DSpace at VNU: Nonlinear torsional buckling and postbuckling of eccentrically stiffened FGM cylindrical shells in thermal environment

DSpace at VNU: Nonlinear torsional buckling and postbuckling of eccentrically stiffened FGM cylindrical shells in therma...

Accepted Manuscript Nonlinear torsional buckling and postbuckling of eccentrically stiffened FGM cylindrical shells in thermal environment Dao Van Dung, Le Kha Hoa PII: DOI: Reference: S1359-8368(14)00463-6 http://dx.doi.org/10.1016/j.compositesb.2014.10.018 JCOMB 3233 To appear in: Composites: Part B Received Date: Revised Date: Accepted Date: 16 April 2014 30 September 2014 October 2014 Please cite this article as: Dung, D.V., Hoa, L.K., Nonlinear torsional buckling and postbuckling of eccentrically stiffened FGM cylindrical shells in thermal environment, Composites: Part B (2014), doi: http://dx.doi.org/10.1016/ j.compositesb.2014.10.018 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain Nonlinear torsional buckling and postbuckling of eccentrically stiffened FGM cylindrical shells in thermal environment Dao Van Dunga, Le Kha Hoaa, b, * a Vietnam National University, Hanoi, Viet Nam b Military Academy of Logistics, Viet Nam * Corresponding author: Tel.: +84 989358315 E-mail address: lekhahoa@gmail.com Abstract: The main aim of this paper is to investigate the nonlinear buckling and post-buckling of functionally graded stiffened thin circular cylindrical shells surrounded by elastic foundations in thermal environments and under torsional load by analytical approach Shells are reinforced by closely spaced rings and stringers in which material properties of shell and the stiffeners are assumed to be continuously graded in the thickness direction The elastic medium is assumed as two-parameter elastic foundation model proposed by Pasternak Based on the classical shell theory with von Karman geometrical nonlinearity and smeared stiffeners technique, the governing equations are derived Using Galerkin method with three-term solution of deflection, the closed form to find critical torsional load and post-buckling load-deflection curves are obtained The effects of temperature, stiffener, foundation, material and dimensional parameters are analyzed Keywords: Torsion; Functionally graded material; B Thermomechanical; A Discontinuous reinforcement; B Buckling; C Analytical modelling Introduction Functionally graded materials (FGMs) are a new group of materials, which were first introduced in 1984 by a group of materials scientists in Japan as a means of preparing thermal barrier materials [1] They are composite materials which have mechanical properties varying continuously from one surface to the other of structure In recent years, FGMs have been developed for use in high temperature environments as for aerospace structures, fusion reactors and other engineering fields [2] Many of FGMs are composed of a ceramic and a metal and can take the advantage of the desirable properties as heat and corrosion resistance of ceramics and high tensile strength, toughness and bonding capability of metals Cylindrical shells are used widely in many engineering applications and they has been attracted attention of many researchers For isotropic and composite shells, many studies have been focused on the buckling and postbuckling as well as vibration of shells Dasgupta [3] investigated the free torsional vibration of thick isotropic incompressible circular cylindrical shell subjected to uniform external pressure Green’s formulation and a perturbation technique are used in his work Using the Galerkin method, Tani and Doki [4] researched vibration and buckling of fluid-filled and cross-ply laminated orthotropic composite cylindrical shells under torsion By Galerkin method and based on the modified Donnell type dynamic stability and compatibility equations, Sofiyev [5] investigated the torsianal buckling of cross-ply laminated orthotropic composite cylindrical thin shells under dynamic loading which is a power function of time Torsional buckling of elastic cylinders with a hard surface coating layer is studied by Zhang and Fu [6] in which deformations of the core and surface layer are obtained analytically though the Navier’s equation and thin shell model, respectively Hui and Du [7] investigated the initial postbuckling behavior of imperfect, and antisymmetric cross-ply cylindrical shells under torsion with the reduced-Badorf parameter ≤ ZH ≤ 20 Using singular perturbation technique, Zhang and Han [8] investigated the buckling and post-buckling of imperfect cylindrical shells subjected to torsion based on the Karman-Donnell-type nonlinear differential equations Jiang et al [9] presented mechanical, electrical and thermal properties of aligned carbon nanotube/polyimide The composite is promising for applications that require high strength, lightweight, or high electrical and thermal conductivities There are many research works on FGM cylinders Batra [10] studied the torsion of cylinders with material modulii varying only in the axial direction Wang et al [11] presented exact solution and transient behavior for torsional vibration of functionally graded finite hollow cylinder Shen [12], based on the higher order shear deformation theory, obtained the results of stability problem of torsion-loaded FGM shells in thermal environments A singular perturbation technique is employed to determine buckling shear load and post-buckling equilibrium paths in that paper Huang and Han [13] studied the nonlinear buckling of torsion-loaded FGM unstiffened cylindrical shells by using the nonlinear large deflection shell theory and Ritz method The nonlinear buckling shape observed in experiment is taken into account in their work Sofiyev and Schnack [14] studied the stability of FGM cylindrical shells under linearly increasing dynamic torsional loading The modified Donnell type dynamic stability equation and Galerkin method were used However, the geometrical relation is linear and the approximate solution was chosen by oneterm Liew et al [15] presented a postbuckling analysis of functionally graded cylindrical shells under axial compression and thermal loads using the element-free kp-Ritz method The formulation was developed to handle problems of small strains and moderate rotations, based on the first-order shear deformation shell theory and von Karman strains For shells resting on elastic foundations, many researches are focused on the buckling and post-buckling analysis of un-stiffened shells Sheng and Wang [16] considered the effect of thermal load on buckling, vibration and dynamic buckling of FGM cylindrical un-stiffened shells embedded in a linear elastic medium based on the first-order shear deformation theory (FSDT) taking into account the rotary inertia and transverse shear strains The post-buckling analysis of tensionless Pasternak FGM un-stiffened cylindrical shells surrounded by an elastic medium under the lateral pressure and axial load are studied by Shen et al [17, 18] using the singular perturbation technique and the higher-order shear deformation shell theory (HDST) Bagherizadeh et al [19] based on the higher-order shear deformation shell theory, investigated the mechanical buckling of FGM un-stiffened cylindrical shells surrounded by Pasternak elastic foundation Many investigations on the stability and vibration analysis of FGM un-stiffened cylindrical and conical shells surrounded by elastic foundations also have been published by Sofiyev et al [20-23] The Galerkin method are used to determine buckling load and frequency of shell Shen and Xiang [24] presented a postbuckling analysis of carbon nanotube-reinforced composite cylindrical panels resting on elastic foundations and subjected to axial compression in thermal environments The cylindrical panels are reinforced by aligned single-walled carbon nanotubes which are assumed to be functionally graded through the thickness direction with different types of distributions Shen and Wang [25] investigated the large amplitude vibration behavior of a shear deformable FGM cylindrical panel resting on elastic foundations in thermal environments Two kinds of micromechanics models, namely, Voigt model and Mori–Tanaka model, were considered The motion equations were based on a higher order shear deformation shell theory that includes shell panel-foundation interaction Based on nonlocal elasticity and piezoelasticity theories, Arani et al [26] investigated electro-thermo-torsional buckling response of a double-walled boron nitride nanotube The effects of surrounding elastic medium such as the spring constant of the Winkler-type and the shear constant of the Pasternak-type were taken into account Sofiyev [27] studied the nonlinear vibration of heterogeneous orthotropic truncated conical shells resting on the Winkler–Pasternak elastic foundations The formulation is based on the Donnell shell theory, exponential-law distribution of orthotropic material properties and von Karman geometric nonlinearity For FGM stiffened cylindrical shell, the stability problem is also very interest subject In 2009, Najafizadeh et al [28] with the linear stability equations in terms of displacements studied the buckling of eccentrically stiffened functionally graded (ESFGM) cylindrical shells under axial compression The stiffeners and skin, in their work, are assumed to be made of functionally graded materials and its properties vary continuously through the thickness direction With the homogeneous stiffeners assumption, Bich et al [29-31] presented an analytical approach to investigate the nonlinear static and dynamic buckling and post-buckling of eccentrically ES-FGM plate, shallow shell and cylindrical shell based on the classical plate and shell theory With FGM stiffeners, Dung and Hoa [32, 33] obtained the results on the nonlinear buckling and post-buckling analysis of FGM circular cylindrical shells under external pressure or torsional load and reinforced by stringers and rings Galerkin method was used to obtain closed-form expressions to determine critical buckling loads and load-deflection curve However, the thermal element still is not taken in to account The objective of this study is to extend the results of the work [33] for torsionally loaded ES-FGM circular cylindrical shell in thermal environments The novelty of this work is that the FGM shells are reinforced by FGM rings and stringers resting on foundation and in thermal environments The elastic foundation is assumed as two- parameter elastic foundation model proposed by Pasternak [34] The theoretical formulations based on the smeared stiffeners technique and the classical shell theory with von Karman’s geometrical nonlinearity, are derived In addition, an approximate three-term solution of deflection including the linear buckling shape and the nonlinear buckling shape are more correctly chosen The closed form expression for determining critical buckling load and postbuckling loaddeflection curves are obtained by Galerkin method The obtained results show that the temperature field affects considerably on critical torsional load and postbuckling behavior of shell The preheated ES-FGM cylindrical shells exhibit a decreasing tendency in postbuckling loading carrying capacity when they are subjected to the added action of torsional load The analysis is carried out also to show the effects of various parameters such as stiffener, foundation, dimensional parameters, and volume fraction index of materials on the stability of ES-FGM cylindrical shell Theoretical derivations 2.1 Torsionally loaded ES-FGM cylindrical shells in thermal environment Consider a thin circular cylindrical shell as shown in Fig 1, with mean radius R, thickness h and length L subjected to torsion load of intensity τ Two butt-ends of shell are assumed to be only deformed in their planes and they still are circular [35] The middle surface of the shells is referred to the coordinates (x, θ , z), y = Rθ The coordinate axis x is chosen in the generatrix direction of the shell, while the coordinate axes y and z respond to the circumferential and thickness directions, respectively Further, assume that the shell is stiffened by closely spaced circular rings and longitudinal stringers attached to the shell skin, and the stiffeners and skin are made of functionally graded materials graded materials varying continuously through the thickness direction of the shell with the power law The ES-FGM cylindrical shells are investigated in this work with two cases Case 1: Cylindrical shell with inner stiffeners Case 2: Cylindrical shell with outer stiffeners x y K1 z K2 h y bs z zs τ τ R ds hs L hr br zr dr h Fig 1: Geometry and coordinate system of a stiffened FGM circular cylindrical shell For case 1, Young’s modulus and thermal expansion coefficient of FGM shell and FGM stiffeners are given by [28, 32] k h h ⎛ 2z + h ⎞ Esh = Em + Ecm ⎜ ⎟ , Ecm = Ec − Em , k ≥ , − ≤ z ≤ , 2 ⎝ 2h ⎠ k ⎛ 2z + h ⎞ α sh = α m + α cm ⎜ ⎟ , α cm = α c − α m , ⎝ 2h ⎠ (1) k ⎛ 2z − h ⎞ h h Es = Ec + Emc ⎜ ⎟ , Emc = Em − Ec , k2 ≥ , ≤ z ≤ + hs , 2 ⎝ 2hs ⎠ k ⎛ 2z − h ⎞ α s = α c + α mc ⎜ ⎟ , α mc = α m − α c , h s ⎠ ⎝ (2) k ⎛ 2z − h ⎞ h h Er = Ec + Emc ⎜ ⎟ , k3 ≥ , ≤ z ≤ + hr , 2 ⎝ 2hr ⎠ k ⎛ 2z − h ⎞ α r = α c + α mc ⎜ (3) ⎟ , ⎝ 2hr ⎠ where ν sh = ν s = ν r = ν = const , k, k2 and k3 are volume fractions indexes of shell, stringer and ring, respectively Subscripts c, m, sh, s and r denote ceramic, metal, shell, longitudinal stringers and circular ring, respectively It is evident that, from Eqs (1)(3), a continuity between the shell and stiffeners is satisfied Note that the thickness of the stringer and the ring are respectively denoted by hs , and hr ; and Ec , Em are Young’s modulus of the ceramic and metal; and αc, αm are thermal expansion coefficient of ceramic and metal, respectively The coefficient ν is Poison’s ratio For case 2, Young’s modulus and thermal expansion coefficient of FGM shell and FGM stiffeners are given in Appendix A To account for the effect of large deflection, the von Karman type nonlinear kinematic relation for the strain components across the shell thickness at a distance z from the middle surface are of the form [36, 37] + zk xy , ε x = ε x0 + zk x , ε y = ε y0 + zk y , γ xy = γ xy (4) k x = − w, xx , k y = − w, yy , k xy = − w, xy , in which ε 0x = u, x + w,2x , ε 0y = v, y − w + w, y , γ 0xy = u, y + v, x + w, x w, y , R (5) where u = u ( x, y ), v = v( x, y ) and w = w( x, y ) are the displacements of the middle surface points along x, y and z axes, and k x , k y and k xy are the change of curvatures and twist of shell, respectively The geometrical compatibility equation deduced from Eqs (5), is given as ε x0, yy + ε y0, xx − γ xy , xy = − w, xx + w,2xy − w, xx w, yy R (6) Unlike [33], this study has used Hooke’s law for cylindrical shell taking into account the temperature as σ xsh = Esh E α ε x + νε y − sh sh ΔT , −ν −ν σ ysh = Esh E α ε y + νε x − sh sh ΔT , −ν −ν σ xysh = E sh γ xy , 2(1 + ν ) ( ( ) ΔT = T − T0 , ) (7) and for stiffeners σ xs = Es ε x − Esα s ΔT , σ yr = Er ε y − Erα r ΔT , (8) where the temperature is assumed to be dependent only on z Taking into account the contribution of stiffeners by the smeared stiffeners technique and omitting the twist of stiffeners and integrating the stress-strain equations and their moments through the thickness of the shell, the expressions for force and moment resultants of an eccentrically stiffened FGM cylindrical shell are defined similarly as [28, 36] N x = C11ε 0x + C12ε 0y + C14 k x + C15 k y − φ1 − φ1Tx , N y = C12ε 0x + C22ε 0y + C24 k x + C25 k y − φ1 − φ1Ty , (9) N xy = C33γ 0xy + C36 k xy , M x = C14ε 0x + C24ε 0y + C44 k x + C45 k y − φ2 − φ2Tx , M y = C15ε 0x + C25ε 0y + C45 k x + C55 k y − φ2 − φ2Ty , (10) M xy = C63γ 0xy + C66 k xy , where the stiffness parameters Cij and thermal parameters φ1 ,φ2 , φijT can be found in Appendix A The relation (10) is most significant contribution and first time it is found in this work in which the thermal element of the both shell and stiffener in equations of Nij, Mij are considered For later use, the reverse relations obtained from Eqs (9) are as * ε 0x = C *22 N x − C12* N y + C14* k x + C15* k y + C16 φ1 + C *22φ1Tx − C12* φ1Ty , * ε 0y = −C12* N x + C11* N y + C *24 k x + C *25 k y + C26 φ1 − C12* φ1Tx + C11* φ1Ty , * * γ 0xy = C 33 N xy − C 36 k xy (11) Introduction of Eqs (11) into Eqs (10), the moment resultants become * * * * * * T * T M x = D14 N x + D24 N y + D44 k x + D45 k y + D46 φ1 + D14 φ1x + D24 φ1 y − φ2 − φ2Tx , * * * * * * T * T M y = D15 N x + D25 N y + D54 k x + D55 k y + D56 φ1 + D15 φ1x + D25 φ1 y − φ2 − φ2Ty , * M xy = D*63 N xy + D66 k xy , (12) where the coefficients Cij* and Dij* are defined in Appendix B 2.2 Nonlinear equilibrium equations taking into account an elastic foundation The nonlinear equilibrium equations of cylindrical shell, taking into account an elastic foundation, based on the classical shell theory are given by [35, 37, 38] N x, x + N xy , y = 0, N xy , x + N y , y = 0, M x , xx + M xy , xy + M y , yy + Ny R + N x w, xx + N xy w, xy + N y w, yy ( (13) ) − K1w + K w, xx + w, yy = 0, where K1 (N/m ) is modulus of subgrade reaction and K (N/m) - the shear modulus of the subgrade, By introducing a stress function f ( x, y ) as N x = f, yy , N y = f, xx , N xy = − f, xy (14) It is obvious that the first two of Eqs (13) are identically satisfied Substituting Eqs (12) and (14) into the third of Eqs (13), taking into account Eq (4) we obtain the following equation α11w, xxxx + α12 w, xxyy + α13 w, yyyy + α14 f, xxxx + α15 f, xxyy + α16 f, yyyy + ( ) f, xx + f , yy w, xx + f, xx w, yy − f, xy w, xy − K1w + K w, xx + w, yy = 0, R (15) where * * * * , α12 = − ( D45 α11 = − D44 + D66 + D54 ) , α13 = − D55* , * * * , α15 = ( D14 α14 = D24 − D*63 + D25 ) , α16 = D15* (16) Eq (15) includes two dependent unknown functions w and f and in order to find a second equation relating to these two functions, the geometrical compatibility equation (6) is used For this aim, substituting Eq (11) into Eq (6), yields β11 f , xxxx + β12 f , xxyy + β13 f , yyyy + β14 w, xxxx + β15 w, xxyy + β16 w, yyyy − w,2xy + w, xx w, yy + w, xx = , R (17) where * β11 = C11* , β12 = C33 − 2C12* , β13 = C *22 , β14 = −C *24 , β15 = − (C14* + C *25 + C 36* ) , β16 = −C15* (18) Eqs (15) and (17) are the nonlinear governing equations used to investigate the nonlinear buckling and postbuckling behavior of eccentrically stiffened FGM cylindrical shells surrounded by elastic foundations and under uniform torsion loads in thermal environment Thermomechanical buckling and postbuckling analysis Consider a torsion-loaded cylindrical shell surrounded by elastic foundations in thermal environment and it is simply supported at two butt-ends x = and x = L In this case, the deflection of shell is expressed by [13, 33, 35] w = w( x, y ) = ξ + ξ1 sin α x sin β ( y − λ x ) + ξ sin α x , (19) in which α = mπ / L, β = n / R and m is the number of axis half waves and n is the number of circumferential waves The first term of w in Eq (19) represents the uniform deflection of points belonging to two butt-ends x=0 and x=L, the second-a linear buckling shape, and the third-a nonlinear buckling shape It is found that the simply supported boundary condition at x=0 and x=L is fulfilled on the average sense Introduction of Eq (19) into Eq (17) obtains β11 f, xxxx + β12 f, xxyy + β13 f, yyyy = G01 cos 2α x + G02 cos 2β ( y − λ x ) ⎡ ⎛α ⎡ ⎛α ⎞ ⎤ ⎞ ⎤ +G03 cos β ⎢ y + ⎜ − λ ⎟ x ⎥ + G04 cos β ⎢ y − ⎜ + λ ⎟ x ⎥ ⎠ ⎦ ⎠ ⎦ ⎣ ⎝β ⎣ ⎝β ⎧⎪ ⎡ ⎛ α ⎡ ⎞ ⎤ ⎛ α ⎞ ⎤ ⎫⎪ +G05 ⎨cos β ⎢ y − ⎜ + λ ⎟ x ⎥ − cos β ⎢ y + ⎜ − λ ⎟ x ⎥ ⎬ , ⎠ ⎦ ⎝ β ⎠ ⎦ ⎪⎭ ⎪⎩ ⎣ ⎝ β ⎣ where ⎡ ⎤ 1⎞ 1 ⎛ G01 = ⎢ 2ξ 2α ⎜ β14α − ⎟ + ξ12α β ⎥ , G02 = ξ12α β , R⎠ 2 ⎝ ⎣ ⎦ (20) Using Eqs (11) and (14), this integral becomes ϕ= 2π RL 2π R L * * ∫ ∫ (C36 w, xy − C33 f, xy − w, x w, y ) dxdy 0 Substituting w and f from Eqs (19) and (22) into this equation obtains ϕ= 2(1 + ν ) n2 λ τ + ξ 12 Em + ( Ec − Em ) / ( k + 1) 4R (34) Combining Eq (27) with Eq (34), the τ − ϕ relation curve of shells will be studied Eq (34) shows that the relation between twist angle φ and shear stress is linear when ξ1 = In addition, ξ1 = and τ = leads to ϕ = therefore the τ − ϕ curve will cross original coordinates Numerical solution and discussion of results 4.1 Comparative study In order to demonstrate the accuracy of present approach, Table compares the static critical load for un-stiffened FGM cylindrical shell without foundation and under torsion with the results given by Huang and Han [13] and Sofiyev and Kuruoglu [23] The input parameters are taken to be h=0.01m, L/R=2, k=1, Ec = 1.68063 × 1011 Pa, Em = 1.05698 × 1011 Pa, ν c = ν m = 0.3 , m=1 As can be seen the good agreement is obtained in this comparison Table 1: Comparisons of static critical torsion load τ cr (MPa) for un-stiffened FGM cylindrical shell without foundation τ crs (MPa) R/h a Huang and Han [13] Sofiyev and Kuruoglu [23] Present study 200 81.70 (10,0.37) 83.67 (8,0.24) 83.4679 (10,0.37)a 300 48.61 (11,0.33) 49.86 (9,0.22) 49.5176 (11,0.33) 400 33.82 (12,0.31) 34.44 (10,0.21) 34.3718 (12,0.31) 500 25.58 (13,0.30) 25.93 (11,0.20) 25.9500 (13,0.30) the numbers in the parentheses denote the buckling mode (n,λ), m=1 In the following subsections, the materials are Alumina with Ec = 380 GPa , α c = 5.4 × 10−6 1/K and Aluminum with Em = 70 GPa , α m = 22.2 × 10−6 1/K and poisson ratio ν = 0.3 The foundation parameters are taken by K1 = 1.5 ×107 N/m3, K = 1.5 × 105 N/m 13 4.2 Effect of temperature field The effect of uniform temperature rise on the buckling and postbuckling is considered in this section The lower and upper torsional load may be determined by minimization critical load with respect to m, n and λ, basing on Eqs (27) and (31) Tables 2a and 2b and Fig show the effect of temperature field on critical loads It can be seen that the critical torsional load of shell reduces when ΔT increases For example, with inner stiffeners and k=1, the lower torsional load τlower =253.8796 MPa corresponding ΔT = 0K is bigger than the lower torsional load corresponding to ΔT = 300K about 10.8205MPa and ΔT = 600K about 22.076MPa Table 2a and 2b also show that the critical load of shell with outer stiffeners is bigger than the critical load of shell with inner stiffeners and the critical load of un-stiffened shell is the smallest The effects of temperature rise on τ - Wmax/h post-buckling curves are shown in Fig The lines 2, and not start at origin of coordinates That means when the shell is preheated, the temperature field makes the shell to be deflected outward (negative deflection) prior to mechanical load acting on it When the shell is subjected the torsional load, its outward deflection is reduced and when torsional load exceeds bifurcation point of load, an inward deflection occurs Table 2a: Effects of ΔT and k on lower torsional load h=0.004m, R/h=200, L/R=2, hs= hr=0.004m, bs=br=0.004m, ns=nr=15, K1=1.5×107N/m3, K2=1.5×105N/m τ (MPa) Un-stiffened Inner Outer a ΔT = K ΔT = 300 K ΔT = 600 K ΔT = K ΔT = 300 K ΔT = 600 K ΔT = K ΔT = 300 K ΔT = 600 K k=0 361.3095 (10,0.41) k=0.5 a k=1 k=5 276.2212 (11,0.43) 235.7449 (12,0.43) 171.2656 (12,0.49) 354.8230 (10,0.41) 266.2685 (11,0.41) 224.3218 (12,0.42) 157.6628 (12,0.46) 348.2363 (10,0.40) 255.9502 (11,0.40) 212.3229 (11,0.39) 143.2822 (12,0.43) 401.4170 (10,0.45) 300.5859 (11,0.46) 253.8796 (11,0.46) 180.9922 (12,0.51) 395.4924 (10,0.44) 291.4347 (11,0.45) 243.0591 (11,0.44) 167.8085 (11,0.46) 389.4510 (10,0.44) 282.0381 (11,0.44) 231.8036 (11,0.42) 153.4360 (11,0.44) 405.2715 (10,0.44) 321.3900 (10,0.45) 279.2830 (11,0.47) 197.7772 (11,0.50) 399.2292 (10,0.44) 312.0045 (10,0.44) 268.8763 (11,0.46) 184.5133 (11,0.48) 393.1560 (10,0.43) 302.3696 (10,0.43) 258.1067 (11,0.44) 170.6088 (11,0.45) Buckling mode (n, λ); m=1 Table 2b: Effects of ΔT and k on upper torsional load h=0.004m, R/h=200, L/R=2, hs= hr=0.004m, bs=br=0.004m, ns=nr=15, K1=1.5×107N/m3, K2=1.5×105N/m τ (MPa) Un-stiffened Inner stiffeners Outer Stiffeners ΔT = K ΔT = 300 K ΔT = 600 K ΔT = K ΔT = 300 K ΔT = 600 K ΔT = K ΔT = 300 K ΔT = 600 K k=0 k=0.5 k=1 k=5 374.0986 (11,0.39) 279.6001 (11,0.40) 235.7769 (12,0.43) 171.2656 (12,0.49) 367.0977 (11,0.38) 268.8476 (11,0.39) 224.3218 (12,0.42) 157.6628 (12,0.46) 360.0155 (11,0.38) 257.8244 (11,0.38) 212.3499 (11,0.38) 143.2822 (12,0.43) 425.0330 (10,0.39) 306.5762 (11,0.42) 255.6699 (11,0.43) 180.9922 (12,0.51) 418.1042 (10,0.39) 296.5269 (11,0.42) 244.2381 (11,0.42) 167.8085 (11,0.46) 411.1756 (10,0.39) 286.2585 (11,0.41) 232.4777 (11,0.41) 153.4360 (11,0.44) 426.0778 (10,0.39) 329.2253 (11,0.43) 281.6150 (11,0.45) 197.7772 (11,0.50) 418.9989 (10,0.38) 319.3484 (11,0.42) 270.5333 (11,0.43) 184.5133 (11,0.48) 411.8938 (10,0.38) 309.2844 (11,0.42) 259.1681 (11,0.42) 170.6088 (11,0.45) 14 520 500 1200 Wmax/h=0 -0.80 h=0.004 m, R/h=100, L/R=2, k=1 τ (MPa) 1000 Inner stiffeners: hs= hr=bs=br=0.004m ns=nr=15 τ (MPa) -0.26 -0.53 480 Foundation: K1=1.5×107 N/m3 K2=1.5×105 N/m 460 440 420 1: 2: 3: 4: ΔT=0K (8,0.52) ΔT=200K (8,0.52) ΔT=400K (8,0.51) ΔT=600K (8,0.51) h=0.004 m, R/h=100 Inner stiffeners: L/R=2, ΔT=600K hs= hr=bs=br=0.004m ns=nr=15 Wmax/h = - 0.43 800 Wmax/h = - 0.68 600 Wmax/h = - 0.80 400 Wmax/h = - 1.06 -2 10 1: k=0, (7,0.49) 2: k=0.5, (8,0.52) 3: k=1, (8,0.51) 4: k=5, (8,0.53) 12 14 16 Foundation: K1=1.5×107 N/m3 K2=1.5×105 N/m 200 Wmax/h 10 Wmax/h Fig 3: Effects of k on τ − Wmax / h curves (m=1) Fig 2: Effects of ΔT on τ − Wmax / h curves (m=1) 450 h=0.004 m R/h=200, L/R=2 400 τ (MPa) 500 300 250 Stiffeners: hs= hr=bs=br=0.004m ns=nr=15 100 -3 10 Foundation: K1=1.5×107 N/m3 K2=1.5×105 N/m 400 300 200 200 150 2a Foundation: K1=1.5×107 N/m3 K2=1.5×105 N/m 10 -2 1b 2b 10 Inner stiffeners: hs= hr=bs=br=0.004m ns=nr=15 h=0.004 m, k=1 600 a) ΔT=0K b) ΔT=600K 350 τ (MPa) 1: Inner stiffeners 2: Outer stiffeners -1 10 k 1a 10 100 10 10 100 1: 2: 3: 4: L/R=1, ΔT=0K L/R=1, ΔT=600K L/R=2, ΔT=0K L/R=2, ΔT=600K 150 200 250 300 350 400 450 R/h k Fig 4: Effects of ΔT on τlower − k curves Fig 5: Effects of ΔT and L/R on τ lower − R / h curves 4.3 Effects of volume fraction index k The effects of volume fraction index k on the critical torsional load of FGM shell is illustrated in Figs and 4, and Tables 2a and 2b It is found that, the critical buckling load decreases with the increase of k The value of critical load with k=0 (ceramic) is the greatest This property appropriate to the real characteristic of material, because the smaller value of k corresponds to the richer ceramic shell and the shell becomes better thermal barrier structure 4.4 Effects of geometric parameters Table 3a and 3b give the effects of the width-to-thickness ratio R/h and L/R ratio on the critical torsional load of FGM shell reinforced by orthogonal stiffeners surrounded by foundation in thermal environment in which the temperature difference ΔT=0K and 600K) Fig shows the effect of L/R ratio on τ lower − R / h curves Fig shows the effects of L/R ratio on τ − ϕ curves The effects of R/h ratio on τ − Wmax / h curves is 15 shown in Fig The obtained results show that the loading carrying capacity of shell is reduced considerably when R/L or R/h ratio increases Table 3a and 3b also indicate that with the same R/h ratio, the critical torsional load reduces when L/R ratio increases For example, with R/h=100 and ΔT= 600K, the critical torsional load τ lower =1093.3052 MPa corresponding to L/R=0.5 is bigger than the critical torsional loads corresponding to L/R=1, 1.5, about 1.71, 2.17, 2.49 times, respectively Table 3a: Effects of geometric parameters on lower torsional load k=1, h=0.004m, hs= hr=0.004m, bs=br=0.004m, ns=nr=15 (inner stiffeners), K1=1.5×107N/m3, K2=1.5×105N/m τ (MPa) L/R=0.5 L/R=1 L/R=1.5 L/R=2 L/R=0.5 L/R=1 L/R=1.5 L/R=2 ΔT = K ΔT = 600 K R/h=100 R/h=200 R/h=300 R/h=400 1095.9510 (10,0.80) 421.9445 (13,0.77) 269.8866 (16,0.73) 211.1693 (19,0.71) 644.1460 (9,0.66) 295.5815 (12,0.60) 218.8618 (15,0.57) 180.4983 (19,0.58) 508.8081 (8,0.56) 265.1730 (11,0.51) 202.3563 (15,0.51) 171.4660 (19,0.56) 443.1430 (8,0.52) 253.8796 (11,0.46) 194.2700 (15,0.49) 167.4829 (18,0.54) 1093.3052 (10,0.80) 410.1812 (13,0.75) 239.2158 (17,0.64) 147.7923 (20,0.51) 640.6520 (9,0.66) 279.2996 (12,0.57) 177.0183 (15,0.47) 101.9994 (18,0.38) 504.5533 (8,0.55) 245.8224 (11,0.48) 155.9333 (15,0.42) 88.9459 (17,0.35) 438.5281 (8,0.51) 231.8036 (11,0.42) 145.9805 (14,0.40) 83.0043 (17,0.34) Table 3b: Effects of geometric parameters on upper torsional load k=1, h=0.004m, hs= hr=0.004m, bs=br=0.004m, ns=nr=15 (inner stiffeners), K1=1.5×107N/m3, K2=1.5×105N/m τ (MPa) L/R=0.5 L/R=1 L/R=1.5 L/R=2 L/R=0.5 L/R=1 L/R=1.5 L/R=2 ΔT = K ΔT = 600 K R/h=100 R/h=200 R/h=300 R/h=400 1116.6520 (10,0.79) 440.7273 (14,0.72) 279.7911 (17,0.68) 214.0419 (20,0.68) 704.5875 (9,0.62) 317.2789 (13,0.55) 220.2843 (16,0.55) 180.4983 (19,0.58) 577.6305 (8,0.48) 275.7974 (12,0.48) 202.3563 (15,0.51) 171.4660 (19,0.56) 501.5850 (8,0.44) 255.6699 (11,0.43) 194.2700 (15,0.49) 167.4829 (18,0.54) 1113.2856 (10,0.79) 425.4406 (14,0.70) 243.8672 (17,0.61) 147.7923 (20,0.51) 700.2618 (9,0.61) 298.5508 (13,0.53) 177.1865 (15,0.46) 101.9994 (18,0.38) 572.2988 (8,0.48) 254.6588 (12,0.45) 155.9333 (15,0.42) 88.9459 (17,0.35) 495.9222 (8,0.44) 232.4777 (11,0.41) 145.9805 (14,0.40) 83.0043 (17,0.34) 600 500 Inner stiffeners: hs= hr=bs=br=0.004m ns=nr=15 550 400 450 400 350 1: R/h=100, 2: R/h=100, 3: R/h=200, 4: R/h=200, τ (MPa) τ (MPa) 500 -0.66 h=0.004 m L/R=2 k=1 Foundation: K1=1.5×107 N/m3 K2=1.5×105 N/m ΔT=0K (8,0.52) ΔT=500K (8,0.51) ΔT=0K (11,0.46) ΔT=500K (11,0.43) 1: 2: 3: 4: L/R=1, ΔT=0K (12,0.60) L/R=1, ΔT=500K (12,0.57) L/R=2, ΔT=0K (11,0.46) L/R=2, ΔT=500K (11,0.43) 300 200 h=0.004 m, R/h=200 k=1 300 -1.30 200 -2 100 250 10 12 14 16 0 18 Inner stiffeners: hs= hr=bs=br=0.004m ns=nr=15 Foundation: K1=1.5×107 N/m3 K2=1.5×105 N/m 0.2 0.4 0.6 0.8 1.2 1.4 1.6 φ (deg) Wmax/h Fig 7: Effects of ΔT and L/R on τ − ϕ curves (m=1) Fig 6: Effects of ΔT and R/h on τ-Wmax/h curves (m=1) 16 4.5 Effects of stiffener and foundation Effects of stiffener and foundation on the upper and lower load of FGM cylindrical shell are illustrated in Table 4a and 4b As can be observed that the critical torsional load increase when the foundation parameters K1 and K2 separately or together increase This point also is noted in Fig and In addition, Tables 4a and 4b, and Figs and 9, show that with the same stiffener numbers, the critical torsional load of FGM shell reinforced by rings is the biggest, the second is shell reinforced by orthogonal stiffeners, and the third is shell reinforced by stringers The critical buckling load of shell without foundation is smallest Figs 10 and 11 show the effects of stiffener, foundation and volume fraction k on τ − ϕ curves Comparing these curves, it can be observed they become to be down gradually when k increases Table 4a: Effects of stiffener and foundation on lower torsional load k=1, ΔT=300K, h=0.004m, R/h=100, L/R=2, hs= hr=0.004m, bs=br=0.004m (Inner stiffeners) τ (MPa) Unstiffened K2=0 N/m K1=0 N/m K1=1.5×10 K1=2.5×10 Stringer (ns=30) Ring (nr=30) Orthogonal (ns=15, nr=15) 301.3699 (8,0.41) 303.9003 (8,0.40) 412.5423 (7,0.49) 368.5542 (7,0.42) 357.4989 (8,0.45) 360.4690 (8,0.44) 464.4150 (7,0.51) 420.8492 (8,0.52) K2=2.5×10 393.2543 (8,0.47) 396.5045 (8,0.46) 498.2776 (7,0.52) 454.0865 (8,0.53) K2=0 N/m 322.0734 (8,0.41) 324.2389 (8,0.41) 436.3041 (7,0.49) 390.0392 (8,0.49) K2=1.5×10 377.7272 (8,0.45) 380.2906 (8,0.44) 487.3353 (7,0.52) 440.8521 (8,0.52) K2=2.5×10 413.2630 (8,0.47) 416.0699 (8,0.47) 520.7023 (7,0.53) 473.9226 (8,0.53) K2=0 N/m 335.0937 (8,0.41) 336.9980 (8,0.41) 451.5810 (7,0.50) 402.9092 (8,0.49) K2=1.5×10 390.4867 (8,0.45) 392.7908 (8,0.45) 502.3105 (7,0.53) 453.5460 (8,0.52) 425.8996 (8,0.47) 428.4295 (8,0.47) 533.7286 (8,0.59) 486.5130 (8,0.53) K2=1.5×10 K2=2.5×10 Table 4b: Effects of stiffener and foundation on upper torsional load k=1, ΔT=300K, h=0.004m, R/h=100, L/R=2, hs= hr=0.004m, bs=br=0.004m (Inner stiffeners) τ (MPa) Unstiffened K2=0 N/m K1=0 N/m3 K1=1.5×10 K1=2.5×10 Ring (nr=30) Orthogonal (ns=15, nr=15) 350.4397 (8,0.34) 350.7795 (8,0.34) 502.8422 (7,0.40) 436.4531 (8,0.40) 412.1370 (8,0.37) 412.5024 (8,0.37) 554.8175 (8,0.48) 490.8181 (8,0.43) K2=2.5×10 450.9582 (8,0.39) 451.3555 (8,0.39) 587.3581 (8,0.50) 525.8447 (8,0.45) K2=0 N/m 360.5838 (8,0.34) 360.9284 (8,0.34) 512.1840 (8,0.47) 444.8631 (8,0.41) K2=1.5×10 Stringer (ns=30) 421.3570 (8,0.38) 421.7411 (8,0.38) 561.8493 (8,0.49) 498.7536 (8,0.44) K2=2.5×10 459.7084 (8,0.40) 460.1310 (8,0.40) 594.2343 (8,0.50) 533.4933 (8,0.45) K2=0 N/m 367.2380 (8,0.35) 367.5914 (8,0.35) 517.0674 (8,0.47) 450.4698 (8,0.41) K2=1.5×10 427.4129 (8,0.38) 427.7998 (8,0.38) 566.5333 (8,0.49) 503.9761 (8,0.44) 464.6362 (9,0.43) 465.3663 (9,0.43) 598.8034 (8,0.51) 538.5641 (8,0.46) K2=1.5×10 K2=2.5×10 17 600 550 1: Stringer (ns=30) 2: Ring (nr=30) 3: Orthogonal (ns=nr=15) τ2upper (8,0.49) 550 500 τ2upper (8,0.44) τ2lower=487.3353 MPa (7,0.52) 450 τ1upper (8,0.38) 400 τ3lower=486.5130 MPa (8,0.53) 450 τ1upper (8,0.40) τ1lower=380.2906 MPa (8,0.44) Foundation: K1=1.5×107 N/m3 K2=1.5×105 N/m 10 τ2lower=440.8521 MPa (8,0.52) 1: K1=0; K2=0 3: K1=2.5*10 ; K2=2.5*10 320 12 2b 2a 3b 3a 1b 1a τ1alower=380.2906 MPa (8,0.44); τ1blower=428.4295 MPa (8,0.47) τ2alower=487.3353 MPa (7,0.52); τ2blower=533.7286 MPa (8,0.59) τ3alower=440.8521 MPa (8,0.52); τ3blower=486.5130 MPa (8,0.53) Inner stiffeners: hs= hr=bs=br=0.004m Foundation: a) K1=1.5×107 N/m3, K2=1.5×105 N/m b) K1=2.5×107 N/m3, K2=2.5×105 N/m 0.5 1: k=0, (7,0.49) 2: k=0.5, (8,0.52 3: k=1, (8,0.52 4: k=5, (8,0.54) h=0.004 m, R/h=100 L/R=2, ΔT=300K 1000 τ (MPa) τ (MPa) 0 12 1200 h=0.004 m, R/h=100 L/R=2, ΔT=300K, k=1 200 100 10 (m =1, τ1upper=436.4531MPa, τ2upper=498.7536MPa, τ3upper=538.5641MPa) 700 300 Fig 9: Effects of foundation on τ − Wmax / h curves (m =1, τ1upper=421.7411MPa, τ2upper=561.8493MPa, τ3upper=498.7536MPa) 400 Wmax/h Fig 8: Effects of stiffeners on τ − Wmax / h curves 500 Wmax/h 600 τ1lower=368.5542 MPa (7,0.42) 2: K1=1.5*107; K2=1.5*105 350 h=0.004 m, R/h=100 L/R=2, ΔT=300K, k=1 400 h=0.004 m, R/h=100 L/R=2, ΔT=300K, k=1 350 300 τ3lower=440.8521 MPa (8,0.52) τ3upper (8,0.44) τ (MPa) τ (MPa) 500 Inner stiffeners: hs= hr=bs=br=0.004m ns=nr=15 τ3upper (8,0.46) 1.5 600 400 1: Stringer (ns=30) 2: Ring (nr=30) 3: Orthogonal (ns=nr=15) 800 200 0 2.5 φ (deg) Foundation: K1=1.5×107 N/m3 K2=1.5×105 N/m 0.5 Inner stiffeners: hs= hr=bs=br=0.004m ns=nr=15 1.5 2.5 3.5 φ (deg) Fig 10: Effects of foundation and stiffeners on τ − ϕ curves (m=1) Fig 11: Effects of k on τ − ϕ curves (m=1) Concluding remarks An analytical investigation to analyze the nonlinear buckling and post-buckling behavior of ES-FGM cylindrical shells under torsion subjected and surrounded by elastic foundations and in thermal environments based on the classical shell theory and the smeared stiffeners technique with geometrical nonlinearity in von Karman sense is presented in this paper Two cases of rings and stringers attached inner and outer of shell are analyzed The material properties of shell and stiffeners varying continuously in the thickness direction are taken into account The thermal elements of shell and stiffeners in equations of Nij and Mij are considered The obtained results show some remarks as: 18 Thermal elements of shell and stiffeners must be considered Expression of deflection with three-term including the linear and nonlinear buckling shape is more correctly chosen By using Galerkin method, the close-form expressions to determine critical buckling loads and nonlinear post-buckling torsional load-deflection curves are obtained Environment temperature affects considerably to the nonlinear static behavior of shell The preheated ES-FGM cylindrical shells exhibit a decreasing tendency in postbuckling loading carrying capacity when they are subjected to added action of torsional load Parameters of foundation, stiffener, volume fraction index k, radius-to-thickness ratio and length-to-radius ratio affect strongly on the critical torsion load and the postbuckling torsional load-deflection curves of ES-FGM cylindrical shells Acknowledgements This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2013.02 Appendix A In Eqs (9) and (10) C11 = E1s bs E2 s bs E1 E2 ν E1 ν E2 C C + , = , = + , C15 = , 12 14 2 ds ds −ν −ν −ν −ν C22 = E1 E1r br E2 E2 r br E1 ν E2 C C C + , = , = + , = , 24 25 33 dr dr (1 + ν ) −ν −ν −ν C36 = E3 E b E3 E b ν E3 E2 , C44 = + 3s s , C45 = , C55 = + 3r r , 2 ds dr +ν −ν −ν −ν C63 = E E2 , C66 = , +ν (1 + ν ) (A.1) in which E1 = Em h + Ecm h kEcm h , E2 = , k +1 2(k + 1)(k + 2) ⎡ Em h3 1 ⎤ + Ecm h3 ⎢ − + E3 = ⎥, 12 ⎣ 4(k + 1) k + k + ⎦ 19 φ1 = 1 −ν h/2 ∫ Eshα sh ΔTdz (A.2) −h / If ΔT = const then φ1 = E α + Ecmα m Ecmα cm ⎤ ⎡ + φ10 hΔT with φ10 = ⎢ Emα m + m cm k +1 2k + ⎥⎦ −ν ⎣ (A.3) Case 1: Inner stiffeners E1s = Ec hs + Emc hs , k2 + E2 s = ⎛ hs Ec ⎞ ⎛h ⎞ + hhs ⎜ s + 1⎟ + Emc hs h ⎜ ⎟, ⎝h ⎠ ⎝ k2 + h 2k2 + ⎠ E3s = ⎡ Ec ⎛ h2 h ⎞ h2 ⎤ h hs ⎜ + , + 1⎟ + Emc hs3 ⎢ + + 2⎥ ⎝ hs hs ⎠ ⎣ k2 + k2 + hs ( k2 + 1) hs ⎦ b φ = s ds T 1x h /2+ hs ∫ Esα s ΔTdz h /2 E1r = Ec hr + Emc hr , k3 + E2 r = ⎛ hr Ec ⎞ ⎛h ⎞ + hhr ⎜ r + 1⎟ + Emc hr h ⎜ ⎟, ⎝h ⎠ ⎝ k3 + h k + ⎠ E3r = ⎡ ⎞ Ec ⎛ h h h h2 ⎤ + + + + + , hr ⎜ E h ⎟ mc r ⎢ 2⎥ ⎝ hr2 hr k k h k + + + h ( ) ⎢ ⎥ 3 r r ⎦ ⎠ ⎣ φ1Ty = br dr h /2+ hr ∫ Erα r ΔTdz (A.4) h /2 If ΔT = const then φ1Tx = φ0 x hs ΔT with φ0 x = bs ds φ1Ty = φ0 y hr ΔT with φ0 y = ⎡ Ecα mc + Emcα c Emcα mc ⎤ + ⎢ Ecα c + ⎥, 2k + ⎦ k2 + ⎣ br dr ⎡ Ecα mc + Emcα c Emcα mc ⎤ + ⎢ Ecα c + ⎥ 2k3 + ⎦ k3 + ⎣ (A.5) Case 2: Outer stiffeners k k k k ⎛ 2z + h ⎞ ⎛ 2z + h ⎞ Es = Em + Ecm ⎜ − ⎟ , −h / − hs ≤ z ≤ −h / ⎟ , α s = α m + α cm ⎜ − ⎝ 2hs ⎠ ⎝ 2hs ⎠ ⎛ 2z + h ⎞ ⎛ 2z + h ⎞ Er = Em + Ecm ⎜ − ⎟ , −h / − hr ≤ z ≤ −h / ⎟ , α r = α m + α cm ⎜ − ⎝ 2hr ⎠ ⎝ 2hr ⎠ k = k3 = k , 20 E1s = Em hs + Ecm E3s = Em φ1Tx = bs ds ⎛ h2 h h + hs hs h ⎞ hs − Ecm ⎜ s + , E2 s = − Em s ⎟, k + k + 2 2 k2 + ⎝ ⎠ ⎛ h3 3hs h + 6hs h + 4hs h 2h h h2 ⎞ + Ecm ⎜ s + s + s ⎟, 12 k + k + k + 2 ⎝ ⎠ − h /2 ∫ Esα s ΔTdz , − h /2− hs E1r = Em hr + Ecm ⎛ hr hr h + hr hh ⎞ hr , E r = − Em − Ecm ⎜ + r ⎟, k3 + ⎝ k3 + 2 k3 + ⎠ ⎛ hr 3hr h + 6hr h + 4hr hr h hr h ⎞ E3r = Em + Ecm ⎜ + + ⎟, 12 ⎝ k3 + k3 + 4k3 + ⎠ b φ = r dr T 1y − h /2 ∫ Erα r ΔTdz (A.6) − h /2− hr if ΔT = const then φ1Tx = φ0 x hs ΔT with φ0 x = bs ⎛ Emα cm + Ecmα m Ecmα cm ⎞ + ⎜ Emα m + ⎟ 2k + ⎠ ds ⎝ k2 + φ1Ty = φ0 y hr ΔT with φ0 y = Emα cm + Ecmα m Ecmα cm ⎞ br ⎛ + ⎜ Emα m + ⎟ 2k3 + ⎠ dr ⎝ k3 + (A.7) in which the bs and br denote widths of stiffeners, respectively Also, d s and d r are the distances between two stringers and rings, respectively, and the eccentricities es and er represent the distance from the shell middle surface to the centroid of the stiffeners cross section (Fig 1b) Appendix B In Eqs (11) and (12) Δ = C22 C11 − C122 , C *22 = C22 / Δ, C12* = C12 / Δ, C14* = (C12 C24 − C22 C14 ) / Δ, C15* = (C12 C25 − C22 C15 ) / Δ, * C16 = (C22 − C12 ) / Δ, C11* = C11 / Δ, * C *24 = (C12 C14 − C11C24 ) / Δ, C *25 = (C12 C15 − C11C25 ) / Δ, C26 = (C11 − C12 ) / Δ, * C 33 = 1/ C33 , * C36 = C36 / C33 (B.1) 21 * * * D14 = C14 C *22 − C24 C12* , D44 = C44 + C24 C *24 + C14 C14* , D24 = C24 C11* − C14 C12* , ( ) * * * * * D45 = C14 C15* + C24 C *25 + C45 , D46 = C14 C16 + C24 C26 , D15 = C15C *22 − C25C12* , * * * D54 = C15C14* + C25C *24 + C45 , D25 = C25C11* − C15C12* , D55 = C15C15* + C25C *25 + C55 , ( ) * * * D56 = C15C16 + C25 C26 , * D*63 = C63C 33 , * * D66 = C66 − C63C 36 , (B.2) Appendix C In Eqs (22)-(24) G1 = G2 = G3 = G4 = G5 = G6 = G01 16β11α = A11ξ + A12ξ12 , G02 16 β ⎡⎣ β11λ + β12 λ + β13 ⎤⎦ 4 = A21ξ12 , G03 ⎡ ⎛α ⎤ ⎞ ⎛α ⎞ β ⎢ β11 ⎜ − λ ⎟ + β12 ⎜ − λ ⎟ + β13 ⎥ ⎢⎣ ⎝ β ⎠ ⎝β ⎠ ⎦⎥ G04 ⎡ ⎛α ⎤ ⎞ ⎛α ⎞ β ⎢ β11 ⎜ + λ ⎟ + β12 ⎜ + λ ⎟ + β13 ⎥ ⎢⎣ ⎝ β ⎠ ⎝β ⎠ ⎥⎦ = A31ξ1 + A32ξ1ξ , = A41ξ1 + A42ξ1ξ , G05 ⎡ ⎛ α ⎤ ⎞ ⎛ α ⎞ β ⎢ β11 ⎜ + λ ⎟ + β12 ⎜ + λ ⎟ + β13 ⎥ ⎢⎣ ⎝ β ⎠ ⎝ β ⎠ ⎦⎥ −G05 ⎡ ⎛ α ⎤ ⎞ ⎛ α ⎞ β ⎢ β11 ⎜ − λ ⎟ + β12 ⎜ − λ ⎟ + β13 ⎥ ⎢⎣ ⎝ β ⎠ ⎝ β ⎠ ⎥⎦ = A5ξ1ξ , = A6ξ1ξ , (C.1) in which A = α + β λ , A11 = β14α − 1/ R β2 α2 , , , A = = A 12 21 8β11α 32 β11α 32 β ⎡⎣ β11λ + β12 λ + β13 ⎤⎦ 1⎧ ⎫ ⎛1 ⎞ ⎡ ⎤ − ⎨− β14 ⎡ A2 + ( 2αβλ ) ⎤ + ⎜ − β15 β ⎟ A − β16 β ⎬ + αβλ ⎢ −2β14 A + − β15 β ⎥ ⎣ ⎦ ⎝R 2⎩ R ⎠ ⎣ ⎦ ⎭ A31 = , ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ α α β ⎢ β11 ⎜ − λ ⎟ + β12 ⎜ − λ ⎟ + β13 ⎥ ⎠ ⎝β ⎠ ⎢⎣ ⎝ β ⎥⎦ 22 α2 A32 = ⎡ ⎛α ⎤ ⎞ ⎛α ⎞ 2 β ⎢ β11 ⎜ − λ ⎟ + β12 ⎜ − λ ⎟ + β13 ⎥ β ⎠ ⎝β ⎠ ⎣⎢ ⎝ ⎦⎥ , ⎛1 ⎞ ⎡ ⎤ − β14 ⎡ A2 + ( 2αβλ ) ⎤ + ⎜ − β15 β ⎟ A − β16 β + 2αβλ ⎢ −2 β14 A + − β15 β ⎥ ⎣ ⎦ ⎝R R ⎠ ⎣ ⎦, A41 = ⎡ ⎤ ⎛α ⎞ ⎛α ⎞ 2β ⎢ β11 ⎜ + λ ⎟ + β12 ⎜ + λ ⎟ + β13 ⎥ ⎠ ⎝β ⎠ ⎢⎣ ⎝ β ⎥⎦ A42 = A5 = A6 = −α ⎡ ⎛α ⎤ ⎞ ⎛α ⎞ 2 β ⎢ β11 ⎜ + λ ⎟ + β12 ⎜ + λ ⎟ + β13 ⎥ β ⎠ ⎝β ⎠ ⎣⎢ ⎝ ⎦⎥ , α2 ⎡ ⎛ α ⎤ ⎞ ⎛ α ⎞ 2 β ⎢ β11 ⎜ + λ ⎟ + β12 ⎜ + λ ⎟ + β13 ⎥ ⎠ ⎝ β ⎠ ⎢⎣ ⎝ β ⎥⎦ −α ⎡ ⎛ α ⎤ ⎞ ⎛ α ⎞ 2 β ⎢ β11 ⎜ − λ ⎟ + β12 ⎜ − λ ⎟ + β13 ⎥ ⎠ ⎝ β ⎠ ⎢⎣ ⎝ β ⎥⎦ ( D1 = α11 ⎡ α + β 2λ ⎢⎣ ) ( , (C.2) ) + ( 2αβλ ) ⎤ + α12 β α + β 2λ + α13 β ⎥⎦ ⎡ ⎛α ⎤ ⎞ ⎛ ⎞ ⎞⎛ α λ α − A31β ⎢α14 ⎜ − λ ⎟ + ⎜ α15 − − + ⎟⎜ 16 ⎥ ⎟ Rβ ⎠ ⎝ β ⎢⎣ ⎝ β ⎥⎦ ⎠ ⎝ ⎠ 4 ⎡ ⎛α ⎤ ⎞ ⎛ ⎞ ⎞⎛ α λ α + A41β ⎢α14 ⎜ + λ ⎟ + ⎜ α15 − + + ⎟⎜ 16 ⎥ , ⎟ Rβ ⎠ ⎝ β ⎢⎣ ⎝ β ⎠ ⎝ ⎠ ⎦⎥ 4 ⎡ ⎛α ⎤ ⎞ ⎛ ⎞ ⎞⎛ α − + λ α D2 = − A32 β ⎢α14 ⎜ − λ ⎟ + ⎜ α15 − ⎟⎜ 16 ⎥ ⎟ Rβ ⎠ ⎝ β ⎢⎣ ⎝ β ⎥⎦ ⎠ ⎝ ⎠ 4 ⎡ ⎛α ⎤ ⎞ ⎛ ⎞ ⎞⎛ α +α β ( − A31 + A41 − A11 ) + A42 β ⎢α14 ⎜ + λ ⎟ + ⎜ α15 − + + λ α ⎟⎜ 16 ⎥ , ⎟ Rβ ⎠ ⎝ β ⎢⎣ ⎝ β ⎥⎦ ⎠ ⎝ ⎠ 2 D3 = −2 ( A21 + A12 )α β , D4 = ( − A32 + A42 − A5 + A6 )α β , ⎡ 1⎞ ⎤ ⎛ D5 = −8α ⎢ −2α11α + ⎜ 4α14α − ⎟ A11 ⎥ , R⎠ ⎦ ⎝ ⎣ 23 1⎞ ⎛ D6 = 8α A12 ⎜ 4α14α − ⎟ + ( A31 − A41 )α β , R⎠ ⎝ D7 = 2α β ( − A32 + A42 − A5 + A6 ) 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shells Mc Graw-Hill, New York; 1975 [37] Reddy JN Mechanics of laminated composite plates and shells: Theory and Analysis Boca Raton: CRS Press 2004 [38] Sofiyev AH, Halilov HM, Kuruoglu N Analytical solution of the dynamic behavior of non-homogenous orthotropic cylindrical shells on elastic foundations under moving loads Journal of Engineering Mathematics 2011;69:359-71 27 ... −C15* (18) Eqs (15) and (17) are the nonlinear governing equations used to investigate the nonlinear buckling and postbuckling behavior of eccentrically stiffened FGM cylindrical shells surrounded... investigate the nonlinear static and dynamic buckling and post -buckling of eccentrically ES -FGM plate, shallow shell and cylindrical shell based on the classical plate and shell theory With FGM. .. nonlinear buckling and post -buckling of ES -FGM cylindrical shell surrounded by elastic foundations and in thermal environment Eliminating ξ and ξ from Eqs (23), (24) and (26) and then solving τ with