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

Corresponding author: Tel.: +84 983992081 E-mail address : quangchan82@gmail.com Abstract This work presents an analytical investigation for analyzing the mechanical buckling of trunca

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

To appear in: Composite Structures

Received Date: 29 May 2016

Revised Date: 24 September 2016

Accepted Date: 4 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

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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 closed- form 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

1 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 Mathieu-type 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 Karman 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

von-in conjunction with the Donnell’s non-lvon-inear equations to derive the bucklvon-ing 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

2 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

3 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, x 0

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 pure-ceramic if it is located at ceramic-rich shell side

d1

d 2

Fig 1. 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

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]

2

12

where εxm and εθm are the normal strains and γx mθ is the shear strain at the middle surface

of the shell, and γxzm, γθzm 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

where the subscripts sh and st denote shell and stiffeners, respectively; E s and E r are Young

elasticity modulus of stiffener in the x-direction and θ-direction, respectively

where N i st, M i st 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

1

0,sin

(xsinαM x),x+M xθ θ, −Mθsinα−xsinαQ x=0,

(xsinαM xθ),x+Mθ θ, +M xθsinα−xsinαQθ =0

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

2

P = p+ qx 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

we obtain

0 0

4 Linearization stability equations

To establish the linearization stability equations, the adjacent equilibrium criterion is used According to this criterion, each of the displacement components on the primary equilibrium path are perturbed infinitesimally to establish a new equilibrium configuration The components of displacement field at the new adjacent equilibrium configuration may

be written as

0 1, 0 1, 0 1,

Similarly, for the force and moment resultants of a neighboring state may be related

to the state of equilibrium also are of the form

where terms with 0 subscripts correspond to the u 0 , v 0 , w 0 displacements and those with 1

subscripts represents the portions of increments of force and moment resultants that are

linear in u 1 , v 1 and w 1 The substitution of Eqs (14) and (15) into Eq.(12) and note that the

terms in the resulting equations with subscript 0 satisfy the equilibrium equations and therefore drop out of the equations, and the nonlinear terms with the subscript 1 are ignored

because they are small compared to the linear terms, leads to

1

0, sin

where P=2πpx0sinα and R ij are differential operators and are determined in Appendix B

Eqs (22-26) are the couple set of five variable coefficient partial differential equations This system is very more complex than the system of stability equations of plates or cylindrical shells This is main reason why the buckling investigation of eccentrically stiffened FGM truncated conical shells still is limitted In this paper, the mentioned difficulty will be got over by Galerkin method

5 Galerkin method and closed-form expression for determining critical buckling load

In this section, an analytical approach is given to investigate the stability of stiffened FGM truncated conical shells Assume that a shell is simply supported at both ends (Fig 2) The boundary conditions in this case, are given by [18,19] as

full-As can be seen the boundary conditions v1=0, w1= and 0 φθ1= at 0 x= x0, x0+ L

are satisfied exactly, but N x1= and 0 M x1= at 0 x=x0, x0+ are fulfilled on the average L

This approach has been presented by Volmir [57, pp 516] for isotropic cylindrical shell subjected to torsional load and Sofiyev at al [56, pp 1136] has applied it for for FGM cylindrical shell subjected to torsional load In this paper, we have developed ideas on the boundary condition type of [57, 56] for stiffened FGM conical shell subjected to axial compressive load and pressure

As above underlined, it is difficult to use the trial function (28) and Eqs (22-26) to obtain directly closed-form of buckling load Therefore, a different procedure is presented here

Namely, in Eqs (22-26) containing variable x in the denominator, so if Galerkin

method immediately is applied, they will lead to have calculate integrals of the functions

L x

,

( 0)3

L x

,

( 0)2

L x

, these integrals have not primary primitives In order to eliminate variable

x in the denominator, we need to multiply both sides of Eqs (22-26) with (x i i =1, 2) to

For this aim, firstly, multiplying Eqs (22, 23, 25) by x and Eqs (24,26) by x 2, then applying Galerkin method for the resulting equations, the following expressions are obtained

3 0

5 0

21 22 23 24 1 25 2 0,

X U+X V +X W +X Φ +X Φ =

31 32 33 36 37 34 1 35 2 0,

X U+X V + X +qX +PX W+X Φ +X Φ = (30)

41 42 43 44 1 45 2 0, X U+X V +X W +X Φ +X Φ = 51 52 53 54 1 55 2 0, X U+X V +X W+X Φ +X Φ = where the coefficients X ij are defined in Appendix C Because the Eqs (30) is a system of linear homogeneous equations for U, V, W and 1 Φ , Φ So for the non-trivial solution, the determinant of its coefficient matrix must be 2 to zero i.e., ( ) 11 12 13 14 15 21 22 23 24 25 31 32 33 36 37 34 35 41 42

X X X X X X X X X X X X X qX PX X X X X + + 43 44 45 51 52 53 54 55 0

X X X X X X X X = Developing this determinant and solving resulting equation for combination of P and q, yields 5 1 2 4 36 37 31 32 34 35 33 3 3 3 3 , + = − D + D + D − D − qX PX X X X X X D D D D (31) where D i (i = 1, 2, 3, 4, 5) are caculated by 12 13 14 15 22 23 24 25 1 42 43 44 45 52 53 54 55

X X X X X X X X D X X X X X X X X = , 11 13 14 15 21 23 24 25 2 41 43 44 45 51 53 54 55

,

X X X X X X X X D X X X X X X X X = 11 12 14 15 21 22 24 25 3 41 42 44 45 51 52 54 55

,

X X X X X X X X D X X X X X X X X = 11 12 13 15 21 22 23 25 4 41 42 43 45 51 52 53 55

,

X X X X X X X X D X X X X X X X X = 11 12 13 14 21 22 23 24 5 41 42 43 44 51 52 53 54

D

=

Note that Eq (31) is the explicit expression for P and q used to determine the critical

buckling load and buckling behaviors analysis of stiffened FGM truncated conical shell subjected to axial compressive load and uniform pressure load

To validate the present study, two following comparisons are carried out

First comparison: Table 1 shows present results compared with those given in the monograph of Brush and Almroth [50, p 217] for an untiffened isotropic truncated conical shell only under external pressure The input parameters are taken as: k=0, h=0.01 ,m

7.1 Effect of stiffener arrangement and pre-compressive load

Consider an ES-FGM conical shell with input parameters as E m =70.109N m/ 2;

b = m h2 =0.02 , m b2 =0.015 m Table 3, by using Eq (31), shows the effect of

stiffener arrangement and pre-compressive load compressive on critical pressures q cr

orthogonal stiffener shell and third for stringer stiffener shell The critical pressure of unstiffened shell is smallest

Table 3 also shows effects of axially pre-loaded compressions P on critical pressure

q cr It is observed that the value of critical pressure q cr decreases in the increase of axial

pre-loaded P This decrease is considerable For example, q cr = 5.4248 kPa (P = 0) is decreases

about 22.99% in comparison q cr = 4.1772 kPa (P = 30 MN) in the case of orthogonal stiffener

7.2 Effect of stiffener number

With the database of the section 7.1, Table 4 presents effects of reinforcement

stiffener number on critical axial compressive load P cr As expected, the obtained results

show that the critical load increases when the number of stiffeners increases and inversely

This increase is considerable For example, P cr = 964.4815 (MN) for ns = nr = 5 in

comparison with P cr = 1058.1 (MN) for ns = nr = 30 increase about 1.1 times The prime reason is that the presence of stiffeners makes the shells to become stiffer, so its doing

capacity is more better

Also, from Table 5 we can see the reinforcing stiffeners also make the critical pressure of the shell increases considerably This is the outstanding advantages of stiffened structures than un-stiffeneded structures

Ring (inside) (nr = n)

Orthogonal (inside) (ns = nr =