Analytical investigation on mechanical buckling of FGM truncated conical shells reinforced by orthogonal stiffeners based on FSDT

Accepted Manuscript Analytical investigation on mechanical buckling of FGM truncated conical shells reinforced by orthogonal stiffeners based on FSDT Dao Van Dung, Do Quang Chan PII: DOI: Reference: S0263-8223(16)30744-9 http://dx.doi.org/10.1016/j.compstruct.2016.10.006 COST 7831 To appear in: Composite Structures Received Date: Revised Date: Accepted Date: 29 May 2016 24 September 2016 October 2016 Please cite this article as: Dung, D.V., Chan, D.Q., Analytical investigation on mechanical buckling of FGM truncated conical shells reinforced by orthogonal stiffeners based on FSDT, Composite Structures (2016), doi: http:// dx.doi.org/10.1016/j.compstruct.2016.10.006 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 Analytical investigation on mechanical buckling of FGM truncated conical shells reinforced by orthogonal stiffeners based on FSDT Dao Van Dung1, Do Quang Chan2* Vietnam National University, Hanoi, Vietnam University of Transport Technology, Hanoi, Vietnam * Corresponding author: Tel.: +84 983992081 E-mail address: quangchan82@gmail.com Abstract This work presents an analytical investigation for analyzing the mechanical buckling of truncated conical shells made of functionally graded materials, subjected to axial compressive load and external uniform pressure Shells are reinforced by closely spaced stringers and rings The change of spacing between stringers in the meridional direction also is taken into account Using the adjacent equilibrium criterion, the first order shear deformation theory (FSDT) and Lekhnitskii smeared stiffener technique, the linealization stability equations have been established The resulting equations which they are the system of five variable coefficient partial differential equations in terms of displacement components are investigated by Galerkin method The closedform expression for determining the critical buckling load is obtained The effects of material properties, dimensional parameters, stiffeners and semi-vertex angle on buckling behaviors of shell are considered Shown that for thick conical shells, the use of FSDT for determining their critical buckling load is necessary and more suitable Keywords: Functionally graded material (FGM); Stiffened truncated conical shell; Buckling; Critical buckling load; First order shear deformation theory Introduction Due to the high strength and thermal resistance, FGM conical shells were applied to many modern technique fields such as military aircraft propulsion system, and rocketry, underwater vehicles, missiles, tanks, pressure vessels, buildings of modern power plants and other applications [1] Therefore, the investigation on buckling characteristics of conical shells under combination of various loads is of great interest for engineering design and manufacture Sofiyev [2-6] investigated the linear stability and vibration of unstiffened FGM truncated conical shells with different boundary conditions The same author [7] presented the nonlinear buckling behavior and nonlinear vibration [8] of FGM truncated conical shells, and considered [9] the buckling of FGM truncated conical shells subjected to axial compressive load and resting on Winkler–Pasternak foundations Sofiyev and Kuruoglu [10] studied the nonlinear buckling behavior of FGM truncated conical shells surrounded by an elastic medium based on the classical shell theory and applying Galerkin method Sofiyev and Kuruoglu [11] investigated the buckling of functionally graded truncated conical shells subjected to external pressures under mixed boundary conditions The basic equations of functionally graded truncated conical shells are derived using Donnell shell theory, new approximation functions and using the Galerkin method Sofiyev and Kuruoglu [12] analyzed the dynamic instability of simply supported, functionally graded truncated conical shells under static and time dependent periodic axial loads Appling Galerkin’s method, the partial differential equations are reduced into a Mathieutype differential equation describing the dynamic instability behavior of the functionally graded conical shell The domains of principal instability are determined by using Bolotin’s method Sofiyev and Kuruoglu [13] are obtained a closed form of the solution for critical combined loads (combined effects of the axial load and lateral pressure or the axial load and hydrostatic pressure) of FGM truncated conical shell in the framework of the shear deformation theory (SDT) The basic equations of FGM truncated conical shell shells subjected to the combined loads are derived in the framework of the SDT By using the Galerkin method to basic equations are obtained the expressions for critical combined loads of FGM truncated conical shell in the framework of the SDT For linear analysis, the general characteristics in his works is that the modified Donnell-type equations are used and Galerkin method is applied to obtain closed-form relations of bifurcation type buckling load or to find expressions of fundamental frequencies, whereas for nonlinear analysis, the large deflection theory with von Karman–Donnell-type of kinetic nonlinearity is used Naj et al [14] based on the first-order shell theory studied the thermal and mechanical instability of FGM truncated conical shells is investigated Bich et al [15] presented results on the buckling of un-stiffened FGM conical panels under mechanical loads The linearized stability equations in terms of displacement components are derived by using the classical shell theory Galerkin method is applied to obtained the explicit expression of buckling load Malekzadeh and Heydarpour [16] investigated the influences of centrifugal and Coriolis forces in combination with the other geometrical and material parameters on the free vibration behavior of rotating FGM truncated conical shells subjected to different boundary conditions based on the first-order shear deformation theory Bagherizadeh et al [17] investigated the mechanical buckling of functionally graded material cylindrical shell that is embedded in an outer elastic medium and subjected to combined axial and radial compressive loads Theoretical formulations are presented based on a higher-order shear deformation shell theory (HSDT) considering the transverse shear strains Using the nonlinear strain–displacement relations of FGMs cylindrical shells, the governing equations are derived The elastic foundation is modelled by two parameters Pasternak model, which is obtained by adding a shear layer to the Winkler model Free vibration analysis of open conical panels made of through-the-thickness functionally graded materials (FGMs) is analyzed by Akbari et al [18] In this research, First order shear deformation theory of shells accompanied with the Donnell type of kinematic assumptions are used to establish the general motion equations and the associated boundary conditions Considering the Lévy type of conical shells, which are simply supported on straight edges, a semi-analytical solution based on the trigonometric expansion through the circumferential direction combined with generalized differential quadrature (GDQ) discretization in meridional direction is developed A linear buckling analysis for nanocomposite conical shells reinforced with single walled carbon nanotubes (SWCNTs) subjected to lateral pressure is presented by Jam et al [19] Material properties of functionally graded carbon nanotube reinforced composite (FG-CNTRC) conical shell are assumed to be graded across the thickness and are obtained based on the modified rule of mixture Governing equilibrium equations of the shell are obtained based on the Donnell shell theory assumptions consistent with the first order shear deformation shell theory General form of the equilibrium equations and the complete set of boundary conditions are obtained based on the concept of virtual displacement principle Weingarten [20] conducted a free vibration analysis for a ring-stiffened simply supported conical shell by considering an equivalent orthotropic shell and using Galerkin method He also carried out experimental investigations Crenwelge and Muster [21] applied an energy approach to find the resonant frequencies of simply supported ring-stiffened, and ring and stringer-stiffened conical shells Mustaffa and Ali [22] studied the free vibration characteristics of stiffened cylindrical and conical shells by applying structural symmetry techniques Some significant results on vibration of FGM conical shells, cylindrical shells and annular plate structures with a four parameter power-law distribution based on the first-order shear deformation theory are analyzed by Tornabene [23] and Tornabene et al [24] Tornabene and Viola [25] investigated static analysis of functionally graded doubly-curved shells and panels of revolution applying the Generalized Differential Quadrature Method Tornabene et al [26] studied stress and strain recovery for functionally graded free-form and doubly-curved sandwich shells using higher-order equivalent single layer theory Srinivasan and Krisnan [27] obtained the results on the dynamic response analysis of stiffened conical shell panels in which the effect of eccentricity is taken into account The integral equation for the space domain and mode superposition for the time domain are used in their work Based on the Donnell–Mushtari thin shell theory and the stiffeners smeared technique, Mecitoglu [28] studied the vibration characteristics of a stiffened truncated conical shell by the collocation method The minimum weight design of axially loaded simply supported stiffened conical shells with natural frequency constraints is considered by Rao and Reddy [29] The influence of placing the stiffeners inside as well as outside the conical shell on the optimum design is studied The expressions for the critical axial (buckling) load and natural frequency of vibration of conical shell also are derived Bagherizadeh et al [30] presented the thermal buckling analysis of FG cylindrical shell on a Pasternak-type elastic foundation In this study, the stability equations of the shell are decoupled to establish an equation in terms of only the out-of-plane displacement component Akbari et al [31] studied thermal buckling of temperature-dependent FGM conical shells with arbitrary edge supports Bifurcation behavior of heated conical shell made of a through-the-thickness functionally graded material is investigated in the present research Properties of the shell are obtained based on a power law form across the thickness Temperature dependency of the constituents is also taken into account The heat conduction equation of the shell is solved based on an iterative generalized differential quadrature method (GDQM) General nonlinear equilibrium equations and the associated boundary conditions are obtained using the virtual displacement principle in the Donnell sense Mirzaei M and Kiani Y [32] studied thermal buckling of temperature dependent FG-CNT reinforced composite conical shells In this research, linear thermal buckling of a composite conical shell made from a polymeric matrix and reinforced with carbon nanotube fibres is investigated Distribution of reinforcements across the shell thickness is assumed to be uniform or functionally graded Thermomechanical properties of the constituents are temperature dependent Under the assumption of first order shear deformation shell theory, Donnell kinematic assumptions and von-Karman type of geometrical nonlinearity, the complete set of equilibrium equations and boundary conditions of the shell are obtained A linear membrane analysis is carried out to obtain the pre-buckling thermal stresses of the shell Adjacent equilibrium criterion is implemented to establish the stability equations associated with the buckling state Sabzikar Boroujerdy [33] based on the Donnell theory of shells combined with the von-Karman type of geometrical nonlinearity, three coupled equilibrium equations for a through-the-thickness functionally graded cylindrical shell embedded in a two parameter Pasternak elastic foundation are obtained Equivalent properties of the shell are obtained based on the Voigt rule of mixture in terms of a power law volume fraction for the constituents Properties of the constituents are considered to be temperature dependent The temperature profile through the shell thickness is obtained by means of the central finite difference method Linear prebuckling analysis is performed to obtain the prebuckling forces of the cylindrical shell Stability equations are derived based on the well-known adjacent equilibrium criterion Three coupled partial differential stability equations are solved with the aid of a hybrid Fourier-GDQ method Thermal bifurcation behavior of cross-ply laminated composite cylindrical shells embedded with shape memory alloy fibers is investigated by Asadi et al [34] Properties of the constituents are assumed to be temperature-dependent Donnell's kinematic assumptions accompanied with the vonKarman type of geometrical non-linearity are used to derive the governing equations of the shell Furthermore, the one-dimensional constitutive law of Brinson is used to predict the behavior of shape memory alloy fibers through the heating process Governing equilibrium equations are established by employing the static version of virtual displacements principle Castro et al [35] studied linear buckling predictions of unstiffened laminated composite cylinders and cones under various loading and boundary conditions using semi-analytical models Semi-analytical models for the linear buckling analysis of unstiffened laminated composite cylinders and cones with flexible boundary conditions are presented The Classical Laminated Plate Theory and the First-order Shear Deformation Theory are used in conjunction with the Donnell’s non-linear equations to derive the buckling equations Castro et al [36] proposed semi-analytical model for the non-linear analysis of simply supported, unstiffened laminated composite cylinders and cones using the Ritz method and the Classical Laminated Plate Theory A matrix notation is used to formulate the problem using Donnell's and Sanders' non-linear equations The approximation functions proposed are capable to simulate the elephant's foot effect, a common phenomenon and a common failure mode for cylindrical and conical structures under axial compression Castro et al [37] presented semi-analytical model to predict the non-linear behavior of unstiffened cylinders and cones considering initial geometric imperfections and various loads and boundary conditions is presented The formulation is developed using the Classical Laminated Plate Theory (CLPT) and Donnell’s equations, solving for the complete displacement field The non-linear static problem is solved using a modified Newton– Raphson algorithm with line-search Khakimova et al [38] investigated the buckling experiments on axially compressed, unstiffened carbon fiber–reinforced polymer (CFRP) truncated cones with an additional lateral load, performed by DLR for validation of the Single Perturbation Load Approach (SPLA) applied to this type of structure Three geometrically identical cones with different layup were designed, manufactured and tested As can be seen that the above introduced works mainly related to unstiffened FGM structures However, in practice, plates and shells including conical shells, usually reinforced by stiffeners system to provide the benefit of added load carrying capability with a relatively small additional weight Thus, the study on static and dynamic behavior of theses structures are significant practical problem In 2009, Najafizadeh et al [39], with the linearized stability equations in terms of displacements studied buckling of FGM cylindrical shell reinforced by rings and stringers 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 Following the direction of FGM stiffeners, Dung and Hoa [40-43] obtained the results on the static and dynamic nonlinear buckling and post-buckling analysis of eccentrically stiffened FGM circular cylindrical shells under external pressure and torsional loads The material properties of shell and stiffeners are assumed to be continuously graded in the thickness direction Galerkin method was used to obtain closed-form expressions to determine critical buckling loads By considering homogenous stiffeners, Bich et al [44, 45] presented an analytical approach to analyze the nonlinear vibration dynamic buckling of eccentrically stiffened imperfect FGM panels and doubly curved thin shallow shells based on the classical shell theory The nonlinear critical dynamic buckling load is found according to the Budiansky-Roth criterion Dung and Nam [46] studied the nonlinear dynamic behavior of eccentrically stiffened functionally graded circular cylindrical thin shells under external pressure and surrounded by an elastic medium by semi-analitycal approach with the deflection function chosen by three-term For eccentrically stiffened FGM conical shells, studies on their buckling and vibration still are limited and should be further studied This may be attributed to the inherent complexity of governing equations of conical shell Those are variable coefficient partial differential equations are limitted Recently, in 2013, Dung et al [47] studied linear buckling of FGM thin truncated conical shells reinforced by homogeneous eccentrical stringers and rings subjected to axial compressive load and uniform external pressure load based on the smeared stiffeners technique and the classical shell theory The new contribution of that paper is the investigation by analytical method on the buckling behavior of shell taking into account the change of distance between stringers in the meridional direction The important highlight is that the authors used the smeared stiffeners technique for establishing correctly the general formula for force and moment resultants of eccentrically stiffened FGM (ES-FGM) truncated conical shells Developing the method in [47], Dung et al [48] obtained results on linear buckling of FGM thin truncated conical shells reinforced by FGM eccentrical stringers and rings resting elastic foundations and subjected to axial compressive load and uniform external pressure Following this direction, Duc et al [49] investigated static linear mechanical and thermal stability of eccentrically stiffened FGM conical shell panels under mechanical and thermal loads on elastic foundations based on the classical shell theory As can be observed that the studies in [47-49] were carried out by using the classical shell theory, so obtained results only are suibtable for thin-walled conical shells However for thicker conical shells, it is necessary to use higher order theories Recently there are some investigations on buckling of truncated conical shells using the first order shear deformation theory [6,16,18,19,35], but these structures are unstiffened conical shells The new contribution of this work is to use the first-order shear deformation theory (FSDT) for investigating the mechanical buckling of FGM thick truncated conical shells reinforced by stringers and rings and subjected to axial compressive load and uniform external pressure load The change of spacing between stringers in the meridional direction is taken into account The general formula for force and moment resultants of eccentrically stiffened FGM (ES-FGM) truncated conical shells are established correctly by the Lekhnitskii smeared stiffeners technique Using the adjacent equilibrium criterion, the linearization stability equations in terms of displacement components are established These couple set of five variable coefficient partial differential equations are investigated by Galerkin method The closed-form expression for determining the static critical buckling load is obtained The influences of various parameters such as stiffener, dimensional parameters and volume fraction index of materials on the stability of shell are clarified in detail Functionally graded material truncated conical shell Assume that a truncated conical shell is made from a mixture of a ceramic and a metal (denoted by c and m, respectively) and the material compositions only vary smoothly along its thickness direction with the power law distribution Then the elasticity modulus and the Poisson ratio of a functionally graded material can be evaluated as following k  2z + h  E ( z ) = EmVm + EcVc = Em + ( Ec − Em )   , ν ( z ) = ν = const  2h  (1) where Vc , Vm being volume-fractions of the ceramic and metal phases respectively, which k  2z + h  are related by Vc + Vm = and Vc is expressed as Vc ( z ) =   , k ≥ is the volume 2h  fraction index Theoretical formulation of FGM truncated conical shell Consider a truncated conical shell of thickness h and semivertex angle a The geometry of shell is shown in Fig 1, where L is the length and R is its small base radius The truncated cone is referred to a curvilinear coordinate system (x, θ, z) whose the origin is located in the middle surface of the shell, x is in the generatrix direction measured from the vertex of conical shell, θ is in the circumferential direction and the axes z being perpendicular to the axes x, lies in the outwards normal direction of the cone Also, x0 indicates the distance from the vertex to small base of the shell Further, assume that the conical shell is reinforced by closely spaced homogeneous longitudinal stringers and rings To guarantee the continuity between the stiffener and shell, the stiffener is taken to be pure-metal if it is located at metal-rich shell side and is pureceramic if it is located at ceramic-rich shell side z O x0 α h x z2 d2 R b1 b2 L h1 h d2 d1 z1 d1 θ x h2 Stringer Ring Fig Geometry of eccentrically stiffened truncated conical shell Based on the Timoshenko-Mindlin assumption, the displacements at distance z from the middle surface of the shell, are represented in the form ux = u ( x,θ ) + zφx ( x,θ ) uθ = v( x,θ ) + zφθ ( x,θ ) , (2) uz = w( x,θ ) where u, v, and w denote the displacement components of a point in the middle surface in the direction x, θ and z; and φx , φθ are the rotations of a transverse normal about the θ and x – axes, respectively The strain-displacement relationship at the middle surface of the shell based on the first-order shear deformatin theory taking into account the gemetrical nonlinearity is given by [52, 53] ε xm = u, x + w,2x , u w v,θ + + cot α + 2 w,2θ , x sin α x x x sin α εθ m = γ xθ m = v u,θ − + v, x + w, x w,θ , x sin α x x sin α (3) γ xzm = w, x + φx , γ θ zm = w,θ + φθ , x sin α k x = φx , x , kθ = 1 1 1  φθ ,θ + φx , k xθ = φθ , x + φx ,θ − φθ  , x sin α x 2 x sin α x  (4) where ε xm and ε θ m are the normal strains and γ xθ m is the shear strain at the middle surface of the shell, and γ xzm , γ θ zm are the transverse shear strains; and k x , kθ and k xθ are the change of curvatures and twist, respectively They are related to the displacement components as [50, 52, 53] The normal and shear strains at distance z from the middle surface of shell are ε x = ε xm + zk x , ε θ = ε θ m + zkθ , γ xθ = γ xθ m + zk xθ , γ xz = γ xzm , (5) γ θ z = γ θ zm Hooke law for the FGM conical shell is defined as σ xsh = σ xshθ = E (z) E ( z) ε +νεθ ) , σ θsh = ( εθ +νε x ) , ( x −ν −ν E ( z) (1 + ν ) γ xθ , σ xzsh = E ( z) (1 + ν ) γ xzm , σ θshz = (6) E ( z) (1 + ν ) γ θ zm , and for stiffeners σ xst = Esε x , σ θst = Er εθ , (7) where the subscripts sh and st denote shell and stiffeners, respectively; Es and Er are Young elasticity modulus of stiffener in the x-direction and θ -direction, respectively The force resultants, moment resultants and transverse force resultants of an eccentrically stiffened FGM conical shell determined by h h N i = ∫ σ ish dz + N ist , M i = ∫ zσ ish dz + M ist , (i=x, θ) − h − h h h − h − h N xθ = ∫ σ xshθ dz , M xθ = ∫ xσ xshθ dz , h Qx = (8) h 5 σ xzsh dz , Qθ = ∫σ θshz dz , ∫ h h − − where N ist , M ist are force resultants and moment resultants of stiffeners, respectivelly Substituting Eqs (3-7) into Eq (8) and using Lekhnitskii smeared stiffener technique, and integrating the above stress–strain equations and their moments through the thickness of the shell, the expressions for force and moment resultants, and transverse force resultants of an eccentrically stiffened FGM conical shell are given by  EA  N x =  A11 + s  ε xm + A12ε θ m +  B11 + c1 ( x )  k x + B12 kθ , d1 ( x )    EA  Nθ = A12ε xm +  A22 + r  ε θ m + B12 k x + ( B22 + c2 ) kθ , d2   (9) N xθ = A66γ xθ m + B66 k xθ ,  EI  M x =  B11 + c1 ( x )  ε xm + B12ε θ m +  D11 + s  k x + D12 kθ , d1 ( x )    EI  M θ = B12ε xm + ( B22 + c2 ) ε θ m + D12 k x +  D22 + r  kθ , d2   (10) M xθ = B66γ xθ m + D66 k xθ , Qx = A44γ xzm = A44 ( w, x + φx ) ,   Qθ = A55γ θ zm = A55  w,θ + φθ  ,  x sin α  (11) where the coefficients Ei , c1 ( x ), c2 , d1 ( x), d , I1 , I , N s , N r , Aij , Bij , Dij are defined in Appendix A Using the first order shear deformation theory, the nonlinear equilibrium equations of truncated conical shells are given as follows [52, 53] xN x , x + N xθ ,θ + N x − Nθ = 0, sin α Nθ ,θ + xN xθ , x + N xθ = 0, sin α xM x , xx + 2M x , x + 2 M xθ , xθ + M xθ ,θ + M θ ,θθ − M θ , x − Nθ cot α sin α x sin α x sin α 1   +  xN x w, x + N xθ w,θ  + sin α   , x sin α    N xθ w, x + x sin α Nθ w,θ  = qx, ,θ (12) ( x sin α M x ),x + M xθ ,θ − Mθ sin α − x sin α Qx = 0, ( x sin α M xθ ), x + Mθ ,θ + M xθ sin α − x sin α Qθ = As can be seen in Eq (12) there are prebuckling force resultants So it is necessary to find these forces For this aim, assume that a shell subjected to the axial compressive load of intensity P = p + qx0 sin α (N) at x = x0 and external uniform pressure q (Pa) (Fig 2) Thus, the prebuckling force resultants of the shell are defined from the membrane form equilibrium equation (12), as N xθ = 0, x dN x + N x − Nθ = 0, dx − Nθ cot α = qx P R q θ Fig The mechanical buckling of eccentrically stiffened functionally graded materials (ES-FGM) Solving this system with the boundary condition N x0 = − P p =− − qx0 tan α at x = x0 , cos α cos α Fig Effects of index volume k on critical load Pcr (q=0) Fig Effects of index volume k on critical load qcr (P=0) 7.7 Comparisons on critical buckling loads based on FSDT and CST This section presents comparisons on critical buckling loads based on FSDT and CST with the database as 22 Em = 70.109 N / m ; Ec = 380.109 N / m ; k = 1; R = 4m; L = × R; α = π ; h1 = 0.02m ; b1 = 0.015m ; h2 = 0.02m; b2 = 0.015m Table 10 gives results on the critical buckling load Pcr (q=0) when the ratio R/h varies from to 200 Table 11 gives results on the critical buckling pressure qcr (P=0) when the ratio R/h varies from to 200 Table 10 Comparisons on critical axial compressive load Pcr for ES-FGM truncated conical shells based on CST and FSDT when the ratio R/h varies Stringer(ns = 30) R/h Pcr (CST) Pcr (FSDT) Ring(nr = 30) Pcr (CST) Pcr (FSDT) Orthogonal(ns = nr = 15) Pcr (CST) Pcr (FSDT) % 385430.00 367240.00 385410.00 367210.00 385420.00 367220.00 4.96 (2,1)* (2,1) (2,1) (2,1) (2,1) (2,1) 10 95552.00 94127.00 95533.00 94108.00 95542.00 94117.00 (2,6) (2,3) (2,6) (2,3) (2,6) (2,3) 23778.00 23585.00 23765.00 23573.00 23772.00 23579.00 (3,8) (3,4) (3,8) (3,4) (3,8) (3,4) 3788.10 3786.70 3786.20 3779.80 3787.10 3783.30 (4,15) (5,6) (4,15) (5,6) (4,15) (5,6) 947.22 945.94 946.39 944.92 947.39 946.36 (5,22) (5,11) (6,21) (7,9) (6,21) (6,10) 237.58 237.57 236.77 236.67 238.35 238.15 20 50 100 200 1.51 0.82 0.10 0.10 0.09 (6,31) (6,16) (10,25) (10,13) (9,28) (6,14) * Buckling mode (m,n) As can be seen that for ES-FGM thin truncated conical shells, the results of critical buckling load using the first order shear deformation theory and using the classical shell theory, have close values for the thin shells However, those values are considerable difference for the thicker shells For example, with R/h=200 and orthogonal stiffeners, the value Pcr = 238.35 (based on CST) bigger than the value Pcr = 238.15 (based on FSDT) about 0.09%, But for thicker shells, results have a big difference, while for thick shell R/h=5, Pcr = 385420.00 (CST, orthogonal) bigger than Pcr = 367220.00 (FSDT, orthogonal) about 4.96% Therefore, the use of FSDT for thicker conical shells is necessary and more suitable 23 Table 11 Comparisons on critical load qcr for ES-FGM truncated conical shells based on the CST and FSDT when the ratio R/h varies Stringer(ns = 30) R/h 10 20 50 qcr (CST) 8.24E+05 (1,7)* 1.42E+05 (1,9) 24576.00 (1,11) 2460.90 qcr (FSDT) 8.13E+05 (1,3) 1.40E+05 (1,4) 24545.00 (1,5) 2457.20 Ring(nr = 30) Orthogonal(ns = nr = 15) qcr (CST) qcr (FSDT) qcr (CST) qcr (FSDT) 8.25E+05 (1,7) 1.42E+05 (1,8) 8.13E+05 (1,3) 1.40E+05 (1,4) 8.26E+05 (1,7) 1.42E+05 (1,9) 8.13E+05 (1,3) 1.40E+05 (1,4) 1.59 24723.00 (1,11) 24660.00 (1,5) 24649.00 (1,11) 24602.00 (1,5) 0.19 2512.70 2508.20 2486.80 2482.50 0.17 % 1.51 (1,14) (1,5) (1,14) (1,7) (1,14) (1,7) 100 445.450 442.448 468.806 466.783 455.184 454.665 0.11 200 (1,17) 78.467 (1,20) (1,8) 78.452 (1,10) (1,17) 94.914 (1,20) (1,8) 94.889 (1,10) (1,17) 86.752 (1,20) (1,8) 86.733 (1,10) 0.02 * Buckling mode (m,n) Table 12 and Fig show comparisons on critical buckling loads Pcr (q=0) and Fig 10 show comparisons on critical uniform pressure qcr (P=0) using FSDT and CST when the index volume k varies As can be seen that results obtained here also describe the same characteristics as results in Tables 10 and 11, that is the clear distinction on critical load between thin and thick shells Table 13 and Fig 11 present comparisons on critical buckling loads based on CST and FSDT when the semi-vertex angle varies The results obtained once again confirme that, for thicker conical shells, a critical buckling load should be determined according to FSDT Table 12 Comparisons on critical axial compressive load Pcr (q=0) based on CST and FSDT when the index volume k varies Stringer(ns = 30) Ring(nr = 30) Orthogonal(ns = nr = 15) k Pcr Pcr Pcr Pcr Pcr Pcr % % % (CST) (FSDT) (CST) (FSDT) (CST) (FSDT) 1743.80 1742.00 0.10 1741.50 1740.00 0.09 1742.70 1741.00 0.10 (6,19)* (6,10) (6,19) (6,10) (6,19) (6,10) 0.5 1198.60 1196.70 0.16 1196.20 1194.30 0.16 1197.40 1195.50 0.16 (6,20) (6,10) (6,20) (6,10) (6,20) (6,10) 947.222 945.942 0.13 946.388 944.918 0.16 947.392 946.358 0.11 (5,22) (5,11) (6,21) (7,9) (6,21) (6,10) 566.581 566.422 0.03 565.031 565.024 0.01 566.686 566.444 0.04 (4,21) (5,10) (6,19) (6,9) (6,19) (5,10) * Buckling mode (m,n) 24 Fig Comparisons critical axial compressive load Pcr (q=0) based on CST and FSDT when the index volume k varies Fig 10 Comparisons on critical uniform pressure qcr (P=0) based on CST and FSDT when the index volume k varies 25 Table 13 Comparisons critical axial compressive load Pcr based on CST and FSDT when the semivertex angle α varies Stringer(ns = 30) Ring(nr = 30) Orthogonal(ns = nr = 15) Α 15o 30o 45o 60o Pcr (CST) Pcr (FSDT) % Pcr (CST) Pcr (FSDT) % Pcr (CST) Pcr (FSDT) % 1176.200 1175.900 0.03 1174.300 1171.400 0.25 1176.800 1174.100 0.23 (5,21)* (7,10) (8,18) (8,9) (7,20) (7,10) 947.222 945.942 946.388 944.918 947.392 946.358 (5,22) (5,11) (6,21) (7,9) (6,21) (6,10) 632.749 632.048 631.620 630.922 632.186 631.487 (5,20) (5,10) (5,20) (5,10) (5,20) (5,10) 316.942 316.710 316.554 316.322 316.749 316.517 (4,18) (4,9) (4,18) (4,9) (4,18) (4,9) 0.14 0.11 0.07 0.16 0.11 0.07 0.11 0.11 0.07 * Buckling mode (m,n) Fig 11 Comparisons on critical axial compressive load Pcr based on CST and FSDT when the semi-vertex angle α varies Conclusions An analytical approach on the mechanical stability of S-FGM thick truncated shell subjected to axial compressive load and uniform pressure is proposed based FSDT The 26 shell is reinforced by stringers and rings in which the change of spacing between stringers in the meridional direction also is taken into account The variable coefficient partial differential equations system is investigated by Galerkin method The closed-form expression for determing the critical buckling load of the shell is obtained The present results shown that reinforcement stiffener, volume fraction index, geometrical parameters and semi-vertex angle affect considerably on the stability of shell The reliability and accuracy of the proposed method is affirmed The numerical calculations pointed out that for thin conical shells, with the input same parameters, the difference between the critical buckling load based on FSDT and CST, has a little However for thicker conical shells, that difference is considerable Therefore, the use of FSDT for determining the critical buckling load of thicker conical shells is necessary and more suitable Acknowledgement This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number No 107.02-2015.11 Appendix A  E − Em  E1 = Em +  c  h,  k +1  ( Ec − Em ) kh2 , E2 = ( k + 1)( k + ) E 1   E3 =  + ( Ec − Em )  − + h ,  k + k + 4k +   12 h + h1 h + h2 2π sinα , A1 = b1h1 , A2 = b2 h2 , z1 = , z2 = , λ0 = 2 ns d1 ( x ) = λ0 x, d = c0 EA z L EAz , c1 = ± s 1 , c1 ( x ) = , c2 = ± r 2 , nr λ0 x d2 b1h13 b h3 + A1 z12 , I = 2 + A2 z22 , 12 12 ν E1 E1 5E1 E1 , A12 = , A44 = A55 = , A66 = , A11 = A22 = 2 −ν −ν 12 (1 + ν ) (1 + ν ) I1 = ν E2 E2 E2 , B12 = , B66 = , 2 −ν −ν (1 + ν ) ν E3 E3 E3 , D12 = , D66 = , D11 = D22 = 2 −ν −ν (1 + ν ) where ns, nr are the number of stringer and ring respectively; h1 and b1 are the thickness and width of stringer (x-direction); h2 and b2 are the thickness and width of ring (θ-direction) Also, d1 = d1 ( x) and d2 are the distance between two stringers and two rings, respectively The quantities A1, A2 are the cross-section areas of stiffeners and I1, I2 are the second moments of inertia of the stiffener cross sections relative to the shell middle surface; and z1, z2 represent the eccentricities of stiffeners with respect to the middle surface of shell B11 = B22 = 27 Appendix B  E A  ∂2 ∂2 ∂  E A 1 R11 =  A11 x + s  + A + A11 −  A22 + r  , 66 2 d2  x λ0  ∂x x sin α ∂θ ∂x   ∂2 1  Er A2  ∂ , R12 = − ( A12 + A66 )  A22 + A66 +  ∂x∂θ x sin α  sin α d  ∂θ ∂ 1 EA  − cot α  A22 + r  , ∂x x d2  1 ∂ ∂2 ∂ R14 = ( B11 x + c1 ) + B66 + B11 − ( B22 + c2 ) , x sin α x ∂x ∂θ ∂x ∂ ∂ 1 R15 = − ( B12 + B66 ) ( B22 + B66 + c2 ) , sin α ∂x∂θ x sin α ∂θ ∂ 1  Er A2  ∂ , R21 = + ( A12 + A66 )  A22 + A66 +  ∂x∂θ x sin α  sin α d  ∂θ R13 = cot α A12  Er A2  ∂ ∂ A +  22  + A66 − A66 , d  ∂θ x ∂x  ∂ EA , R23 = ( A22 + r ) cot α d2 x sin α ∂θ R22 = A66 x ∂2 + x sin α ∂x ∂2 ∂ 1 + ( B12 + B66 ) ( B22 + B66 + c2 ) , sin α ∂x∂θ x sin α ∂θ 2 1 ∂ ∂ ∂ R25 = B66 x + ( B22 + c2 ) + B66 − B66 , x sin α x ∂x ∂θ ∂x 3 ∂ ∂ ∂2 R31 = ( B11 x + c10 ) + B + B + B ( 12 66 ) 11 ∂x x sin α ∂x∂θ ∂x ∂2  1  ∂ + 2 ( B22 + c2 ) −  A12 cot α + ( B22 + c2 )  x sin α ∂θ x  ∂x   EA    −   A22 + r  cot α − ( B22 + c2 )  , d2  x  x   R24 = ∂3 ∂3 ∂2 1 + B + c − B + c ( 22 ) ( 22 ) sin α ∂x 2∂θ x sin α ∂θ x sin α ∂x∂θ   ∂ EA   , −   A22 + r  cot α − ( B22 + c2 )  d2  x  x sin α ∂θ   R32 = ( B12 + B66 ) ∂2 ∂2 1 ∂ + B + c cot α − ( B22 + c2 ) cot α ( ) 22 2 2 ∂x x sin α ∂θ x ∂x    EA  −   A22 + r  cot α − ( B22 + c2 )  cot α , d2  x  x   R33 = B12 cot α  ∂3 ∂2 E I  ∂3 R34 =  D11 x + s  + D + D + D ( 66 ) 11 ∂x∂θ ∂x λ0  ∂x x sin α 12  28  ∂2 EI  +  D22 + r  2 − d  x sin α ∂θ   EI  − ( B22 + c2 ) cot α −  D22 + r d2    Er I )  B12 cot α + ( D22 + d2 x      ,  x  x  ∂   ∂x  ∂3 ∂3 Er I  R35 = ( D12 + D66 ) +  D22 +  sin α ∂x 2∂θ  d  x sin α ∂θ   ∂2 EI  EI −  D22 + r  − ( B22 + c2 ) cot α − (D22 + r ) d  x sin α ∂x∂θ d2 x    ∂ ,   x sin α ∂θ 1 ∂2 ∂2 ∂  , R36 = −  x tan α + + x tan α ∂x sin 2α ∂θ ∂x  2 x0 ∂ , R37 = − cos α ∂x 1 ∂2 ∂2 ∂ R41 = ( B11 x + c10 ) sin α + B66 + B11 sin α − ( B22 + c2 ) sin α , x sin α ∂θ x ∂x ∂x ∂ ∂ , R42 = ( B12 + B66 ) − ( B22 + B66 + c2 ) ∂x∂θ x ∂θ ∂ R43 = ( B12 cos α − A44 x sin α ) − ( B22 + c2 ) cos α , x ∂x 2  ∂ ∂ ∂ EI  R44 =  D11 x + s  sin α + D66 + D11 sin α ∂x x sin α ∂θ ∂x λ0     E I   −  A44 x +  D22 + r   sin α , d  x     ∂2 E I 1 ∂ , R45 = ( D12 + D66 ) −  D22 + D66 + r  ∂x∂θ  d  x ∂θ ∂2 ∂ , R51 = ( B12 + B66 ) + ( B22 + B66 + c2 ) ∂x∂θ x ∂θ 1 ∂2 ∂2 ∂ R52 = B66 x sin α + ( B22 + c2 ) + B66 sin α − B66 sin α , x sin α x ∂x ∂θ ∂x  ∂  , R53 = −  A55 − ( B22 + c2 ) cot α  x  ∂θ   ∂2 E I 1 ∂ , R54 = ( D12 + D66 ) +  D22 + D66 + r  ∂x∂θ  d  x ∂θ ∂2  EI  ∂2 ∂   R55 = D66 x sin α +  D22 + r  + D66 sin α −  A55 x + D66  sin α ∂x  d  x sin α ∂θ ∂x  x  29 Appendix C  ( x0 + L ) − x04 3L3 ( x0 + L ) m 2π X 11 = − A11 sin α  + L 8m 2π      n2 EA  L ( x0 + L ) sin α  A22 + A66 + r  sin α d2    ( x0 + L )3 − x03  π m 2π Es A1 L3 sin α  − +  + L ( x0 + L ) A11 sin α , λ0 L 4m 2π    ( x0 + L )3 − x03  nL2  Er A2  mnπ L3 X 12 = − + − ( A12 + A66 )   A22 + A66 + , 2 L 4m π  4m  d2   − π  ( x0 + L )3 − x03 mπ L3 cot α sin α A12  X13 = + L 4m 2π   Er A2  L2   + cot α sin α  A22 + , 4m  d2    ( x0 + L )4 − x04 3L3 ( x0 + L )  m 2π X 14 = − B11 sin α  +  L 8m2π   n2 B66 + c2 ) sin α  ( x0 + L )3 − x03  π m 2π L3 − c1 sin α  +  + L ( x0 + L ) B11 sin α , 2 L 4m π   − π L ( x0 + L ) sin α ( B22 + mnπ ( x0 + L )( B12 + B66 ) , sin α  ( x0 + L )3 − x03 mnπ L3 X 21 = − − ( A12 + A66 )  L 4m 2π  X15 = − X 22  nL2  Er A2  −  A22 + A66 + , d2   4m   ( x0 + L ) − x04 3L3 ( x0 + L )  π m 2π = − A66 sin α  −  − L ( x0 + L ) A66 sin α L 8m 2π     Er A2   A22 +  + A66  , d2     nπ EA  = L ( x0 + L ) cot α  A22 + r  , d2   − X 23 X 24  n L ( x0 + L ) sin α   sin α π  ( x0 + L )3 − x03 mnπ L3 =− − ( B12 + B66 )  L 4m 2π   nL2 − ( B22 + B66 + c2 ) ,  4m  ( x0 + L )3 − x03  n 2π L m 2π L3 π X 25 = − B66  − − ( B22 + c2 ) + LB66 , 2 L 4m π  2sin α      m3π 3L5   ( x0 + L ) − x0 L  x0 − ( x0 + L )  X 31 = B11 sin α  + +  L 10 2m 2π 4m 4π    30  m3π   ( x0 + L ) − x04 3L3 ( x0 + L )  mπ +  c1 sin α + cot α sin α A12   −  L 8m 2π   L    mπ   ( x0 + L ) − x03 mn 2π L3 sin α ( 3B11 + B22 + c2 ) + + − ( B12 + B66 )   L sin α 4m 2π  L     n2  E A  L ( x0 + L ) L2 sin α ( B22 + c2 )  −  , + cot α sin α  A22 + r  + d2  4m 4m  sin α   X 32  ( x0 + L )4 − x04 3L3 ( x0 + L ) m nπ = − ( B12 + B66 )  L2 8m 2π        3   Er A2   ( x0 + L ) − x0 L3 + nπ cot α  A22 + −   d   4m 2π   nπ L ( x0 + L )  n2  + ( B22 + c2 )  −  ,  sin α   ( x0 + L ) − x04 3L3 ( x0 + L ) m 2π X 33 = − cot α sin α B12  − L 8m2π  3  Er A2   ( x0 + L ) − x0 L3 −π cot α sin α  A22 + −   d   4m 2π   n2  π − L ( x0 + L ) cot α sin α ( B22 + c2 )  −  ,  sin α  X 34           m3π 3L5   ( x0 + L ) − x0 L  x0 − ( x0 + L )  = D11 sin α  + +  L 10 2m 2π 4m 4π     m3π Es I1   ( x + L ) − x04 3L3 ( x0 + L )  mπ sin α + cot α sin α B12   + −  λ0 L 8m 2π   L    mπ   ( x0 + L ) − x03  Er I  mn 2π L3 sin α 3D11 + D22 + + − ( D12 + D66 )   + d  L sin α 4m 2π   L   L2 ( x0 + L )   L2 E I   n2 cot α sin α ( B22 + c2 ) + sin α  D22 + r   −  , + 4m 4m d   sin α    ( x0 + L )3 − x03 m nπ L3 X 35 = D + D −  ( ) 12 66 L2 sin α 4m 2π     nπ nπ L ( x0 + L ) cot α ( B22 + c2 ) − L ( D12 + D66 ) 4sin α sin α   Er I   n EI nπ nπ + L  D22 + L(2 D12 + D22 + D66 + r ),  −2 − 2sin α  d   sin α d2  sin α + 31    X 36 X 37    m 2π  ( x0 + L ) − x0 5L  x0 − ( x0 + L )  15 L ( x0 + L ) tan α sin α  = + + L2 12 8m 2π 8m 4π   2n 2π   ( x0 + L ) − x04 3L3 ( x0 + L )  sin α + 2π tan α sin α   + − , 8m 2π  sin 2α     ( x0 + L ) − x04 3L3 ( x0 + L ) m2π = x0 tan α  − L 8m 2π       ,   ( x0 + L ) − x04 3L3 ( x0 + L )  m 2π X 41 = − B11 sin α  +  L 8m 2π   3 m 2π  ( x0 + L ) − x0 L3 − c1 sin α  + L 4m 2π  +    π L ( x0 + L ) sin α ( B11 − B22 − c2 ) − n B66  , X 42  ( x0 + L )3 − x03 mnπ L3 sin α ( B12 + B66 )  =− + L 4m 2π  X 43  ( x0 + L ) − x04 3L3 ( x0 + L ) mπ 2 =− A44 sin α  + L 8m 2π   nL2 sin α ( B22 + B66 + c2 ) , −  4m     ( x0 + L )3 − x03 mπ L3 sin α cos α B12  + + L 4m2π  X 44  L2 sin α cos α ( B22 + c2 ) , +  4m  m 2π   ( x0 + L ) − x04 3L3 ( x0 + L )  2 = −  D11 sin α + π A44 sin α   +  8m 2π   L   3  m 2π Es I1  ( x0 + L ) − x0 L3 sin α  − +  2 L λ0 4m π      π EI  + L ( x0 + L ) sin α  D11 − D22 − r  − n D66  , d2     mnπ ( x0 + L )( D12 + D66 ) ,  ( x0 + L )3 − x03 mnπ L3 X 51 = − − ( B12 + B66 )  L 4m 2π  X 45 = − X 52  nL2 − ( B22 + B66 + c2 ) ,  4m  ( x0 + L ) − x04 3L3 ( x0 + L )  m 2π = − B66 sin α  −  L 8m 2π    n2  − L ( x0 + L )  ( B22 + c2 ) + B66 sin α  ,  2sin α  π 32  ( x0 + L )3 − x03  nπ L3 X 53 = − nπ A55  L ( x0 + L ) cot α ( B22 + c2 ) , − + 4m 2π    ( x0 + L )3 − x03  nL2  Er I  mnπ L3 X 54 = − − − ( D12 + D66 )   D22 + D66 + , 2 L 4m π  4m  d2       ( x0 + L ) − x03 m 2π L3 X 55 = −π  A55 + D66   −  L 4m 2π     + 2n  L  D66 −  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