DSpace at VNU: Buckling Analysis of Eccentrically Stiffened Functionally Graded Toroidal Shell Segments under Mechanical Load

DSpace at VNU: Buckling Analysis of Eccentrically Stiffened Functionally Graded Toroidal Shell Segments under Mechanical...

Buckling Analysis of Eccentrically Stiffened Functionally Graded Toroidal Shell Segments under Mechanical Load

Bich Huy Dao1; Ninh Gia Dinh2; and Thinh Ich Tran3

Abstract: An analytical approach is presented to investigate the linear buckling of eccentrically stiffened functionally graded thin toroidal shell segments subjected to axial compression, lateral pressure, and hydrostatic pressure On the basis of classical thin shell theory, the smeared stiffener technique and the adjacent equilibrium criterion, the governing equations of buckling of eccentrically stiffened functionally graded toroidal shell segments are derived The functionally graded toroidal shell segments with simply supported edges are reinforced by a ring and stringer stiffener system on an external surface The resulting equations in the case of compressive and pressure loads are solved directly The obtained results show the effects of stiffeners and input factors on the buckling behavior of these structures In this paper, the results are also compared with the solutions published in the literature for the specific cases of a toroidal shell DOI:10.1061/(ASCE)EM 1943-7889.0000964 © 2015 American Society of Civil Engineers

Author keywords: Functionally graded material; Toroidal shell segments; Stiffeners; Critical buckling loads; Axial compression; Lateral pressure; Hydrostatic pressure

Introduction

Functionally graded materials (FGMs) are composite materials that

have mechanical properties varying continuously from one surface

of a structure to another Nowadays, the application of FGMs is

becoming so widespread that many problems related to new

struc-tures as well as the static and dynamic behaviors of strucstruc-tures are

being noticed

Many studies on the buckling analysis of FGMs have been

reported in the literature Expressions for the buckling load and

postbuckling equilibrium path of an axially compressed thin

homo-geneous cylindrical shell have been obtained by Karman and Tsien

(1941) Shen (2002) investigated the postbuckling analysis of

ax-ially loaded functionally graded cylindrical shells in thermal

envi-ronments using classical shell theory with a von Kármán–Donnell

type of kinematic nonlinearity Shen and Leung (2003) investigated

the postbuckling of a pressure-loaded FGM cylindrical panel

subjected to lateral pressure in a thermal environment based on

Reddy’s higher-order shear deformation shell theory with a von

Karman–Donnell type of kinematic nonlinearity The postbuckling

of shear deformation FGM cylindrical shells surrounded by a

Pasternak foundation with two kinds of micromechanics models,

the Voigt model and the Mori–Tanaka model, was researched by

Shen (2013) The governing equations were based on a

higher-order shear deformation shell theory, and the material properties of

FGMs were assumed to be temperature dependent

The nonlinear dynamic buckling of functionally graded cylin-drical shells subjected to a time-dependent axial load was re-searched by Huang and Han (2010) These authors, whose work was mentioned previously (Huang and Han 2008), researched the buckling of imperfect functionally graded cylindrical shells under axial compression Sofiyev (2010) analyzed the buckling of func-tionally graded material circular truncated conical and cylindrical shells subjected to combined axial extension loads and hydrostatic pressure and resting on a Pasternak-type elastic foundation The buckling of a simply supported three-layer circular cylindrical shell under axial compressive load was studied by Li and Batra (2006)

A postbuckling analysis was performed by Shen and Noda (2007) for a functionally graded cylindrical shell with piezoelectric actua-tors subjected to lateral or hydrostatic pressure combined with elec-tric loads in thermal environments Bich et al (2013) investigated the nonlinear static and dynamic buckling of imperfect eccentri-cally stiffened functionally graded thin circular cylindrical shells subjected to axial compression based on classical thin shell theory with geometrical nonlinearity in a von Karman–Donnell sense Moreover, Bich et al (2012) researched the linear problem of func-tionally graded conical panels buckling under mechanical loads using classical thin shell theory in which the Galerkin method was applied to obtain closed-form relations of bifurcation-type buckling loads Dung and Hoa (2013) researched nonlinear torsional buck-ling and postbuckbuck-ling of eccentrically stiffened functionally graded thin circular cylindrical shells A linear buckling analysis of eccen-trically stiffened functionally graded circular cylindrical thin shells under mechanical load was conducted by Phuong and Bich (2013) Stein (1964) was the first to recognize the importance of nonlinear prebuckling deformations on the buckling load of perfect cylindri-cal shells Weaver (2000) pointed out that the load-bearing effi-ciency of cylindrical shells derived from both the properties of the material of which they are made and from the shape itself Toroidal shells have been proposed for use in such applications

as fusion reactor vessels, satellite support structures, underwater toroidal pressure hulls, rocket fuel tanks, and diver oxygen tanks Nowadays FGMs consisting of metal and ceramic components have received considerable attention in structural applications The smooth and continuous changes in material properties enable

1 Professor, Vietnam National Univ., No 144 Xuan Thuy St., Cau Giay

District, Hanoi 10000, Vietnam.

2 Lecturer, Hanoi Univ of Science and Technology, No 1 Dai Co Viet

St., Hai Ba Trung District, Hanoi 10000, Vietnam (corresponding author).

E-mail: ninhdinhgia@gmail.com; ninh.dinhgia@hust.edu.vn

3 Professor, Hanoi Univ of Science and Technology, No 1 Dai Co Viet

St., Hai Ba Trung District, Hanoi 10000, Vietnam.

Note This manuscript was submitted on December 23, 2014; approved

on April 6, 2015; published online on May 26, 2015 Discussion period

open until October 26, 2015; separate discussions must be submitted

for individual papers This paper is part of the Journal of Engineering

Mechanics, © ASCE, ISSN 0733-9399/04015054(10)/$25.00.

FGMs to avoid interface problems and unexpected thermal stress

concentrations Some of the aforementioned structural components

may be made of FGMs Thus, buckling analysis has become an

important part of the design process for such shells

Stein and McElman (1965) researched the buckling of

ho-mogeneous toroidal shell segments In a NASA technical note,

McElman (1967) investigated eccentrically stiffened shallow shells

of double curvature Furthermore, Hutchinson (1967) presented a

study on the initial postbuckling behavior of homogeneous toroidal

shell segments

The present paper is concerned with studies on the linear

buck-ling analysis of eccentrically stiffened functionally graded toroidal

shell segments under axial compression, lateral pressure, and

hy-drostatic pressure In this study, shells are assumed to be perfect,

and the analysis does not consider geometric or material

imperfec-tions The fundamental equations for the buckling analysis of

stiff-ened FGM toroidal shell segments based on classical shell theory

using the smeared stiffener technique and adjacent equilibrium

cri-terion are derived

Governing Equations

Functionally Graded Material (FGM)

FGMs are microscopically inhomogeneous materials whose

prop-erties vary smoothly and continuously from one surface of a

material to another These materials are made from a mixture of

ceramic and metal or a combination of different materials Such a

mixture of ceramic and metal with a continuously varying volume

fraction can be manufactured In particular, FGM thin-walled

struc-tures with ceramic in the inner surface and metal in the outer

sur-face are widely used in practice It is assumed that the elasticity

modulus E changes in the thickness direction z, while the Poisson

ratioν is constant

Denote by Vmand Vcthe volume fractions of metal and ceramic

phases, respectively These parameters are related by Vmþ Vc¼ 1,

in which Vcis expressed as VcðzÞ ¼ ½ð2z þ hÞ=ð2 hÞk, where h is

the thickness of a thin-walled structure, k is the volume-fraction

exponent (k ≥ 0), and z is the thickness coordinate and varies from

−h=2 to h=2 The subscripts m and c refer to the metal and ceramic

constituents, respectively According to the aforementioned law,

the Young’s modulus can be expressed in the form

EðzÞ ¼ EmVmþ EcVc¼ Emþ ðEc− EmÞ

2z þ h

2 h

k ð1Þ and the Poisson ratioν is assumed to be constant

Constitutive Relations and Governing Equations

Consider a functionally graded toroidal shell segment of thickness

h and length L formed by the rotation of a plane circular arc of

radius R around an axis in the plane of the curve as shown in Fig.1

The geometry and coordinate system of a stiffened FGM toroidal

shell segment are described in Fig.2 For the middle surface of the

toroidal shell, from the figure we have

r ¼ a − Rð1 − sin φÞ where a is the equator radius and φ is the angle between the axis of

revolution and the normal to the shell surface For a sufficiently

shallow toroidal shell in the region of the equator of the torus,

the angle φ is approximately equal to π=2; thus, sin φ ≈ 1,

cosφ ≈ 0, and r ¼ a (Stein and McElman 1965) The form of

the governing equation is simplified by setting

dx1¼ Rdφ; dx2¼ adθ The radius of arc R is positive with the convex toroidal shell segment and negative with the concave one Suppose the FGM toroidal shell segment is reinforced by closely spaced stringer and ring stiffeners To provide continuity between the shell and the stiffeners and to make it easier to manufacture them, homo-geneous stiffeners can be used Because the pure ceramic stiffeners are brittle, metal stiffeners are used and arranged on the metal-rich side of the shell Applying the law indicated by Eq (1), the outer

Fig 1 Configuration and coordinate system of eccentrically stiffened toroidal shell segments: (a and b) convex shell; (c and d) concave shell

Fig 2 Geometry and coordinate system of stiffened FGM toroidal shell segments: (a) stringer stiffeners; (b) ring stiffeners

surface of the shell is metal-rich, so the external metal stiffeners

are arranged on this side

In this paper, classical shell theory and the Lekhnitsky smeared

stiffener technique are used to obtain the equilibrium and

compat-ibility equations, as well as expressions for the buckling loads of

stiffened FGM toroidal shell segments

The strains across the shell thickness at a distance z from the

midsurface are

ε1¼ ε0− zχ1; ε2¼ ε0− zχ2; γ12¼ γ0

12− 2zχ12 ð2Þ whereε0andε0= normal strains;γ0

12= shear strain at middle

sur-face of shell; andχij = the curvatures

According to classical shell theory, the strains at the middle

sur-face and the curvatures are related to the displacement components

u, v, and w in the x1, x2, and z coordinate directions as (Brush and

Almroth 1975)

ε0¼∂x∂u

1w

R; ε0¼∂x∂v

2þw

12¼∂x∂u

2þ∂x∂v

1;

χ1¼∂∂x2w2; χ2¼∂∂x2w2; χ12¼∂x∂2w

1∂x2 ð3Þ From Eqs (3), the strains must satisfy the deformation

compat-ibility equation

∂2ε0

∂x2 þ∂2ε0

∂x2 − ∂2γ012

∂x1∂x2¼ ∂2w

R∂x2þ ∂2w

The constitutive stress–strain equations determined by Hooke’s

law for the shell material are omitted here for brevity The

contri-bution of the stiffeners can be calculated using the Lekhnitsky

smeared stiffener technique

Integrating the stress–strain equations and their moments

through the thickness of the shell, the expressions for the force

and moment resultants of an eccentrically stiffened FGM toroidal

shell are obtained:

N1¼

A11þEmA1

s1



ε0þ A12ε0− ðB11þ C1Þχ1− B12χ2

N2¼ A12ε0þ

A22þEmA2

s2



ε0− B12χ1− ðB22þ C2Þχ2

N12¼ A66γ0

12− 2B66χ12 ð5Þ

M1¼ ðB11þ C1Þε0þ B12ε0−

D11þEmI1

s1



χ1− D12χ2

M2¼ B12ε0þ ðB22þ C2Þε0− D12χ1−

D22þEmI2

s2



χ2

M12¼ B66γ0

12− 2D66χ12 ð6Þ where Aij; Bij; Dijði; j ¼ 1; 2; 6Þ are the extensional, coupling, and

bending stiffnesses of a shell without stiffeners,

A11¼ A22¼ E1

1 − ν2; A12¼ E1:ν

1 − ν2; A66¼ E1

2ð1 þ νÞ;

B11¼ B22¼1 − νE22; B12¼1 − νE2:ν2; B66¼2ð1 þ νÞE2 ;

D11¼ D22¼ E3

1 − ν2; D12¼ E3:ν

1 − ν2; D66¼ E3

with

E1¼

EmþEc− Em

k þ 1

 h; E2¼ ðEc− EmÞkh2

2ðk þ 1Þðk þ 2Þ

E3¼



Em

12þ ðEc− EmÞ

k þ 3− 1

k þ 2þ4k þ 41



and

C1¼ −EmA1z1

s1 ; C2¼ −EmA2z2

In Eqs (5), (6) and (9), the spacing of the stringer and ring striff-eners are denoted by s1and s2, respectively The quantities A1and

A2are the cross-sectional areas of the stiffeners, and I1, I2, z1, and

z2are the second moments of the cross-sectional areas and

eccen-tricities of the stiffeners with respect to the middle surface of the shell, respectively The minus sign for C1and C2 depends on the

external stiffeners

Remark: Conversely, if the inner side of the stiffened FGM shell

is metal-rich, then all calculated expressions can be used, but Ec

and Emmust be replaced each other in Eqs (8) and the plus sign

is taken in Eqs (9)

The eccentrically stiffened FGM shell is subjected to mechani-cal loads in two cases:

1 Case 1: An axial compression load p uniformly distributed on the two end edges of the shell and a lateral pressure q uniform distributed on the surface; and

2 Case 2: Hydrostatic pressure load

The nonlinear equilibrium equations of a toroidal shell based on classical shell theory are given by (Brush and Almroth 1975)

∂N1

∂x1 þ∂N12

∂x2 ¼ 0; ∂N12

∂x1 þ∂N2

∂x2 ¼ 0;

∂2M1

∂x2 þ 2∂x∂2M12

1∂x2þ∂∂x2M22þ N1∂2w

∂x2þ 2N12 ∂2w

∂x1∂x2

þ N2∂2w

∂x2∓N1

R −N2

The stability equations of eccentrically stiffened functionally graded shells may be established by the adjacent equilibrium cri-terion It is assumed that the equilibrium state of an eccentrically stiffened functionally graded shell under applied load is represented

by the displacement components u0, v0, and w0 The state of ad-jacent equilibrium differs that of stable equilibirum by increments

u1, v1, w1, and the total displacement component of a neighboring configuration are

u ¼ u0þ u1; v ¼ v0þ v1; w ¼ w0þ w1 ð11Þ Similarly, the force and moment resultants of a neighboring state are represented by

N1¼ N0þ N1; N2¼ N0þ N1; N12¼ N0

12þ N1

12;

M1¼ M0þ M1; M2¼ M0þ M1; M12¼ M0

12þ M1

12 ð12Þ where terms with subscripts 0 correspond to the u0, v0, and w0

displacements and those with subscript 1 represent the increments

of force and moment resultants that are linear in u1, v1, and w1.

Subsequently, inserting Eqs (11) and (12) into Eq (10) and subtracting from the resulting equations the term relating to the stable equilibrium state, neglecting the nonlinear term in u1, v1,

and w1 or their counterparts in the form of, for example,

N1, N1, N112,: : : and prebuckling the rotations yields the following

stability equations:

∂N1

∂x1 þ∂N112

∂x2 ¼ 0 ∂N112

∂x1 þ∂N∂x1

2 ¼ 0

∂2M1

∂x2 þ 2∂2M112

∂x1∂x2þ∂∂x2M21∓N1

R −N1

a þ N0∂2w1

∂x2

þ 2N0

12 ∂2w1

∂x1∂x2þ N0∂2w1

Considering the first two of Eqs (13), a stress function may be

defined as

N1¼∂∂x2F2; N1¼∂∂x2F2; N112¼ −∂x∂2F

1∂x2 ð14Þ The reverse relations are obtained from Eqs (5) for increments

ε01

1 ¼ A

22N1− A

12N1þ B

11χ1þ B

12χ1

ε01

2 ¼ A

11N1− A

12N1þ B

21χ1þ B

22χ1

γ01¼ A

66N112þ 2B

66χ1

where

A11¼Δ1

A11þE0A1

s1



; A22¼Δ1

A22þE0A2

s2



;

A12¼AΔ12; A66¼ 1

A66

Δ ¼

A11þE0A1

s1

 :

A22þE0A2

s2



− A2 12

B11¼ A

22ðB11þ C1Þ − A

12B12;

B22¼ A

11ðB22þ C2Þ − A

12B12

B12¼ A

22B12− A

12ðB22þ C2Þ;

B21¼ A

11:B12− A

12ðB11þ C1Þ; B66¼B66

A66

Substituting Eq (15) into Eq (6) for increments yields

M1¼ B

11N1þ B

21N1− D

11χ1− D

12χ1;

M1¼ B

12N1þ B

22N1− D

21χ1− D

22χ1;

M112¼ B

66N112− 2D

66χ1

where

D11¼ D11þE0I1

s1 − ðB11þ C1ÞB

11− B12B21

D22¼ D22þE0I2

s2 − B12B21− ðB22þ C2ÞB

22

D12¼ D12− ðB11þ C1ÞB

12− B12B22

D21¼ D12− B12B11− ðB22þ C2ÞB

21 D66¼ D66− B66B66 The substitution of Eqs (15) into the compatibility Eqs (4) and

substituting Eqs (16) into the third of Eqs (13), taking into account

Eqs (3) and (14), yields a system of equations

A11∂4F

∂x4þ ðA

66− 2A

12Þ∂x∂24∂xF2þ A

22∂4F

∂x4þ B

21∂4w1

∂x4

þ ðB

11þ B

22− 2B

66Þ ∂4w1

∂x2∂x2þ B

12∂4w1

∂x4 −1 R

∂2w1

∂x2

−1 a

∂2w1

D11∂4w1

∂x4 þ ðD

12þ D

21þ 4D

66Þ∂x∂42w∂x12þ D

22∂4w1

∂x4 − B

21∂4F

∂x4

− ðB

11þ B

22− 2B

66Þ∂x∂24∂xF2− B

12∂4F

∂x41 R

∂2F

∂x2þ1 a

∂2F

∂x2

− N0∂2w1

∂x2 − 2N0

12 ∂2w1

∂x1∂x2− N0∂2w1

Eqs (17) and (18) are the basic equations used to investigate the stability of eccentrically stiffened functionally graded toroidal shell segments They are linear equations in terms of two dependent un-knowns, w1 and F

Buckling Analysis of Eccentrically Stiffened Functionally Graded Toroidal Shell Segments Subjected to Axial Compressive Load

and Lateral Pressure

In this paper, an eccentrically stiffened FGM shell is free and is simply supported at two end edges and subjected to mechanical loads in two cases In Case 1, an axial compression load p is uniformly distributed on the two end edges of the shell and a lateral pressure q is uniformly distributed on the external surface

In Case 2, there is a hydrostatic pressure load

By solving the membrane form of the equilibrium equations, prebuckling force resultants are determined as (Stein and McElman

1965) Case1∶ N0¼ −ph; N0¼ −qa; N012¼ 0 ð19Þ Case2∶ N0¼ −qa2; N0¼ −qa

1∓2Ra



; N012¼ 0

ð20Þ The boundary conditions considered in the present study are written for increments as follows:

Conditions (21) can be satisfied if the buckling mode shape is represented by

w1¼X

m

X

n

Wmnsinmπx1

L sin

nx2

where Wmnis a maximum deflection; m, n are the numbers of half waves in axial and circumferential direction respectively Substitut-ing Eqs (22) into Eq (17) and solving the obtained equation for unknown F leads to

m

X

n

fmnsinmπx1

L sin

nx2

where

fmn¼ −½B21m4π4þ ðB

11þ B

22− 2:B

66Þm2n2π2λ2þ B

12n4λ4− am2π2λ2∓a2n2λ4=R

A11m4π4þ ðA

66− 2A

12Þm2n2π2λ2þ A

22n4λ4 Wmn ð24Þ

Case 1: Axial Compression Load p Uniformly Distributed on

Two End Edges of Shell and Lateral Pressure q Uniformly

Distributed on External Surface

Inserting expressions (22), (23) and (19) into Eqs (18) leads to

X

m

X

n



D þB

2

A þ ð−phm2π2− qan2λ2ÞL2

:Wmn

× sinmπx1

L sin

nx2

where denote

A ¼ A11m4π4þ ðA

66− 2A

12Þm2n2π2λ2þ A

22n4λ4; λ ¼ L=a;

B ¼ B21m4π4þ ðB

11þ B

22− 2:B

66Þm2n2π2λ2

þ B

12n4λ4− am2π2λ2− a2n2λ4=R;

D ¼ D11m4π4þ ðD

12þ D

21þ 4D

66Þm2n2π2λ2þ D

22n4λ4

Eq (25) satisfies for all x1, x2 so that

D þB

2

A þ ð−phm2π2− qan2λ2ÞL2¼ 0 ð26Þ Now investigate the linear buckling of reinforced FGM toroidal

shell segments subjected to only axial compression (q ¼ 0), The

Eq (26) becomes

D þB

2

From Eq (27) the compressive buckling load can be obtained

hm2π2L2

D þB

2

A



ð28Þ Introducing parameters

¯D ¼ D

h3; ¯B ¼ B

into Eq (28), the compressive buckling load can be obtained

m2π2ðL=hÞ2

The critical axial compression load of eccentrically stiffened

FGM toroidal shell is determined by condition pcr¼

minpvs.ðm; nÞ

If the toroidal shell is subjected to only lateral pressure (p ¼ 0),

Eq (26) becomes

D þB

2

The pressure buckling load can be determined

an2λ2L2

D þB

2

A



ð32Þ or

ða=hÞn2λ2ðL=hÞ2

¯D þ¯B¯A2 ð33Þ The critical lateral pressure of eccentrically stiffened FGM toroidal shell is determined by condition qcr¼ minqvs.ðm; nÞ Case 2: Hydrostatic Pressure Load

The substitution of expressions (22), (23) and (20) into Eqs (18) leads to

X

m

X

n



D þB

2

Aþ

−q:a2m2π2− q:a

1∓2Ra



n2λ2



L2

 :Wmn

× sinmπx1

L sin

nx2

where

A ¼ A11m4π4þ ðA

66− 2A

12Þm2n2π2λ2þ A

22n4λ4; λ ¼ L=a;

B ¼ B21m4π4þ ðB

11þ B

22− 2:B

66Þm2n2π2λ2

þ B

12n4λ4þ am2π2λ2∓a2n2λ4=R;

D ¼ D11m4π4þ ðD

12þ D

21þ 4D

66Þm2n2π2λ2þ D

22n4λ4

Eq (34) is satisfied by all x1 and x2, so that

D þB

2



−qa2m2π2− qa

1∓2Ra



n2λ2

L2¼ 0 ð35Þ

From Eq (35) the buckling hydrostatic pressure load can be obtained:

a

2m2π2þ a

1∓2Ra



n2λ2

L2

D þB

2

A

 ð36Þ or

a h



m2π2

1∓2Ra



n2λ2

L h

2

¯D þ¯B2 A

 ð37Þ

The critical hydrostatic pressure load of an eccentrically stiff-ened FGM toroidal shell is determined by the condition qcr¼ minqvs.ðm; nÞ

Numerical Result and Discussions

Validation Studies Cylindrical shells are a specific case of toroidal shell segments (when R → þ∞) The buckling of a simply supported FGM cy-lindrical shell without stiffeners under axial compression is consid-ered To demonstrate the formulation presented here, calculations

of a FGM cylindrical shell under mechanical loads are compared with the results of Huang and Han (2010) Numerical results are given for cylindrical shells made of zirconia (ZrO2) and titanium (Ti-6Al-4 V) The elasticity moduli of zirconia and titanium at the

initial temperature To ¼ 300 K are taken to be 168.08 GPa and

105.69 GPa, respectively The Poisson ratio is chosen to be 0.3

As can be seen in Table1, good agreement is obtained in this

comparison study

It can be shown that for simply supported FGM cylindrical

shells with stiffeners under axial compression, the compressive

buckling load determined by Eq (30) (when R → þ∞) in the text

coincides completely with Eq (32), deduced by Bich et al (2013),

and Eq (26) of Phuong and Bich (2013) For shells under lateral

pressure the present buckling load determined by Eq (33)

coin-cides identically with Eq (27) of Phuong and Bich (2013)

This fact shows the accuracy of the present approach

Secondly, the results in this paper are compared with those in the

monograph of Brush and Almroth (1975) for a stiffened

homo-geneous cylindrical shell under axial compression, as shown in

Table2 The comparison shows that good results are achieved

Another comparison is established for homogeneous isotropic

toroidal shell segments under lateral pressure, based on the formula

of a critical buckling load given by Hutchinson (1967) and the

formula in the present paper

The results presented in Table3show the good agreement of the

present formulation

Results of Buckling Analysis of FGM Toroidal Shell

Segments

To illustrate the present formulation, the FGM toroidal shell

seg-ments are made of aluminum Em¼ 7 × 1010N=m2 and alumina

Ec¼ 38 × 1010N=m2 The Poisson ratioν is chosen to be 0.3 for

simplicity The height of the stiffener is equal to 0.005 m, and its width is 0.002 m The Young’s moduli of the external stringer stiff-eners and external ring stiffstiff-eners is E0¼ Em The stiffeners include

50 ring and 50 stringer stiffeners distributed regularly in the axial and circumferential directions, respectively

Effects of R=h Ratio and Volume-Fraction Index k Tables 4 and 5 show the critical buckling load of stiffened and unstiffened toroidal segments versus four different values of the volume-fraction index k (0.5, 1, 5, 10) and four values of the R=h ratio [R > 0 (convex shell) and R < 0 (concave shell)], respectively, with the following geometric properties: L=a ¼ 2, a=h ¼ 100,

h ¼ 0.002 m, d1¼ d2¼ 0.002 m, h1¼ h2¼ 0.005 m, n1¼

n2¼ 50, where n1 and n2 are the number of stringer and ring

stiffeners, respectively

Table 1 Comparison of Present Critical Load (MPa) with Theoretical Results Reported by Huang and Han (T o ¼ 300 K, L=a ¼ 2)

Reference

Huang and Han ( 2010 )

ðσ scr ¼ σ dcr =τ cr Þ

189.262 (2, 11) a 164.352 (2, 11) 144.471 (2, 11) 236.578 (5, 15) 157.984 (3, 14) 118.849 (2, 12) Present 189.324 (2, 11) 164.386 (2, 11) 144.504 (2, 11) 236.464 (5, 15) 158.022 (3, 14) 118.898 (2, 12)

a

Numbers in parentheses indicate buckling mode (m, n).

Table 2 Critical Buckling Load per Unit Length ¯p cr ¼ p cr hð106 N =mÞ of

Externally Stiffened Homogeneous Cylindrical Shells under Axial

Com-pression versus a=h Ratio

Ratio

Brush and Almroth ( 1975 )

Difference (%)

Note: 50 rings, 50 stringers; L ¼ 1 m; a ¼ 0.5 m; E ¼ 7 × 1010ðN=m 2 Þ;

v ¼ 0.3; d1¼ d2¼ 0.0025 m; h1¼ h ¼ 0.01 m.

Table 3 Critical Buckling Load ¯q of Homogeneous Toroidal Shells under Lateral Pressure Load versus R=h Ratio

R=h

Hutchinson ( 1967 ) Present Difference (%) Hutchinson ( 1967 ) Present Difference (%)

400 111.1143 (1, 10) 111.1147 (1, 10) 0.0003 24.6953 (2, 5) 24.6953 (2, 5) 0.0000

500 136.8904 (1, 11) 136.8910 (1, 11) 0.0004 24.7356 (2, 5) 24.7356 (2, 5) 0.0000 Note: L ¼ 0.603 m; a ¼ 0.243; R ¼ 0.972 m; E ¼ 72.4 × 109ðN=m 2 Þ; v ¼ 0.3.

Table 4 Critical Buckling Load of Stiffened FGM Convex Toroidal Shell Segment (R > 0) under Axial and Lateral Pressure Load

R=h k

p cr × 10 3(MPa) q cr (MPa) Unstiffened

External stiffener Unstiffened

External stiffener

200 0.5 1.5824 (12, 1)a 2.7229 (8, 7) 8.1357 (1, 13) 25.0926 (1, 8)

1 1.2512 (12, 1) 2.2917 (8, 7) 6.4275 (1, 14) 22.6557 (1, 8)

5 0.7502 (11, 1) 1.5205 (7, 6) 3.8432 (1, 13) 16.4929 (1, 7)

10 0.6411 (11, 1) 1.3539 (7, 6) 3.3082 (1, 13) 14.4822 (1, 7)

300 0.5 1.5818 (12, 1) 2.6075 (8, 7) 5.5897 (1, 11) 17.5799 (1, 7)

1 1.2506 (12, 1) 2.1962 (7, 7) 4.4197 (1, 11) 15.9629 (1, 7)

5 0.7497 (11, 1) 1.4542 (7, 6) 2.6503 (1, 11) 11.6756 (1, 6)

10 0.6407 (11, 1) 1.2741 (6, 6) 2.2885 (1, 11) 10.2494 (1, 6)

400 0.5 1.5814 (12, 1) 2.5450 (7, 7) 4.3565 (1, 10) 14.2583 (1, 7)

1 1.2503 (12, 1) 2.1291 (7, 7) 3.4379 (1, 10) 12.8183 (1, 6)

5 0.7495 (11, 1) 1.4077 (6, 6) 2.0760 (1, 10) 9.5365 (1, 6)

10 0.6406 (11, 1) 1.2343 (6, 6) 1.7946 (1, 9) 8.4624 (1, 5)

500 0.5 1.5812 (12, 1) 2.4964 (7, 7) 3.6487 (1, 9) 12.0645 (1, 6)

1 1.2502 (12, 1) 2.0899 (7, 7) 2.8813 (1, 9) 10.9962 (1, 6)

5 0.7494 (11, 1) 1.3785 (6, 6) 1.7358 (1, 9) 8.2607 (1, 5)

10 0.6405 (11, 1) 1.2111 (6, 6) 1.5029 (1, 9) 7.2204 (1, 5)

a

Numbers in parentheses indicate buckling mode (m, n).

Fig.3shows the effect of the R=h ratio and the volume-fraction

index k on the critical buckling load of FGM convex toroidal shell

segments reinforced by external stiffeners Fig.4illustrates the

ef-fect of the R=h ratio and stiffener positions of FGM convex toroidal

shell segments (R > 0) on the critical buckling load

From the aforementioned tables and figures, the following ob-servations may be made:

1 The critical buckling load of stiffened FGM toroidal shell segments is always higher than that of unstiffened ones Thus, stiffeners enhance the stability of a structure

2 When the volume-fraction index k is increased (raising the volume of the metal), the critical buckling load decreases because the modulus of a metal is lower than that of a ceramic

3 For a convex shell (R > 0), when R=h increases, the critical buckling load decreases Nevertheless, the critical buckling load of a concave shell (R < 0) has no stable tendency In par-ticular, when the R=h ratio increases from −200 to − 300, the axial critical buckling load increases, but it drops at the R=h ¼ − 300 to − 400 For the ratio of R=h ¼ − 400 to

−500, the axial critical buckling load increases again Thus,

Table 5 Critical Buckling Load of Stiffened FGM Concave Toroidal Shell

Segment (R < 0) under Axial and Lateral Pressure Load

p cr × 10 3 (MPa) q cr (MPa) Unstiffened

External stiffeners Unstiffened

External stiffeners

−200 0.5 0.2994 (2, 4) a 0.6414 (2, 4) 1.7268 (2, 5) 4.2839 (1, 2)

1 0.2367 (2, 4) 0.5586 (2, 4) 1.3557 (2, 5) 3.5641 (1, 2)

5 0.1420 (2, 4) 0.3265 (1, 2) 0.8336 (2, 5) 2.1314 (1, 2)

10 0.1227 (2, 4) 0.2749 (1, 2) 0.7305 (2, 5) 1.7840 (1, 2)

−300 0.5 0.3323 (2, 5) 0.8183 (1, 3) 0.9927 (1, 3) 2.4306 (1, 3)

1 0.2579 (2, 5) 0.7188 (1, 3) 0.7927 (1, 3) 2.2507 (1, 3)

5 0.1649 (2, 5) 0.5479 (1, 3) 0.4588 (1, 3) 1.8325 (1, 3)

10 0.1414 (1, 3) 0.5184 (1, 3) 0.3877 (1, 3) 1.6684 (1, 3)

−400 0.5 0.1605 (1, 3) 0.5932 (1, 3) 0.4401 (1, 3) 1.8090 (1, 3)

1 0.1252 (1, 3) 0.5312 (1, 3) 0.3433 (1, 3) 1.7292 (1, 3)

5 0.0787 (1, 3) 0.4339 (1, 3) 0.2158 (1, 3) 1.5094 (1, 3)

10 0.0698 (1, 3) 0.4203 (1, 3) 0.1915 (1, 3) 1.3906 (1, 3)

−500 0.5 0.4883 (1, 4) 0.9303 (1, 3) 0.7529 (1, 4) 2.7306 (1, 3)

1 0.3832 (1, 4) 0.8048 (1, 3) 0.5910 (1, 4) 2.4750 (1, 3)

5 0.2280 (1, 3) 0.5793 (1, 3) 0.3636 (1, 4) 1.9018 (1, 3)

10 0.1903 (1, 3) 0.5360 (1, 3) 0.3188 (1, 4) 1.7027 (1, 3)

a

Numbers in parentheses indicate buckling mode (m, n).

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

p cr

3 (MPa)

R/h

6

8

10

12

14

16

18

20

22

24

26

q cr

R/h

(a)

(b)

Fig 3 Effect of R=h ratio on buckling load of externally stiffened

FGM toroidal shell segments under (a) axial compression and (b) lateral

pressure

(a)

(b)

1 1.5 2 2.5 3

p cr

3 (MPa)

R/h

2 6 10 14 18 22 26

q cr

R/h

Fig 4 Effect of R=h ratio and stiffener positions on buckling load of FGM toroidal shell segments under (a) axial compression and (b) lateral pressure, with k ¼ 0.5

Table 6 Critical Buckling Load of Externally Stiffened FGM Toroidal Shell Segment under Lateral Pressure Load (L=h ¼ 200; a=h Changes)

q cr × 10 7N=m 2

a=h

k

20 11.3048 (1, 2) 10.4236 (1, 2) 9.0094 (1, 2) 6.6660 (1, 2)

30 6.6154 (1, 2) 5.4504 (1, 2) 4.1032 (1, 2) 2.9067 (1, 2)

40 4.1271 (1, 3) 3.6104 (1, 3) 2.9370 (1, 3) 2.1261 (1, 3)

50 3.2130 (1, 4) 2.7538 (1, 3) 2.0310 (1, 3) 1.4165 (1, 3)

100 1.3120 (1, 6) 1.0996 (1, 6) 0.8261 (1, 5) 0.5581 (1, 5)

concave toroidal shell segments need to be thoroughly

inves-tigated

4 The critical buckling load of the concave shell is lower than

that of the convex shell

Effect of a=h and L=a Ratios

Tables6and7show the effect of the a=h and L=a ratios on the

critical load of stiffened FGM toroidal shell segments with various

values of the volume-fraction index under a pressure load The geo-metric properties are as follows: R=h ¼ 500, h ¼ 0.002 m, d1¼

d2¼ 0.002 m, h1¼ h2¼ 0.005 m, n1¼ n2¼ 50, R > 0

As can be seen, the critical load of a stiffened FGM toroidal shell segment under a pressure load decreases when the toroidal shell equator curvature or length rises

Effect of a=R Ratio The critical buckling load of an externally stiffened FGM toroidal shell segment under axial load and hydrostatic pressure are inves-tigated for R > 0 and R < 0 in Tables 8and 9, respectively The geometric properties are L=a ¼ 2, L=h ¼ 200, h ¼ 0.002 m,

d1¼ d2¼ 0.002 m, h1¼ h2¼ 0.005 m, n1¼ n2¼ 50, with a=R varying

As can be seen, for a convex shell (R > 0), the decrease in the a=R ratio leads to an increase in the critical loads But for a concave shell (R < 0), the tendency is unstable The load-carrying capacity

of a concave shell is lower than that of a convex shell

Effect of Ratio L=h Table10indicates the critical buckling load of stiffened FGM toroi-dal shell segments under hydrostatic pressure with four configura-tions: L=h ¼ 100, 200, 300, and 400 When the L=h ratio rises, the critical buckling hydrostatic pressure will decrease (≈10 ÷ 36%) The geometric properties of this problem are R=h ¼ 400, a=h ¼

100, h ¼ 0.002 m, d1¼ d2¼ 0.002 m, h1¼ h2¼ 0.005 m, n1¼

n2¼ 50, and k ¼ 0.2

Internal Metal Stiffeners Case Now the same toroidal shell segment, made of functionally graded material such that the inner side is metal-rich and internal metal stiffeners are arranged on this side, is considered It is sufficient

to compare critical loads of both types of stiffened FGM toroidal shells with the volume-fraction index k ¼ 1

First, the critical buckling loads of internally stiffened FGM toroidal shell segments under axial and pressure loads with four different values of the R=h ratio [R > 0 (convex shell) and R < 0 (concave shell)] are given in Tables 11and 12, respectively The geometric properties are as follows: L=a ¼ 2, a=h ¼ 100, h ¼ 0.002 m, d1¼ d2¼ 0.002 m, h1¼ h2¼ 0.005 m, n1¼ n2¼ 50, and k ¼ 1 The corresponding results of the critical loads of exter-nally stiffened FGM shells are taken from Tables4and5 respec-tively for comparison

From Table11, it can be seen that for convex (R > 0) internally and externally stiffened FGM toroidal shells, both critical axial and pressure loads decrease when the R=h ratio increases The critical axial load of internally stiffened shells is greater than that of externally stiffened shells, but the critical pressure load

of internally stiffened shells is lower than that of externally stiff-ened ones

As can be seen for concave (R < 0) stiffened toroidal shells (Table12), the unstable tendency of both critical loads occurs when R=h changes The critical pressure load of internally stiffened shells

Table 7 Critical Buckling Load of Externally Stiffened FGM Toroidal

Shell Segment under Lateral Pressure Load (a=h ¼ 50; L=a Changes)

q cr × 10 7N=m 2

L=a

k

1 13.3021 (1, 5) 10.7771 (1, 4) 7.3409 (1, 4) 4.7992 (1, 4)

2 6.1931 (1, 4) 5.3438 (1, 4) 4.1184 (1, 4) 2.6718 (1, 3)

3 4.1465 (1, 4) 3.6982 (1, 4) 2.6835 (1, 3) 1.8075 (1, 3)

4 3.2310 (1, 4) 2.7538 (1, 3) 2.0309 (1, 3) 1.4165 (1, 3)

5 2.7129 (1, 3) 2.2205 (1, 3) 1.6733 (1, 3) 1.1908 (1, 3)

Table 8 Critical Buckling Load of Externally Stiffened FGM Toroidal

Shell Segment under Axial Load

p cr × 10 9 N=m 2

a=R

k

0.125 2.8355 (7, 7) 2.0327 (7, 7) 1.1773 (6, 6) 0.9434 (5, 6)

0.25 2.9699 (8, 7) 2.1291 (7, 7) 1.2343 (6, 6) 1.0054 (6, 6)

0.5 3.1688 (8, 6) 2.2917 (8, 7) 1.3539 (7, 6) 1.0898 (6, 6)

0.8 3.3321 (9, 6) 2.4483 (8, 6) 1.4563 (7, 6) 1.2016 (7, 6)

−0.125 1.9290 (1, 4) 1.5845 (1, 4) 0.9553 (1, 3) 0.7813 (1, 3)

−0.25 0.6571 (1, 3) 0.5312 (1, 3) 0.4203 (1, 3) 0.4044 (1, 3)

−0.5 0.7281 (2, 4) 0.5586 (2, 4) 0.2749 (1, 2) 0.2113 (1, 2)

−0.8 0.7892 (3, 5) 0.5981 (2, 3) 0.3111 (2, 3) 0.2476 (2, 3)

Table 9 Critical Buckling Load of Externally Stiffened FGM Toroidal

Shell under Hydrostatic Pressure

Hydrostatic pressure, q cr × 10 6N=m 2

a=R

k

0.125 8.8290 (1, 6) 7.0010 (1, 6) 4.5094 (1, 5) 4.0033 (1, 5)

0.25 12.8772 (1, 7) 10.0517 (1, 6) 6.4396 (1, 6) 5.5368 (1, 5)

0.5 20.9663 (1, 9) 16.1853 (1, 8) 9.6095 (4, 8) 7.7603 (3, 7)

0.8 26.3947 (5, 10) 19.2260 (5, 10) 11.0045 (5, 8) 8.8875 (5, 8)

−0.125 2.9319 (1, 4) 2.4083 (1, 4) 1.9364 (1, 4) 1.8781 (1, 4)

−0.25 1.7800 (1, 3) 1.4389 (1, 3) 1.1384 (1, 3) 1.0954 (1, 3)

−0.5 4.2435 (2, 4) 3.2552 (2, 4) 1.6024 (1, 2) 1.2316 (1, 2)

−0.8 6.7142 (3, 5) 5.2717 (3, 5) 2.9707 (2, 3) 2.3646 (2, 3)

Table 10 Critical Buckling Load of Stiffened FGM Toroidal Shell under Hydrostatic Pressure

Hydrostatic pressure, q cr × 10 6N=m 2

Configuration

L=h

Rings external and stringers external 19.9483 (1, 7) 12.8772 (1, 7) 10.2969 (1, 7) 8.9958 (1, 7)

is lower than that of externally stiffened ones (except for R=h ¼

−200) Consequently, the pressure-bearing capacity of externally

stiffened convex and concave FGM shells is greater than that of

internally stiffened ones

Second, Table13illustrates the critical buckling load of

stiff-ened FGM toroidal shells under hydrostatic pressure The

geomet-ric properties of this problem are as follows: L=a ¼ 2, L=h ¼ 200,

h ¼ 0.002 m, d1¼ d2¼ 0.002 m, h1¼ h2¼ 0.005 m, n1¼ n2¼

50, k ¼ 1, with a=R varying The corresponding critical loads of

externally stiffened shells are taken from Table9

It is observed that for convex stiffened FGM toroidal shells the

critical buckling hydrostatic pressure load increases when the a=R

ratio increases and the critical load of externally stiffened shells

is higher than that of internally stiffened ones It is impossible to

determine the behavior of concave stiffened FGM toroidal shells

under such conditions This problem will be investigated further

Conclusions

A formulation of the governing equations for the investigation of

the linear buckling of eccentrically stiffened functionally graded

toroidal shell segments subjected to axial compression, lateral

pressure, and hydrostatic pressure based on classical shell theory, the smeared stiffener technique, and the adjacent equilibrium cri-terion was presented Analytical solutions in the form of Fourier series are assumed to satisfy the simply supported boundary con-ditions to derive the closed-form relations of the buckling load The buckling behavior of FGM stiffened toroidal shell segments can be investigated using the method presented The effects of stiffeners, volume-fraction index, and dimensional parameters on the buckling

of FGM stiffened toroidal shell segments were observed, illustrat-ing specific characteristics of such a structure

This study has demonstrated the following points:

1 There exists a definite trend of variation in critical compres-sive, pressure, and hydrostatic pressure loads versus the var-iation in the volume-fraction index k and stiffeners for both convex shells (R > 0) and concave shells (R < 0)

2 For convex toroidal shell segments, there exists a definite trend toward variation in critical loads versus dimensional ratios, but for concave shells this trend is unstable The critical buckling load of concave toroidal shell segments is lower than that of convex ones

3 Stiffeners, volume-fraction index k, and dimensional ratios strongly affect the critical buckling load of toroidal shell segments

Acknowledgments

This research was funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant 107.02-2013.25

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Table 11 Critical Buckling Loads of Stiffened FGM Convex Toroidal

Shell Segment (R > 0) under Axial and Lateral Pressure Load

R=h

p cr × 10 3(MPa) q cr (MPa) Internally

stiffened

Externally stiffened

Internally stiffened

Externally stiffened

200 2.4213 (9, 1) 2.2917 (8, 7) 17.5997 (1, 8) 22.6557 (1, 8)

300 2.4192 (9, 1) 2.1962 (7, 7) 12.4664 (1, 7) 15.9629 (1, 7)

400 2.4181 (9, 1) 2.1291 (7, 7) 10.1513 (1, 7) 12.8183 (1, 6)

500 2.4175 (9, 1) 2.0899 (7, 7) 8.7075 (1, 6) 10.9962 (1, 6)

Table 12 Critical Buckling Loads of Stiffened FGM Concave Toroidal

Shell Segment (R < 0) under Axial and Lateral Pressure Load

R=h

p cr × 10 3(MPa) q cr (MPa) Internally

stiffened

Externally stiffened

Internally stiffened

Externally stiffened

−200 0.6499 (2, 4) 0.5586 (2, 4) 4.0091 (2, 4) 3.5641 (1, 2)

−300 0.6924 (1, 3) 0.7188 (1, 3) 1.8984 (1, 3) 2.2507 (1, 3)

−400 0.5419 (1, 3) 0.5312 (1, 3) 1.4858 (1, 3) 1.7292 (1, 3)

−500 0.8378 (1, 3) 0.8048 (1, 3) 2.2969 (1, 3) 2.4750 (1, 3)

Table 13 Critical Buckling Loads of Stiffened FGM Toroidal Shell under

Hydrostatic Pressure

Hydrostatic pressure, q cr × 10 6N=m 2

a=R

Externally stiffened

Internally stiffened

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