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Accepted ManuscriptOn the stability of functionally graded truncated conical shells reinforced by functionally graded stiffeners and surrounded by an elastic medium Dao Van Dung, Le Kha

Accepted Manuscript

On the stability of functionally graded truncated conical shells reinforced by

functionally graded stiffeners and surrounded by an elastic medium

Dao Van Dung, Le Kha Hoa, Nguyen Thi Nga

To appear in: Composite Structures

Please cite this article as: Dung, D.V., Hoa, L.K., Nga, N.T., On the stability of functionally graded truncated conical shells reinforced by functionally graded stiffeners and surrounded by an elastic medium, Composite Structures (2013), doi: http://dx.doi.org/10.1016/j.compstruct.2013.09.002

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On the stability of functionally graded truncated conical

shells reinforced by functionally graded stiffeners and

surrounded by an elastic medium

Dao Van Dung, Le Kha Hoa * , Nguyen Thi Nga

Vietnam National University, Hanoi, Viet Nam

* Corresponding author: Tel.: +84 989358315

E-mail address: lekhahoa@gmail.com

Abstract:

This paper presents an analytical approach to investigate the mechanical buckling load of eccentrically stiffened functionally graded truncated conical shells surrounded by elastic medium and subjected to axial compressive load and external uniform pressure Shells are reinforced by stringers and rings in which material properties of shell and stiffeners are graded in the thickness direction according to a volume fraction power-law distribution The elastic medium is assumed

as two- parameter elastic foundation model proposed by Pasternak The change of spacing between stringers in the meridional direction is taken into account The equilibrium and linearized stability equations for stiffened shells are derived based on the classical shell theory and smeared stiffeners technique The resulting equations which they are the couple set of three variable coefficient partial differential equations in terms of displacement components are investigated by Galerkin method and the closed-form expression for determining the buckling load is obtained Four cases of stiffener arrangement are analyzed Carrying out some computations, effects of foundation, stiffener and input factors on stability of shell have been studied The effectiveness

of FGM stiffeners in enhancing the stability of cylindrical shells comparing with homogenuos stiffener is sh own.

Keywords: Stiffened truncated conical shell; Stiffener ; Elastic medium ; Functionally graded material; Critical buckling load.

1 Introduction

Functionally graded shells involving conical shells, in recent years, are widely used in space vehicles, aircrafts, nuclear power plants and many other engineering applications. These structures are usually laid on or placed in a soil medium modeled as an elastic foundation. To increase the resistance of shells to buckling, they are strengthened by stiffeners and thus the critical load can be increased considerably with only a little addition of material As a results stability and vibration analysis of those strutures are very important problems and have attracted increasing research effort

In static analysis of conical shells without foundation and stiffener, many studies have been focused on the buckling behavior analysis of shells under mechanic and thermal loading Seide [1] analyzed the buckling of conical shells

under the axial loading Singer [2] investigated the buckling of conical shells subjected to the axisymmetrical external pressure Chang and Lu [3] examined the themoelastic buckling of conical shells based on nonlinear analyses They used Galerkin method for integrating the equilibrium equation Tani and Yamaki [4] obtained the results of truncated conical shells under axial compression Using the Donnell-type shell theory, the linear buckling analysis of laminated conical shells, with orthotropic stretching-bending coupling, under axial compressive load and external pressure, are studied by Tong and Wang [5] Wu and Chiu [6] studied a three-dimensional solution for the thermal buckling of the laminated composite conical shells

For shells resting on elastic foundations, many significant results on the vibration and dynamic buckling of isotropic and anitropic cylindrical shells are obtained Paliwal et al.[7] studied free vibration of circular cylindrical shell on Winkler and Pasternak foundations Amabili and Dalpiaz [8] investigated free vibration of cylindrical shells with non-asymmetric mass ditribution on elastic bed Ng and Lam [9] considered effects of elastic foundations on the dynamic stability of cylindrical shells The same authors [10] analyzed free vibration of rotating circular cylindrical shell on an elastic foundation Naili and Oddou [12] investigated the buckling of short cylindrical shells surrounded by an elastic medium while Fok [12] studied the buckling of long cylindrical shells embedded

in an elastic medium using the energy method Effects of elastic foundations on the vibration of laminated non-homogenous orthotropic circular cylindrical shells

is reported by Sofiyev et al [13] Geometrically nonlinear dynamic analysis of doubly curved isotropic shells resting on elastic foundation by a combination of hamonic differential quadrature-finite difference method is presented by Civalek [14] Solution of axisymmetric dynamic problems for cylindrical shells on an elastic foundation is considered by Golovko et al [15] Sheng and Wang [16] considered the effect of thermal load on buckling, vibration and dynamic buckling

of FGM cylindrical shells embedded in a linear elastic medium based on the order shear deformation theory (FSDT) taking into account the rotary inertia and transverse shear strains The post-buckling analysis of tensionless Pasternak FGM cylindrical shells surrounded by an elastic medium under the lateral pressure and axial load are studied by Shen [17] and Shen et al [18] using the singular perturbation technique and the higher-order shear deformation shell theory (HDST) The mechanical buckling of FGM cylindrical shells surrounded by Pasternak elastic foundation is studied by Bagherizadeh et al [19] in which the equilibrium and stability equations are derived based on the higher-order shear

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 explicit expression

of buckling load

As can be seen that the above introduced works only relate to unstiffened 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 Weingarten [34] 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 [35] applied an energy approach to find the resonant frequencies of simply supported ring-stiffened, and ring and stringer-stiffened conical shells Mustaffa and Ali [36] 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 [37] and Tornabene et al [38]

Srinivasan and Krishnan [39] 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 [40] 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 [41] 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

In 2009, Najafizadeh et al [42] 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 this direction, Dung and Hoa [43, 44] obtained the results on the static nonlinear buckling and post-buckling analysis of eccentrically stiffened FGM circular cylindrical shells under external pressure and torsional load 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 [45] presented an analytical approach to investigated the nonlinear post-buckling of eccentrically stiffened FGM plates and shallow shells based on the classical shell theory in which the stiffeners are assumed to be homogeneous Bich et al [46] obtained the results on the nonlinear dynamic analysis of eccentrically stiffened FGM cylindrical panels The governing equations of motion were derived by using the smeared stiffeners technique and the classical shell theory with von Karman geometrical nonlinearity The same authors [47] investigated the nonlinear vibration dynamic buckling of eccentrically stiffened imperfect FGM

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 et al [48] studied a mechanical buckling of eccentrically stiffened functionally graded (ES-FGM) thin truncated conical shells subjected to axial compressive load and uniform external pressure load based on the smeared stiffeners technique and the classical shell theory and considering homogenous stiffeners

The objective of this study is to extend mechanical buckling results of the work [48] for stiffened FGM thin truncated conical shells surrounded by an elastic medium The present novelty is that an analytical approach to investigate the buckling load of stiffened functionally graded truncated conical shells surrounded by elastic foundations is presented Shells under combined load are reinforced by rings and stringers in which their material properties are graded in the thickness direction according to a volume fraction power-law distribution The change of spacing between stringers in the meridional direction is taken into account The theoretical formulations based on the smeared stiffeners technique and the classical shell theory, are derived The resulting equations which they are the couple set of three variable coefficient partial differential equations in terms of displacement components are solved by Galerkin method The closed-form expressions to determine critical buckling loads are obtained Four cases of stiffener arrangement are investigated The influences of various parameters such

as foundation, stiffener, dimensional parameters and volume fraction index of materials on the stability of shell are considered The effectiveness of FGM stiffeners in enhancing the stability of cylindrical shells comparing with homogenuos stiffener is shown

2 ES-FGM truncated conical shell and derivations

2.1 FGM truncated conical shell

Consider a thin truncated conical shell of thickness h and semi-vertex angle α 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 plane (x,θ), lies

in the outwards normal direction of the cone Also, x0 indicates the distance from

Case 1: Conical shell with ceramic outside surface and metal inside surface and outside stiffener

Case 2: Conical shell with ceramic outside surface and metal inside surface and inside stiffener

Case 3: Conical shell with metal outside surface and ceramic inside surface

where νsh =νs =νr = =ν const , k, k 2 and k 3 are volume fractions indexes of shell,

stringer and ring, respectively and 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 h , and s h ;and r

,

E E are Young’s modulus of the ceramic and metal, and E , E sh s and E r are

Young moduli of shell, of stiffener in the x-direction and θ -direction, respectively The coefficient ν is Poison’s ratio

Young’s modulus for remaining cases are given in Appendix I

2

1,2

where the subscripts sh and st denote shell and stiffeners, respectively

Taking into account the contribution of stiffeners by the smeared stiffener technique and omitting the twist of stiffeners because these torsion constants are smaller more than the moments of inertia [49] In addition, the change of spacing between stringers in the meridional direction also is taken into account Integrating the above stress-strain equations and their moments through the thickness of the shell, we obtain the expressions for force and moment resultants of an eccentrically stiffened FGM conical shells

The stability equations of conical shell are derived using the adjacent equilibrium criterion [49] Assume that the equilibrium state of ES-FGM conical shell under mechanical loads is defined in terms of the displacement components

0, v and0 0

u w We give an arbitrarily small increments u1, v and1 w to the 1

displacement variables, so the total displacement components of a neighboring state are

u=u + , u v=v0+ , v1 w=w0+ w1 (12) Similarly, the force and moment resultants of a neighboring state may be related

to the state of equilibrium as

with 1 subscripts represents the portions of increments of force and moment resultants that are linear in u1, v and1 w1 The stability equations may be obtained by substituting Eqs (12) and (13) into Eqs (11) and note that the terms in the resulting equations with subscript 0 satisfy the equilibrium equations and therefore drop out

of the equations In addition, the nonlinear terms with subscript 1 are ignored because they are small compared to the linear terms The remaining terms form the stability equations as follows

1

0sin

3 Buckling analysis of ES-FGM truncated conical shell surrounded by elastic foundation

An analytical approach is given, in this section, to investigate the stability of FGM truncated conical shells Assume that a shell is simply supported at both ends Thus the boundary conditions are expressed by

x ≤ ≤x x + ; L 0≤θ ≤2π and for sake of convenience in integration, Eqs (18),

(19) are multiplied by x and Eq (20) by x , and applying Galerkin method for the 2

resulting equations, that are

where the coefficients s ij are defined in Appendix III

Because the solutions of Eq (25) are nontrivial, the determinant of coefficient matrix of this system must be zero Developing that determinant and solving

resulting equation for combination of P and q, leads to

uniform pressure load The buckling loads P and q still depend on values of m and

n , therefore must minimize these expressions with respect to m and n, we obtain the critical values of P and q respectively

( )

4

* 10 cr/

q = Rq Eh , where q is found from Eq (26) cr

Next, Table 2 compares the present results with those of Naj et al [32] and Baruch et al [50] for a pure isotropic un-stiffened truncated conical shell The input parameters are: k= , 0 h=0.01m, R=100× , h ν =0.3, q=0, P*=P cr/P cl with

R Naj et al.[32]

Buckling mode (m,n)

Finaly, Table 3 compares the present results with those of Seide [1] and Sofiyev [27] for a pure isotropic un-stiffned truncated conical shell (without foundation) The input parameters are: E m=E c = =E 210GPa, ν =0.3, h=0.0005m, ' 400

Table 3

Comparisons σcr(MPa) with results of Seide [1] and Sofiyev [27] for un-stiffened isotropic

truncated conical shells

As can be seen that good agreements are obtained in these three comparisons

4.2 ES-FGM truncated conical shell with and without elastic foundation

In the following subsections, the materials used are Alumina with E c =380GPa and Aluminum with E m =70GPa and ν =0.3 The geometrical parameters are

taken as h=0.005m, R/h=150, L=2R, h s =h r =0.005m, b s =b r =0.0025m The stiffener is made of FGM material

a Effect of foundation (without stiffener)

Using Eq (29) with q=0,α =300, k=1, the critical axial compressive load of

FGM truncated conical shell may be found in Table 4 It shows that when q= the 0foundation parameters K and1 K affect strongly for the critical load P Table 5 2presents effects of foundation on critical pressures qcr (P=0) As can be observed that

the critical buckling load corresponding to the presence of the both foundation parameters K1= ×5 107N/m3 and K2 = ×5 105N/m is the biggest The critical buckling load of shell without foundation is the smallest For example, the critical

buckling load of shell without foundation P =14.75584 MN, m=7, n=26 when cr

compared in Table 4 with P =18.92762 MN, m=11, n=1, incerases about 28.3% cr

b Effect of stiffener (without foundation)

The effects of stiffeners (without foundation) on critical compression load P and critical external pressure load q are given in Tables 6 and 7 Table 6 shows that in the case of only compression load P, with the same stiffener numbers, the critical

buckling load of stiffened shell by orthogonal is the biggest, the second is stiffened shell by stringers and the critical load of the stiffened shell by ring stiffeners is the smallest In additrion, the critical compressive load increases with the increase of stiffeners number The effect of pre-loaded axial compression P on critical pressure

q is illustrated in Table 7 As can be seen that the value of q decreases in the cr increase of axial load P Table 7 also shows that the critical pressure q (with P≠ 0)

of stiffened shell by ring is the biggest

14.93088 (4,26)

14.96462 (4,26)

14.99822 (4,26)

15.03168 (4,26)

Ring

nr=nst

14.81156 (9,20)

14.81922 (9,20)

14.82689 (9,20)

14.83455 (9,20)

14.84221 (9,20)

Orthogonal

ns=nr=nst/2

14.92844 (8,24)

15.08314 (8,23)

15.23399 (8,23)

15.38450 (8,23)

15.53467 (8,23)

c Effect of stiffener and foundation

Table 8 and 9 illustrate the effects of stiffeners and foundation on critical

compression load P and critical external pressure load q Fig 2 and Table 8 show

that when q= with the same stiffener numbers, the critical buckling load of 0stiffened shell by stringer is the biggest (P xcr =18.27572MN reached at m=9, n=19),

the second is stiffened shell by orthogonal stiffeners and the critical load of ring

stiffened shell is the smallest Fig 3 and Table 9 ( with P=0) show that the critical

buckling load of stiffened shell by ring is the biggest (q xcr=709.4372 kPa for m=1, n=21)