¼ Em The stiffeners include 50 ring and 50 stringer stiffeners distributed regularly in the axial and circumferential directions, respectively Effects of R=h Ratio and Volume-Fraction Index k Tables and show the critical buckling load of stiffened and unstiffened toroidal segments versus four different values of the volume-fraction index k (0.5, 1, 5, 10) and four values of the R=h ratio [R > (convex shell) and R < (concave shell)], respectively, with the following geometric properties: L=a ¼ 2, a=h ¼ 100, h ¼ 0.002 m, d1 ¼ d2 ¼ 0.002 m, h1 ¼ h2 ¼ 0.005 m, n1 ¼ n2 ¼ 50, where n1 and n2 are the number of stringer and ring stiffeners, respectively Table Critical Buckling Load of Stiffened FGM Convex Toroidal Shell Segment (R > 0) under Axial and Lateral Pressure Load pcr × 103 (MPa) qcr (MPa) External stiffener k Unstiffened 200 0.5 10 1.5824 (12, 1)a 1.2512 (12, 1) 0.7502 (11, 1) 0.6411 (11, 1) 2.7229 2.2917 1.5205 1.3539 (8, (8, (7, (7, 7) 7) 6) 6) 8.1357 6.4275 3.8432 3.3082 (1, (1, (1, (1, 13) 14) 13) 13) 25.0926 22.6557 16.4929 14.4822 (1, (1, (1, (1, 8) 8) 7) 7) 300 0.5 10 1.5818 1.2506 0.7497 0.6407 (12, (12, (11, (11, 1) 1) 1) 1) 2.6075 2.1962 1.4542 1.2741 (8, (7, (7, (6, 7) 7) 6) 6) 5.5897 4.4197 2.6503 2.2885 (1, (1, (1, (1, 11) 11) 11) 11) 17.5799 15.9629 11.6756 10.2494 (1, (1, (1, (1, 7) 7) 6) 6) 400 0.5 10 1.5814 1.2503 0.7495 0.6406 (12, (12, (11, (11, 1) 1) 1) 1) 2.5450 2.1291 1.4077 1.2343 (7, (7, (6, (6, 7) 7) 6) 6) 4.3565 3.4379 2.0760 1.7946 (1, (1, (1, (1, 10) 14.2583 (1, 7) 10) 12.8183 (1, 6) 10) 9.5365 (1, 6) 9) 8.4624 (1, 5) 500 0.5 10 1.5812 1.2502 0.7494 0.6405 (12, (12, (11, (11, 1) 1) 1) 1) 2.4964 2.0899 1.3785 1.2111 (7, (7, (6, (6, 7) 7) 6) 6) 3.6487 2.8813 1.7358 1.5029 (1, (1, (1, (1, 9) 9) 9) 9) a Unstiffened External stiffener R=h 12.0645 10.9962 8.2607 7.2204 (1, (1, (1, (1, 6) 6) 5) 5) Numbers in parentheses indicate buckling mode (m, n) Table Critical Buckling Load q¯ of Homogeneous Toroidal Shells under Lateral Pressure Load versus R=h Ratio R>0 R=h 100 200 300 400 500 Hutchinson (1967) 32.5033 58.6545 85.0677 111.1143 136.8904 (1, (1, (1, (1, (1, 5) 7) 8) 10) 11) R 0) on the critical buckling load © ASCE From the aforementioned tables and figures, the following observations may be made: The critical buckling load of stiffened FGM toroidal shell segments is always higher than that of unstiffened ones Thus, stiffeners enhance the stability of a structure When the volume-fraction index k is increased (raising the volume of the metal), the critical buckling load decreases because the modulus of a metal is lower than that of a ceramic For a convex shell (R > 0), when R=h increases, the critical buckling load decreases Nevertheless, the critical buckling load of a concave shell (R < 0) has no stable tendency In particular, when the R=h ratio increases from −200 to − 300, the axial critical buckling load increases, but it drops at the R=h ¼ − 300 to − 400 For the ratio of R=h ¼ − 400 to −500, the axial critical buckling load increases again Thus, 04015054-7 J Eng Mech J Eng Mech Table Critical Buckling Load of Externally Stiffened FGM Toroidal Shell Segment under Lateral Pressure Load (a=h ¼ 50; L=a Changes) qcr × 107 N=m2 k L=a Downloaded from ascelibrary.org by National Taiwan University on 06/28/15 Copyright ASCE For personal use only; all rights reserved 0.2 13.3021 6.1931 4.1465 3.2310 2.7129 (1, (1, (1, (1, (1, 5) 4) 4) 4) 3) ∞ 10.7771 5.3438 3.6982 2.7538 2.2205 (1, (1, (1, (1, (1, 4) 4) 4) 3) 3) 7.3409 4.1184 2.6835 2.0309 1.6733 (1, (1, (1, (1, (1, 4) 4) 3) 3) 3) 4.7992 2.6718 1.8075 1.4165 1.1908 (1, (1, (1, (1, (1, 4) 3) 3) 3) 3) Table Critical Buckling Load of Externally Stiffened FGM Toroidal Shell Segment under Axial Load pcr × 109 N=m2 k a=R 0.125 0.25 0.5 0.8 −0.125 −0.25 −0.5 −0.8 0.2 2.8355 2.9699 3.1688 3.3321 1.9290 0.6571 0.7281 0.7892 (7, (8, (8, (9, (1, (1, (2, (3, 7) 7) 6) 6) 4) 3) 4) 5) 2.0327 2.1291 2.2917 2.4483 1.5845 0.5312 0.5586 0.5981 ∞ 10 (7, (7, (8, (8, (1, (1, (2, (2, 7) 7) 7) 6) 4) 3) 4) 3) 1.1773 1.2343 1.3539 1.4563 0.9553 0.4203 0.2749 0.3111 (6, (6, (7, (7, (1, (1, (1, (2, 6) 6) 6) 6) 3) 3) 2) 3) 0.9434 1.0054 1.0898 1.2016 0.7813 0.4044 0.2113 0.2476 (5, (6, (6, (7, (1, (1, (1, (2, 6) 6) 6) 6) 3) 3) 2) 3) values of the volume-fraction index under a pressure load The geometric properties are as follows: R=h ¼ 500, h ¼ 0.002 m, d1 ¼ d2 ¼ 0.002 m, h1 ¼ h2 ¼ 0.005 m, n1 ¼ n2 ¼ 50, R > As can be seen, the critical load of a stiffened FGM toroidal shell segment under a pressure load decreases when the toroidal shell equator curvature or length rises Effect of a=R Ratio The critical buckling load of an externally stiffened FGM toroidal shell segment under axial load and hydrostatic pressure are investigated for R > and R < in Tables and 9, respectively The geometric properties are L=a ¼ 2, L=h ¼ 200, h ¼ 0.002 m, d1 ¼ d2 ¼ 0.002 m, h1 ¼ h2 ¼ 0.005 m, n1 ¼ n2 ¼ 50, with a=R varying As can be seen, for a convex shell (R > 0), the decrease in the a=R ratio leads to an increase in the critical loads But for a concave shell (R < 0), the tendency is unstable The load-carrying capacity of a concave shell is lower than that of a convex shell Effect of Ratio L=h Table 10 indicates the critical buckling load of stiffened FGM toroidal shell segments under hydrostatic pressure with four configurations: L=h ¼ 100, 200, 300, and 400 When the L=h ratio rises, the critical buckling hydrostatic pressure will decrease (≈10 ÷ 36%) The geometric properties of this problem are R=h ¼ 400, a=h ¼ 100, h ¼ 0.002 m, d1 ¼ d2 ¼ 0.002 m, h1 ¼ h2 ¼ 0.005 m, n1 ¼ n2 ¼ 50, and k ¼ 0.2 Internal Metal Stiffeners Case Table Critical Buckling Load of Externally Stiffened FGM Toroidal Shell under Hydrostatic Pressure Hydrostatic pressure, qcr × 106 N=m2 k a=R 0.2 ∞ 10 0.125 8.8290 (1, 6) 7.0010 (1, 6) 4.5094 (1, 5) 4.0033 (1, 0.25 12.8772 (1, 7) 10.0517 (1, 6) 6.4396 (1, 6) 5.5368 (1, 0.5 20.9663 (1, 9) 16.1853 (1, 8) 9.6095 (4, 8) 7.7603 (3, 0.8 26.3947 (5, 10) 19.2260 (5, 10) 11.0045 (5, 8) 8.8875 (5, −0.125 2.9319 (1, 4) 2.4083 (1, 4) 1.9364 (1, 4) 1.8781 (1, −0.25 1.7800 (1, 3) 1.4389 (1, 3) 1.1384 (1, 3) 1.0954 (1, −0.5 4.2435 (2, 4) 3.2552 (2, 4) 1.6024 (1, 2) 1.2316 (1, −0.8 6.7142 (3, 5) 5.2717 (3, 5) 2.9707 (2, 3) 2.3646 (2, 5) 5) 7) 8) 4) 3) 2) 3) concave toroidal shell segments need to be thoroughly investigated The critical buckling load of the concave shell is lower than that of the convex shell Effect of a=h and L=a Ratios Tables and show the effect of the a=h and L=a ratios on the critical load of stiffened FGM toroidal shell segments with various Now the same toroidal shell segment, made of functionally graded material such that the inner side is metal-rich and internal metal stiffeners are arranged on this side, is considered It is sufficient to compare critical loads of both types of stiffened FGM toroidal shells with the volume-fraction index k ¼ First, the critical buckling loads of internally stiffened FGM toroidal shell segments under axial and pressure loads with four different values of the R=h ratio [R > (convex shell) and R < (concave shell)] are given in Tables 11 and 12, respectively The geometric properties are as follows: L=a ¼ 2, a=h ¼ 100, h ¼ 0.002 m, d1 ¼ d2 ¼ 0.002 m, h1 ¼ h2 ¼ 0.005 m, n1 ¼ n2 ¼ 50, and k ¼ The corresponding results of the critical loads of externally stiffened FGM shells are taken from Tables and respectively for comparison From Table 11, it can be seen that for convex (R > 0) internally and externally stiffened FGM toroidal shells, both critical axial and pressure loads decrease when the R=h ratio increases The critical axial load of internally stiffened shells is greater than that of externally stiffened shells, but the critical pressure load of internally stiffened shells is lower than that of externally stiffened ones As can be seen for concave (R < 0) stiffened toroidal shells (Table 12), the unstable tendency of both critical loads occurs when R=h changes The critical pressure load of internally stiffened shells Table 10 Critical Buckling Load of Stiffened FGM Toroidal Shell under Hydrostatic Pressure Hydrostatic pressure, qcr × 106 N=m2 L=h Configuration Rings external Stringers external Rings external and stringers external © ASCE 100 200 300 400 19.9214 (1, 7) 5.6678 (1, 11) 19.9483 (1, 7) 12.7601 (1, 7) 4.7663 (1, 10) 12.8772 (1, 7) 10.1562 (1, 7) 4.5831 (1, 10) 10.2969 (1, 7) 8.8472 (1, 7) 4.5154 (1, 9) 8.9958 (1, 7) 04015054-8 J Eng Mech J Eng Mech Table 11 Critical Buckling Loads of Stiffened FGM Convex Toroidal Shell Segment (R > 0) under Axial and Lateral Pressure Load pcr × 103 (MPa) Internally stiffened R=h Downloaded from ascelibrary.org by National Taiwan University on 06/28/15 Copyright ASCE For personal use only; all rights reserved 200 300 400 500 2.4213 2.4192 2.4181 2.4175 (9, (9, (9, (9, qcr (MPa) Externally stiffened 1) 1) 1) 1) 2.2917 2.1962 2.1291 2.0899 (8, (7, (7, (7, Internally stiffened 7) 7) 7) 7) 17.5997 12.4664 10.1513 8.7075 (1, (1, (1, (1, Externally stiffened 8) 7) 7) 6) 22.6557 15.9629 12.8183 10.9962 (1, (1, (1, (1, 8) 7) 6) 6) Table 12 Critical Buckling Loads of Stiffened FGM Concave Toroidal Shell Segment (R < 0) under Axial and Lateral Pressure Load pcr × 103 (MPa) R=h −200 −300 −400 −500 Internally stiffened 0.6499 0.6924 0.5419 0.8378 (2, (1, (1, (1, 4) 3) 3) 3) qcr (MPa) Externally stiffened 0.5586 0.7188 0.5312 0.8048 (2, (1, (1, (1, 4) 3) 3) 3) Internally stiffened 4.0091 1.8984 1.4858 2.2969 (2, (1, (1, (1, 4) 3) 3) 3) Externally stiffened 3.5641 2.2507 1.7292 2.4750 (1, (1, (1, (1, 2) 3) 3) 3) Table 13 Critical Buckling Loads of Stiffened FGM Toroidal Shell under Hydrostatic Pressure 0.125 0.25 0.5 0.8 −0.125 −0.25 −0.5 −0.8 Externally stiffened 7.0010 10.0517 16.1853 19.2260 2.4083 1.4389 3.2552 5.2717 (1, (1, (1, (5, (1, (1, (2, (3, 6) 6) 8) 10) 4) 3) 4) 5) Internally stiffened 6.2528 8.8259 13.8659 19.1813 2.3667 1.4681 3.7878 4.8206 (1, (1, (1, (1, (1, (1, (2, (2, 6) 7) 8) 10) 4) 3) 4) 4) is lower than that of externally stiffened ones (except for R=h ¼ −200) Consequently, the pressure-bearing capacity of externally stiffened convex and concave FGM shells is greater than that of internally stiffened ones Second, Table 13 illustrates the critical buckling load of stiffened FGM toroidal shells under hydrostatic pressure The geometric properties of this problem are as follows: L=a ¼ 2, L=h ¼ 200, h ¼ 0.002 m, d1 ¼ d2 ¼ 0.002 m, h1 ¼ h2 ¼ 0.005 m, n1 ¼ n2 ¼ 50, k ¼ 1, with a=R varying The corresponding critical loads of externally stiffened shells are taken from Table It is observed that for convex stiffened FGM toroidal shells the critical buckling hydrostatic pressure load increases when the a=R ratio increases and the critical load of externally stiffened shells is higher than that of internally stiffened ones It is impossible to determine the behavior of concave stiffened FGM toroidal shells under such conditions This problem will be investigated further Conclusions A formulation of the governing equations for the investigation of the linear buckling of eccentrically stiffened functionally graded toroidal shell segments subjected to axial compression, lateral © ASCE Acknowledgments This research was funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant 107.02-2013.25 Hydrostatic pressure, qcr × 106 N=m2 a=R pressure, and hydrostatic pressure based on classical shell theory, the smeared stiffener technique, and the adjacent equilibrium criterion was presented Analytical solutions in the form of Fourier series are assumed to satisfy the simply supported boundary conditions to derive the closed-form relations of the buckling load The buckling behavior of FGM stiffened toroidal shell segments can be investigated using the method presented The effects of stiffeners, volume-fraction index, and dimensional parameters on the buckling of FGM stiffened toroidal shell segments were observed, illustrating specific characteristics of such a structure This study has demonstrated the following points: There exists a definite trend of variation in critical compressive, pressure, and hydrostatic pressure loads versus the variation in the volume-fraction index k and stiffeners for both convex shells (R > 0) and concave shells (R < 0) For convex toroidal shell segments, there exists a definite trend toward variation in critical loads versus dimensional ratios, but for concave shells this trend is unstable The critical buckling load of concave toroidal shell segments is lower than that of convex ones Stiffeners, volume-fraction index k, and dimensional ratios strongly affect the critical buckling load of toroidal shell segments References Bich, D H, Phuong, N T., and Tung, H V (2012) “Buckling of functionally graded conical panels under mechanical loads.” Compos Struct., 94(4), 1379–1384 Bich, D H., Dung, D V., Nam, V H., and Phuong, N T (2013) “Nonlinear static and dynamic buckling analysis of imperfect eccentrically stiffened functionally graded circular cylindrical thin shells under axial compression.” Int J Mech Sci., 74, 190–200 Brush, D O., and Almroth, B O (1975) Buckling of bars, plates and shells, McGraw-Hill, New York Dung, D V., and Hoa, L K (2013) “Research on nonlinear torsional buckling and post-buckling of eccentrically stiffened functionally graded thin circular cylindrical shells.” Compos Part B, 51, 300–309 Huang, H., and Han, Q (2008) “Buckling of imperfect functionally graded cylindrical shells under axial compression.” Eur J Mech A/Solids, 27(6), 1026–1036 Huang, H., and Han, Q (2010) “Nonlinear dyamic buckling of functionally graded cylindrical shells 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(2010) “Buckling analysis of FGM circular shells under combined loads and resting on the Pasternak type elastic foundation.” Mech Res Commun., 37(6), 539–544 Stein, M (1964) “The influence of prebuckling deformations and stresses on the buckling of perfect cylinders.” TR R-190, NASA, Washington, DC Stein, M., and McElman, J A (1965) “Buckling of segments of toroidal shells.” Am Inst Aeronaut Astronaut J., 3(9), 1704–1709 Weaver, P M (2000) “Design of laminated composite cylindrical shells under axial compression.” Compos Part B, 31(8), 669–679 Downloaded from ascelibrary.org by National Taiwan University on 06/28/15 Copyright ASCE For personal use only; all rights reserved Shen, H S (2013) “Thermal postbuckling of shear deformable FGM cylindrical shells surrounded by an elastic medium.” J Eng Mech., 10.1061/(ASCE)EM.1943-7889.0000439, 979–991 Shen, H S., and Leung, A (2003) “Postbuckling of pressure-loaded functionally graded cylindrical panels in thermal environments.” J Eng Mech., 10.1061/(ASCE)0733-9399(2003)129:4(414), 414–425 Shen, H S., and Noda, N (2007) “Postbuckling of pressure-loaded FGM hybrid cylindrical shells in thermal environments.” Compos Struct., 77(4), 546–560 © ASCE 04015054-10 J Eng Mech J Eng Mech ... critical load of a stiffened FGM toroidal shell segment under a pressure load decreases when the toroidal shell equator curvature or length rises Effect of a=R Ratio The critical buckling load of an... lower than that of a convex shell Effect of Ratio L=h Table 10 indicates the critical buckling load of stiffened FGM toroidal shell segments under hydrostatic pressure with four configurations: L=h... good agreement of the present formulation Results of Buckling Analysis of FGM Toroidal Shell Segments To illustrate the present formulation, the FGM toroidal shell segments are made of aluminum