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Full length articleNonlinear thermal vibration of eccentrically stiffened Ceramic-FGM-Metal layer toroidal shell segments surrounded by elastic foundation Dinh Gia Ninha,n, Dao Huy Bichb

Full length article

Nonlinear thermal vibration of eccentrically stiffened Ceramic-FGM-Metal

layer toroidal shell segments surrounded by elastic foundation

Dinh Gia Ninha,n, Dao Huy Bichb

a

School of Mechanical Engineering, Hanoi University of Science and Technology, Hanoi, Vietnam

b Vietnam National University, Hanoi, Vietnam

a r t i c l e i n f o

Article history:

Received 11 February 2016

Received in revised form

15 March 2016

Accepted 15 March 2016

Keywords:

Toroidal shell segments

FGM core

FGM sandwich shell

Non-linear vibration

Thermal environment

Stiffeners

a b s t r a c t

The eccentrically stiffened Ceramic-FGM-Metal layer toroidal shell segments which applied for heat-resistant, lightweight structures in aerospace, mechanical, and medical industry and so forth are the new structures Thus, the nonlinear vibration of eccentrically stiffened (ES) Ceramic-FGM-Metal (C-FGM-M) layer toroidal shell segments surrounded by an elastic medium in thermal environment is investigated in this paper Based on the classical shell theory with the geometrical nonlinear in von Karman-Donnell sense, Stein and McElman assumption and the smeared stiffeners technique, the governing equations of motion of ES-C-FGM-M layer toroidal shell segments are derived The dynamical characteristics of shells

as natural frequencies, nonlinear frequency-amplitude relation, and nonlinear dynamic responses are considered Furthermore, the effects of characteristics of geometrical ratios, ceramic layer, elastic foun-dation, pre-loaded axial compression and temperature on the dynamical behavior of shells are studied

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1 Introduction

Toroidal shell has been applied in practicalfields as rocket fuel

tanks, fusion reactor vessels, diver's oxygen tanks, satellite support

structures, and underwater toroidal pressure hull There are many

studies with vibration and buckling problems in this structure

such as Jiang and Redekop[1], Buchanan and Liu[2], Wang et al

[3,4]and Tizzi[5] One of the special structures of toroidal shell is

toroidal shell segment Stein and McElman [6] carried out the

homogenous and isotropic toroidal shell segments about the

buckling problem Moreover, the initial post-buckling behavior of

toroidal shell segments subjected to several loading conditions

based on the basic of Koiter's general theory was performed by

Hutchinson[7] Parnell[8]gave a simple technique for the analysis

of shells of revolution applied to toroidal shell segments Recently,

there have had some new publications about toroidal shell

seg-ment structure Bich et al [9–11] has studied the buckling and

nonlinear buckling of functionally graded toroidal shell segment

under lateral pressure based on the classical thin shell theory, the

smeared stiffeners technique and the adjacent equilibrium

criter-ion Furthermore, the nonlinear buckling and post-buckling of

ES-FGM toroidal shell segments under torsional load based on the

analytical approach are investigated by Ninh et al.[12,13]

Today, sandwich FGM structures have received mentionable attention in structural applications The smooth and continuous change in material properties enables sandwich FGMs to avoid interface problems and unexpected thermal stress concentrations Furthermore, the sandwich structures also have the remarkable properties, especially thermal and sound insulation Sofiyev and Kuruoglu [14] investigated the parametric instability of simply-supported sandwich cylindrical shell with a FGM core under static and time dependent periodic axial compressive loads The gov-erning equations of sandwich cylindrical shell with an FGM core were derived to reduce the second order differential equation with the time-dependent periodic coefficient or Mathieu-type equation

by using the Garlerkin's method and the equation was solved by Bolotin's method Moreover, the dynamic instability of three-layered cylindrical shells containing a FG interlayer under static and time dependent periodic axial compressive loads was studied

by Sofiyev and Kuruoglu[15] The expressions for boundaries of unstable regions of three-layered cylindrical shell with an FG in-terlayer were found The bending response of the sandwich panel with FG skins using Fourier conduction equation and obtaining temperature distribution under thermal mechanical load based on higher order sandwich plate theory was studied by Sadighi et al [16] Taibi et al.[17]analyzed the deformation behavior of shear deformable FG sandwich plates resting on Pasternak foundation under thermo-mechanical loads The influences of shear stresses and rotary inertia on the vibration of FG coated sandwich cylind-rical shells resting on Pasternak elastic foundation based on the

Contents lists available atScienceDirect

journal homepage:www.elsevier.com/locate/tws Thin-Walled Structures

http://dx.doi.org/10.1016/j.tws.2016.03.018

0263-8231/& 2016 Elsevier Ltd All rights reserved.

n Corresponding author.

E-mail addresses: ninhdinhgia@gmail.com ,

ninh.dinhgia@hust.edu.vn (D.G Ninh).

modification of Donnell type equations of motion were examined

by Sofiyev et al.[18] The basic equations were reduced to an

al-gebraic equation of the sixth order and numerically solving this

algebraic equation gave the dimensionless fundamental frequency

Xia and Shen [19] dealt with the small and large-amplitude

vi-bration of compressively and thermally post-buckled sandwich

plates with FGM face sheets in thermal environment The

for-mulations were based on a higher-order shear deformation plate

theory and a general von-Karman-type equation that includes a

thermal effect and the equations of motion were solved by an

improved perturbation technique The refined hierarchical

kine-matics quasi-3D Ritz models for free vibration analysis of doubly

curved FGM shells and sandwich shells with FGM core were

in-vestigated by Fazzolari and Carrera[20] Sburlati[21]presented an

analytical solution in the framework of the elasticity theory to

describe the elastic bending response of axisymmetric circular

sandwich panels with functionally graded material cores and

homogeneous face-sheets The elastic solution was obtained using

a Plevako representation, which reduced the problem to the

search of potential functions satisfying linear fourth-order partial

differential equations The effect of continuously gradingfiber

or-ientation face sheets on free vibration of sandwich panels with

functionally graded using generalized power-law distribution was

investigated by Aragh and Yas [22] Woodward and Kashtalyan

[23]analyzed a three-dimensional elasticity for a sandwich panel

with stiffness of the core graded in the thickness direction, on the

basic of the developed 3D elasticity solution subjected to

dis-tributed and concentrated loadings

The vibration problem about the shell structures have been

attracted a large number of studies The free vibration analysis of

function analysis of FGM cylindrical shell with holes was studied

with the variational equation and the unified displacement

mode-shape function of the shells with various boundary conditions by

Cao and Wang[24] Sofiyev and Kuruoglu[25]carried out the

vi-bration and stability of FGM orthotropic cylindrical shells under

external pressures using the shear deformation shell theory The

basic equations of shear deformable FG shell were derived using

Donnell shell theory and solved using the Galerkin method Bich

et al [26] investigated the characteristics of free vibration and

nonlinear responses, using the governing equations of motion of

eccentrically stiffened functionally graded cylindrical panels with

geometrically imperfections based on the classical shell theory

with the geometrical nonlinearity in von Karman-Donnell sense

and smeared stiffeners technique Moreover, Bich and Nguyen[27]

presented the study of the nonlinear vibration of a functionally

graded cylindrical shell subjected to axial and transverse

me-chanical loads based on improved Donnell equations The

non-linear forced vibration of infinitely long FGM cylindrical shell using

the Lagrangian theory and multiple scale method was presented

by Du et al.[28] The energy approach was applied to derive the

reduced low-dimensional nonlinear ordinary differential

equa-tions of motion Sheng et al.[29]investigated the nonlinear

vi-brations of FGM cylindrical shell based on Hamilton's principle,

von-Karman nonlinear theory and first-order shear deformation

theory The vibration of FGM cylindrical shells under various

boundary conditions with the strain displacement relations form

Love's shell theory were studied by Pradhan et al.[30] Based on

the Rayleigh method, the governing equations were derived and

the natural frequencies were investigated depending on the

con-stituent volume fractions and boundary condition The

strains-displacement relations from Love's shell theory and energy

func-tional with the Rayleigh–Ritz method to solve the governing

equation were used Strozzi and Pellicano[31]analyzed the

non-linear vibrations of FGM circular cylindrical shells using the

San-ders-Koiter theory The displacement fields were expanded by

means of double mixed series based on Chebyshev orthogonal

polynomials for the longitudinal variable and harmonic functions for the circumferential variable Sofiyev[32]studied the dynamic behavior of FGM structures such as dynamic response of an FGM cylindrical shell subjected to combined action of the axial tension, internal compressive load and ring-shaped compressive pressure with constant velocity based on the von Karman-Donnell type nonlinear kinematics using the superposition and Galerkin methods The nonlinear vibration of simply supported FGM cy-lindrical shells with embedded piezoelectric layers using a semi analytical approach was addressed by Jafari et al [33] The gov-erning differential equations of motion of the FG cylindrical shell were derived using the Lagrange equations under the assumption

of the Donnell's nonlinear shallow-shell theory Firooz and Seyed [34] investigated the nonlinear free vibration of prestressed cir-cular cylindrical shells placed on Pasternak foundation using the nonlinear Sanders-Koiter shell theory to derive strain-displace-ment The governing equations in linear state were solved by the Rayleigh–Ritz procedure Based on strain-displacement relations from the Love's shell theory and the eigenvalue governing equa-tion using Rayleigh–Ritz method, Loy et al.[35]gave the study at vibration filed of functionally cylindrical shells The dynamic be-havior of FGM truncated conical shells subjected to asymmetric internal ring-shaped moving loads using Hamilton's principle based on thefirst order shear deformation theory was studied by Malekzadeh and Daraie [36] The vibration of FGM cylindrical shells on elastic foundations using wave propagation to solve dy-namical equations were analyzed by Abdul et al [37] The shell was assumed to be simply supported with movable edges and the equations of motion were reduced using Galerkin method to a system of infinite nonlinear ordinary differential equations with quadratic and cubic nonlinearities Haddadpour et al [38] per-formed free vibration analysis of functionally graded cylindrical shells using the equations of motion based on Love's shell theory and the von Karman-Donnell type of kinematic nonlinearity for the thermal effects investigated by specifying arbitrary high tem-perature on the outer surface and ambient temtem-perature on the inner surface of the cylindrical Based on the first-order shear deformation theory of shells, the influences of centrifugal and Coriolis forces in combination with the other geometrical and material parameters on the free vibration behavior of rotating functionally graded truncated conical shells subjected to different boundary conditions were investigated by Malekzadeh and Hey-darpour[39]

Noda[40], Praveen et al.[41]firstly realized the heat-resistant FGM structures and studied material properties dependent on temperature in thermo-elastic analyses Sheng and Wang [42] researched the nonlinear response of functionally graded cylind-rical shells under mechanical and thermal loads using con Karman nonlinear theory The coupled nonlinear partial differential equa-tions are discretized based on a series expansion of linear modes and a multiterm Galerkin's method Furthermore, Shen[43]took into account the nonlinear vibration of shear deformable FGM cylindrical shells offinite length embedded in a large outer elastic medium and in thermal environments The motion equations were based on a higher order shear deformation shell theory that in-cluded shell-foundation interaction The transient thermoelastic analysis of functionally graded cylindrical shells under moving boundary pressure and heat flux was presented by Malekzadeh and Heydarpour [44] The hyperbolic heat conduction equations were used to include the influence of finite heat wave speed The resulting system of differential equations was solved using New-mark's time integration scheme in the temporal domain Kiani

et al.[45]investigated the thermoelastic dynamic behavior of an FGM doubly curved panel under the action of thermal and me-chanical loads based on thefirst-order shear deformation theory

of modified Sanders assumptions applying the Laplace

transformation Furthermore, Malekzadeh et al.[46,47]studied the

influences of thermal environment on the free vibration

char-acteristics of functionally graded shells and panels based on the

first-order shear deformation theory By taking into account both

the temperature dependence of material properties, which were

assumed to be graded in the thickness direction, and the initial

thermal stresses, the equations of motion and the related

bound-ary conditions were derived using Hamilton's principle

To the best of the authors' knowledge, there has not been any

study to the nonlinear thermal vibration of eccentrically stiffened

Ceramic-FGM-Metal layer toroidal shell segments surrounded by

an elastic foundation

In the present paper, the nonlinear vibration of eccentrically

stiffened FGM sandwich toroidal shell segments on elastic medium

in thermal environment are investigated Based on the classical

shell theory with the nonlinear strain-displacement relation of large

deflection, the Galerkin method, Stein and McElman assumption,

Volmir's assumption and the numerical method using fourth-order

Runge-Kutta are performed for dynamic analysis of shells to give

expression of natural frequencies and nonlinear dynamic responses

2 Governing equations 2.1 Ceramic-FGM-Metal layer (C-FGM-M) The coordinate system (x1, x2, z) is located on the middle sur-face of the shell, x1and x2is the axial and circumferential direc-tions, respectively and z is the normal to the two axes Consider the sandwich toroidal shell segment of thickness h, length L, which

is formed by rotation of a plane circular arc of radius R about an axis in the plane of the curve as shown inFig 1in a coordinate system (x1, x2, z) consists of ceramic, FGM and metal described in Fig 1 The thickness of the shell is defined in a coordinate system (x2, z) in Fig 2 The inner layer (z¼h/2) and the outer layer (z¼ h/2) are isotropic homogenous with ceramic and metal, re-spectively Suppose that the material composition of the shell varies smoothly along the thickness in such a way that the inner surface is ceramic, the outer surface is metal and the core is FGM The thickness of the shell, ceramic-rich and metal rich are h, hc,

hm, respectively Thus, the thickness of FGM core is hhchm The subscripts m and c are referred to the metal and ceramic

constituents respectively Denote Vmand Vcas volume - fractions

of metal and ceramic phases respectively, where VmþVc¼1

Ac-cording to the mentioned law, the volume fraction is expressed as

⎧

⎨

⎪

⎪

⎪⎪

⎩

⎪

⎪

⎪

⎪

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

≥

( )

V z z h h

h h h

h

h z h h

k

0,

/2

,

1,

1

k

According to the mentioned law, the Young modulus and

thermal expansion coefficient of C-FGM-M shell are expressed of

the form

( ) = ( ) + ( ) = + ( − ) ( )

E z E V z m m E V z c c E m E c E V z , m c

α( ) =z α m m V +α c c V =α m+ (α c−α m) ( )V z , c

the Poisson's ratioνis assumed to be constant

2.2 Constitutive relations and governing equations

For the middle surface of a toroidal shell segment, from the

Fig 1we have:

φ

r a R 1 sin ,

where a is the equator radius andφis 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

angleφis approximately equal toπ/2, thussinφ ≈1;cosφ ≈0 and

r¼a[6] The form of governing equation is simplified by putting:

dx1 Rd , dx2 ad

The radius of arc R is positive with convex toroidal shell

seg-ment and negative with concave toroidal shell segseg-ment The shell

is surrounded by an elastic foundation with Winkler foundation

modulus K1 (N/m3) and the shear layer foundation stiffness of

Pasternak model K2(N/m)

Suppose the eccentrically stiffened C-FGM-M (ES-C-FGM-M)

toroidal shell segment is reinforced by string and ring stiffeners In

order to provide continuity within the shell and stiffeners and

easier manufacture, the homogeneous stiffeners can be used

Be-cause pure ceramic ones are brittleness we used metal stiffener

and put them at metal side of the shell With the law indicated in

(1) the outer surface is metal, so the external metal stiffeners are

put at outer side of the shell.Fig 3depicts the geometry and

co-ordinate system of stiffened C-FGM-M shell on elastic foundation

The von Karman type nonlinear kinematic relation for the

strain component across the shell thickness at a distance z from

the middle surface are of the form[48]:

ε1=ε1 −z χ1; ε2=ε2 −z χ2; γ12=γ120−2z χ12, ( )2

where ε1 and ε2 are normal strains, γ120 is the shear strain at the middle surface of the shell and χ1and χ2are the curvatures and χ12

is the twist

According to the classical shell theory the strains at the middle surface and curvatures are related to the displacement compo-nents u,v, w in the x1, x2, z coordinate directions as[48]:

⎛

⎝

⎠

⎝

⎠

⎟

= ∂

∂

∂

∂

∂

= ∂

∂

∂

∂

∂

∂

∂

∂

u x

w R

w x

v x

w a

w x u

x

v x

w x

w x

w x

w x

w

x x

1

1

3

1

2

2

2

12 0

2

1 2 2

2 12 2

1 2

Hooke's law for toroidal shell segment is defined as

σ

ν ε νε

α

ν Δ Δ σ

ν ε νε

α

ν Δ σ

ν γ

= ( )

( ) ( )

= ( )

( ) ( )

−

T T T T

T

E z

1

1

sh

sh

sh

and for metal stiffeners

σ1st=E m ε1−E m m α Δ T; σ2st=E m ε2−E m m α Δ T

where T0 is initial value of temperature at which the shell is thermal stress free

By integrating the stress–strain equations and their moments through the thickness of the shell and using the smeared stiffeners technique; the expressions for force and moment resultants of a C-FGM-M toroidal shell segment can be obtained as[13,48]:

⎛

⎝

⎞

⎠

⎛

⎝

⎠

⎟

N A E A

, ,

m

a a

m

a a

1

1 12 20 11 1 1 12 2

2 12 1 0 22 2

2

2 12 1 22 2 2 0

Fig 2 The material characteristic of C-FGM-M.

Fig 3 Geometry and coordinate system of a stiffened C-FGM-M toroidal shell segment on elastic foundation (a) stringer stiffeners; (b) ring stiffeners.

⎝

⎠

⎟

⎛

⎝

⎠

⎟

s

, ,

m

m

1 11 1 1

0

12 2 0

1 1 12 2

2 2

12 66 12

0

66 12

where A ij,B D ij, ij (i, j¼1, 2, 6) are extensional, coupling and

bending stiffnesses of the shell without stiffeners

ν

ν

ν

ν

ν

ν

where

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

⎡

⎣

⎛

⎝

⎞

⎠

⎤

⎦

⎡

⎣

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

⎤

⎦

∫

∫

∫

+

−

( + )( + )( − − )

+

( + )( + )( + )( − − ) +

−

−

−

k

k

E

k

h

h

E

E h

E

1 2

2

2

3

h

h

m cm c cm c m

h

h cm c cm c cm

c c m

cm

c m

h

cm

c m c c

c c

m

1

/2

/2

2

3

/2

3

3

in which Ecm¼EcEm

C E A z

s C

E A z s

1

2

A1 h d1 1, A2 h d2 2, 8

I d h A z I d h A z

1 1 1

3

1 12 2 2 2

3

2 22

and

**

**

Φ

=

*

*

−

− −

−

−

− −

−

− −

−

9

d

d

d

1

h

a h h h

m m

h

h

m

h h

h

h h

h

m m

/2

1 /2 1 /2

2

2 /2 2

/2

/2 /2

1

1 /2 1

2 /2 2 /2

IfΔT¼const

**

Φ

ν

=

+

( − − ) +

P T

k

k

k

d h

d h

1

1

a

m m c c c m m c m cm c m

cm m c m cm cm c m

a 1 1 m m a m m

1

2 2 2

in which

The spacings of the stringer and ring stiffeners are denoted by

s1and s2respectively The quantities A1, A2are the cross section

areas of the stiffeners and I1, I2, z1, z2are the second moments of

cross section areas and eccentricities of the stiffeners with respect

to the middle surface of the shell respectively

The nonlinear equilibrium equations of a toroidal shell segment under a lateral pressure q, an axial compression p and surrounded

by an elastic foundation based on the classical shell theory are given by[48]:

⎛

⎝

⎠

⎟⎟

( )

ρ

ρ

∂

∂ +

∂

∂ =

∂

∂

∂

∂ +

∂

∂ =

∂

∂

∂

∂ + ∂

∂ ∂ +

∂

∂ + ∂

∂

∂ ∂ +

∂

∂

− ∂

∂ +

∂ +∂

∂

= ∂

∂

N x N x u t N

x N x v t M

x

M

x x M

w

w

w

w x

N R N

w x w x

w t

w t

, ,

1 1 12

2 2 12

1 2

2 2 2

1 1

2 12

1 2

2 2 2 1 2 1 12 2

1 2 2 2 2 2 1 1

2

2 1 2 2 1 2

where K1(N/m3) is linear stiffness of foundation, K2(N/m) is the shear modulus of the sub-grade,ε is damping coefficient and

⎛

⎝

⎞

⎠

ρ =ρ +ρ + ρ ( − − ) ρ

k

A s

A s

m cm c

m

1

1

1 2

2

By substitutingEq (3)into(5)and(6)and then intoEq (11), the term of displacement components are expressed as follows:

ρ

ρ

( ) + ( ) + ( ) + ( ) = ∂

∂

( ) + ( ) + ( ) + ( ) = ∂

∂

∂

∂

Y u Y v Y w P w u

t

Y u Y v Y w P w v

t

Y u Y v Y w P w Q u w R v w ph w

x q

w t

w t

,

,

13

2

2

2

2

2

1

1 2

where the linear operators Y ij( )(i j, = 1, 2, 3)and the nonlinear operators P i( )( =i 1, 2, 3), Q3 and R3 are demonstrated in Ap-pendix A

Eq (13) are the nonlinear governing equations used to in-vestigate the nonlinear dynamical responses of ES-C-FGM-M tor-oidal shell segments surrounded by elastic foundation in thermal environment

3 Nonlinear analysis

In the present paper, the simply supported boundary condi-tions are considered

w 0, v 0, M1 0 at x1 0and x1 L 14

The approximate solutions of the system ofEq (13)satisfying the conditions (14) can be expressed as:

π

( )

u U t m x

L

nx

a v V t

m x L

nx a

w W t m x

L

nx a

mn

where Umn, Vmn, Wmn are the time depending amplitudes of vi-bration, m and n are numbers of half wave in axial direction and wave in circumferential direction, respectively

SubstitutingEq (15)intoEq (13)and then applying the Ga-lerkin method leads to:

⎣⎢

⎤

⎦⎥

( )

**

ρ

ρ

π

δ δ

π

16

dt

dt

n V W

d W

dt

dW dt

,

,

2 2

2 2

2 2

2

1

2

where yij; niare given inAppendix B

Otherwise, the Volmir's assumption[49] can be used in the

dynamic analysis Taking the inertia forces ρ (1d U dt2 / 2) →0 and

ρ (1d V dt2 / 2) →0into consideration because ofu< <w v, < <w,Eq

(16)can be rewritten as follows:

⎡

⎣⎢

⎤

⎦⎥

( )

**

π

δ δ π

*

17

L

mn q

d W

dt

dW

dt

0, 0,

2

mn mn mn mn

mn mn mn mn

2 2

2

1

2

Solving the first and the second obtained equations with

re-spect to Umnand Vmnand then substituting the results into the

third equation yields

⎡

⎣⎢

⎤

⎦⎥

δ δ

( )

d W

dt

dW

dt g W g W g W

2

18

1

2

2 3 3

1 2

2

where

( )

π

−

−

m L

,

,

.

1 33 31 12 23 22 13

11 22 12 21

32 21 13 11 23

11 22 12 21

2 2 2

2 3 31 12 2 22 1

11 22 12 21

32 12 1 11 2

11 22 12 21

5 12 23 22 13

11 22 12 21

6 21 13 11 23

11 22 12 21

3 4 5 12 2 22 1

11 22 12 21

6 21 1 11 2

11 22 12 21

3.1 Natural frequencies

Taking linear parts of the set ofEq (16)and putting p¼0, q¼0

andε¼0 the natural frequencies of the shell can be directly

cal-culated by solving determinant

ρ ω

ρ ω

ρ ω

+

+

+

=

( )

0

20

11 1

2

2 23

2

Solving Eq.(20)leads to three angular frequencies of the

tor-oidal shell segment in the axial, circumferential and radial

direc-tions, and the smallest one is being considered

In other hand, the fundamental frequencies of the shell can be

approximately determined by explicit expression inEq (18)

ω ρ

=

( )

g

21

1

.SolvingEq (20)leads to exact solution but implicit expression whileEq (21)performs approximate frequencies but explicit ex-pression and simpler

3.2 Frequency amplitude curve Consider nonlinear vibration of a toroidal shell segment under

a uniformly distributed transverse load q=QsinΩ t including thermal effects Assuming pre-loaded compression p, Eq (18) takes the form

⎡

⎣⎢

⎤

⎦⎥

δ δ

( )

d W dt

dW

dt g W g W g W

2 4

22

1 2

2 3 3

1 2 2

Eq (22)can be rewritten as

( )

23

d W dt

dW

mn mn

mn mn mn mn

2 2

where ω =

ρ mn g

1

is the fundamental frequency of linear vi-bration of the toroidal shell segment and H¼g2/g1, K¼g3/g1,

= δ δ

π ρ

mn

4 1 2

2 , = δ δ ⎡⎣ (Φ +Φ*) + (Φ +Φ**)⎤⎦

π ρ

G

For seeking amplitude-frequency relation of nonlinear vibra-tion we substitute

Ω

intoEq (23)to give

K A t F t G

mn

2 3 3

Integrating over a quarter of vibration period

π Ω

Ysin tdt 0,

0 /2

the frequency-amplitude relation of nonlinear vibration is ob-tained

⎛

⎝

⎞

⎠

Ω ε

( )

HA K A F

A

G A

4

3

3 4

4

26

mn

By denoting α2=Ω ω2/ mn2 Eq (26)is rewritten as

( )

HA K A F

A

G A

4

3

3 4

4

27

For the nonlinear vibration of the shell without damping (ε = 0), this relation has of the form

α

( )

HA K A F

A

G A

3

3 4

4

28

If F¼0, G¼0 i.e no force excitation and thermal effect acting on the shell, the frequency-amplitude relation of the free nonlinear vibration without damping is obtained

⎛

⎝

⎞

⎠

ω ω

π

( )

HA K

3

3

where ω is the nonlinear vibration frequency

3.3 Nonlinear vibration responses

Consider an eccentrically stiffened functionally graded toroidal

shell segment acted on by a uniformly distributed transverse load

Ω

( ) =

q t Q sin t and a pre-loaded axial compression p, the set of

motionEq (16)has of the form

⎡

⎣⎢

⎤

⎦⎥

( )

**

ρ ρ π

δ δ

π

*

30

dt

dt

d W

dt

dW

dt

, ,

4

2

2 2

2 2

2 2

2

1

2

And the motion Eq (18)by the use of Volmir's assumption

becomes

⎡

⎣⎢

⎤

⎦⎥

δ δ

( )

d W

dt

dW

dt g W g W g W

2

4

31

1

2

2 3 3

1 2

2

Using the fourth-order Runge-Kutta method intoEq (30)orEq

(31)combined with initial conditions, the nonlinear vibration

re-sponses of ES-FGM toroidal shell segment can be investigated

4 Results and discussion

4.1 Validation

Up to now, to the best of the authors' knowledge, there is no

publication about nonlinear vibration of ES-C-FGM-M toroidal

shell segment, that is reason to compare the results in this paper

with homogenous and FGM cylindrical shell (i.e a toroidal shell

segment with R-1 and hc¼hm¼0)

Firstly, the results of natural frequencies in present will be

compared with results for the un-stiffened isotropic cylindrical

shell studied by Lam and Loy[50], Li[51]and Shen[43]and can be

seen inTable 1

As can be seen inTable 1that good agreements are obtained in

this comparison Moreover the frequencies calculated byEq (20)

(the full order equation ODE) andEq (21)(the Volmir's

assump-tion) are quite close to each other

Secondly, the natural frequencies of FGM cylindrical shell

illu-strated inTable 2are computed and compared with the results of

Loy et al.[35]using Rayleigh–Ritz method and Shen[43]with two

kinds of micromechanics models: Voigt model and Mori–Tanaka

model based on a higher order shear deformation shell theory A

FGM cylindrical shell is made of stainless steel and nickel material

in initial temperature T0¼300 K is considered with the following

material properties

ENi¼205.09 GPa; υNi¼0.31; ρNi¼8900 kg/m3; ESS¼207.7877 GPa;υ¼0.32; ρNi¼8166 kg/m3

As can be seen, a very good agreement is obtained in the com-parison with the results of Ref.[35], but there is a little difference with those of Ref.[43]because the author used other theories

In the following sections, the materials consist of Aluminum and Alumina with E m=70×109N m/ 2; ρ = 2702 m kg/m3;

α m=23×10−60C−1; E c=380×109N m/ 2; ρ = 3800 c kg/m3;

α = c 5.4×10−60C−1 and Poisson's ratio is chosen to be 0.3 The elastic foundation parameters are taken as K1¼2.5  108N/m3,

K2¼5  105N/m with Pasternak foundation The parameters

n1¼50 and n2¼50 are the number of stringer and ring stiffeners, respectively

4.2 The fundamental frequencies The natural frequencies of ES-C-FGM-M toroidal shell segment

in three cases usingEq (20)are illustrated inTable 3 The datum of problem: h¼0.01 m; hc¼0.3 h; hm¼0.1 h; a¼300 h; R¼500 h;

L¼2a; d1¼d2¼h/2; h1¼h2¼h/2; n1¼n2¼50; k¼1 It can be seen that the natural frequencies of the shell on Pasternak foundation are the highest while the natural frequencies with pre-loaded axial compression (p¼1 GPa) are the lowest It means that when the shell is subjected to pre-loaded axial compression, the natural frequencies will lessen

4.3 Frequency-amplitude curve For investigating the dynamic responses, we can use with ar-bitrary mode (m, n), for instance mode number (m, n)¼(1, 7) The frequency-amplitude curve of nonlinear free vibration of the shell and the effects of pre-loaded axial compression, elastic foundation are indicated inFig 4 It can be observed that the lowest frequency will increase when the shell is on elastic foundation Whereas, it will decrease when the shell bears the pre-loaded axial com-pression without elastic foundation

The effect of amplitude of external force on the frequency-amplitude curve in case of forced vibration is illustrated inFig 5 The line 1 is corresponding to the free vibration case (F¼0, p¼0)

of the shell without elastic foundation The lines 2 and 3 corre-spond to the forced vibration cases of the shell with pre-loaded axial compression (p¼2.5 GPa) and without elastic foundation under excited forces with F¼5  105

and F¼8  105

, respectively Finally, the lines 4 and 5 correspond to the free vibration and the forced vibration cases of the shell on Pasternak foundation, re-spectively It can be seen, the frequency-amplitude curve trend further from the curve of the free vibration case when the am-plitude of external force increases The frequency-amam-plitude curves of the shell on elastic foundation move ahead in the in-creasing frequency direction in comparison with those curves of the shell without elastic foundation

4.4 Nonlinear vibration responses The comparison of the nonlinear response of the shell calcu-lated by the approximateEq (31)(Volmir's assumption) and the full order systemEq (30)is shown inFig 6

FromTables 1and2andFig 6, we conclude that the Volmir's assumption can be used to investigate nonlinear dynamical ana-lysis with an acceptable accuracy

In the next sections, the full order systemEq (30)is used to investigate nonlinear vibration responses As following the effects

of the characteristics of functionally graded materials, the pre-loaded axial compression, the dimensional ratios, the elastic foundation and thermal loads on the nonlinear dynamic responses

Table 1

Comparison of dimensionless frequencies ω¯ =Ω(h/π) 2 1 ( + )ν ρ/Efor an isotropic

cylindrical shell (a/L¼2; h/a¼0.06, E¼210 GPa, υ¼0.3, ρ = 7850 kg/m 3

).

(m, n) Lam and Loy

[50]

Li [51] Shen [43] Present

( Eq 21 )

Present ( Eq 20 )

of the ES-FGM sandwich toroidal shell segments are analyzed.

The effects of material and geometric parameters, elastic

foundation, thermal environment and the beating vibration

phe-nomenon on the non-linear vibration of FGM sandwich toroidal

shell are considered inFigs 7–21

Figs 7and8depict the effect of R/h ratio on nonlinear vibration

of convex and concave stiffened FGM sandwich toroidal shell

segment, respectively It can be seen that when increasing R/h

ratio, the amplitudes of nonlinear vibration of both stiffened

C-FGM-M shell also increase and the frequency does not modify much Furthermore, the amplitudes of nonlinear vibration of convex ES-FGM sandwich shell are smaller than ones of concave ES-FGM sandwich shell

The effect of L/R ratio is described inFigs 9and10 It can be seen that when L/R ratios increase, the amplitudes of nonlinear vibration of convex ES-FGM core shell also go up while those of concave ES-C-FGM-M shell decrease It means that the amplitudes

Table 2

Comparisons of natural frequencies f=2Ω π(Hz) for FGM cylindrical shells (L/a¼20, a/h¼20, h¼0.05 m, T¼300 K).

SUS304/Ni

Ni/SUS304

Table 3

The fundamental frequencies (s 1 ) using Eq (20) in various cases of ES-C-FGM-M

toroidal shell segment.

Cases ω 1 (1, 1) * ω 2 (1, 2) ω 3 (1, 3) ω 4 (2, 1) ω 5 (2, 2)

Natural frequencies 2362.216 2214.457 2096.708 2805.679 2727.048

Natural frequencies of

shell on Pasternak

foundation

2932.162 3037.544 3116.408 3846.529 3788.811

Natural frequencies of

shell with

pre-loa-ded axial

compres-sion (p¼1 GPa)

2351.385 2201.753 2081.748 2750.674 2670.454

n The numbers in brackets indicate the vibration buckling mode (m, n).

Fig 4 Effects of elastic foundation and pre-loaded axial compression on

fre-quency-amplitude curve of ES-C-FGM-M toroidal shell segment in case of free

vi-bration and no damping.

Fig 5 The frequency-amplitude curve in case of forced vibration.

Fig 6 The comparison of the nonlinear dynamical response on Pasternak foun-dation calculated by Eq (31) (Volmir's assumption) and Eq (30) (the full order equation system).

Fig 7 Effect of R/h ratio on nonlinear vibration response of convex ES-FGM

sandwich toroidal shell segment on elastic medium.

Fig 8 Effect of R/h ratio on nonlinear vibration response of concave ES-FGM

sandwich toroidal shell segment on elastic medium.

Fig 9 Effect of L/R ratio on nonlinear vibration response of convex ES-FGM

sandwich toroidal shell segment on elastic medium.

Fig 10 Effect of L/R ratio on nonlinear vibration response of concave ES-FGM

sandwich toroidal shell segment on elastic medium.

Fig 11 Effect of L/a on nonlinear vibration response of convex ES-FGM sandwich toroidal shell segment on elastic medium.

Fig 12 Effect of L/a on nonlinear vibration response of concave ES-FGM sandwich toroidal shell segment on elastic medium.

Fig 13 Effect of thickness of ceramic and metal layer on nonlinear vibration re-sponse of convex ES-FGM sandwich toroidal shell segment on elastic medium.

Fig 14 Effect of thickness of ceramic and metal layer on nonlinear vibration re-sponse of concave ES-FGM sandwich toroidal shell segment on elastic medium.

of the more convex shells are greater than that of the less convex ones whereas this feature of concave shells is completely on the contrary On the other hand, the amplitudes of nonlinear vibration

of convex shell are lower than ones of concave shell

As can be observed inFigs 11and12, the influence of ratio L/a

on the nonlinear response of the shell is similar as one of ratio L/R

In addition, the amplitude of nonlinear vibration response of ES-FGM sandwich concave shell is unequal

Based on Figs 13 and 14, as can be seen, the amplitude of nonlinear vibration of both ES-FGM core shells decrease when the thickness of ceramic layer increases It means that the sandwich structures will be stiffer than FGM structures with the same geo-metry parameters Thus, the amplitudes of nonlinear vibration of sandwich structure will be lower than those of FGM structure

Fig 15 Effect of volume-fraction k on nonlinear vibration response of ES-FGM

sandwich toroidal shell segment on elastic medium.

Fig 16 Effect of elastic foundation on nonlinear vibration response of ES-FGM

sandwich toroidal shell segment.

Fig 17 Effect of pre-loaded axial compression on nonlinear vibration response of

ES-FGM sandwich toroidal shell segment.

Fig 18 Effect of thermal environment on nonlinear vibration response of ES-FGM

Fig 19 Effect of thermal environment on nonlinear vibration response of ES-FGM sandwich concave toroidal shell segment.

Fig 20 Nonlinear response of FGM convex toroidal shell segment on elastic medium with k¼1(ω = 3089.0091 rad/s).

Fig 21 Nonlinear response of FGM concave toroidal shell segment on elastic medium with k¼1 (ω = 2701.6441 rad/s).