Thin-Walled Structures 104 (2016) 198–210

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Nonlinear thermal vibration of eccentrically stiffened Ceramic-FGM-Metallayer toroidal shell segments surrounded by elastic foundation

Dinh Gia Ninh a,n, Dao Huy Bich b

a School of Mechanical Engineering, Hanoi University of Science and Technology, Hanoi, Vietnamb Vietnam National University, Hanoi, Vietnam

a r t i c l e i n f o

Article history:Received 11 February 2016Received in revised form15 March 2016Accepted 15 March 2016

Keywords:Toroidal shell segmentsFGM coreFGM sandwich shellNon-linear vibrationThermal environmentStiffeners

x.doi.org/10.1016/j.tws.2016.03.01831/& 2016 Elsevier Ltd. All rights reserved.

esponding author.ail addresses: ninhdinhgia@gmail.com,hgia@hust.edu.vn (D.G. Ninh).

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 newstructures. 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 inthis paper. Based on the classical shell theory with the geometrical nonlinear in von Karman-Donnellsense, Stein and McElman assumption and the smeared stiffeners technique, the governing equations ofmotion of ES-C-FGM-M layer toroidal shell segments are derived. The dynamical characteristics of shellsas natural frequencies, nonlinear frequency-amplitude relation, and nonlinear dynamic responses areconsidered. 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.

& 2016 Elsevier Ltd. All rights reserved.

1. Introduction

Toroidal shell has been applied in practical fields as rocket fueltanks, fusion reactor vessels, diver's oxygen tanks, satellite supportstructures, and underwater toroidal pressure hull. There are manystudies with vibration and buckling problems in this structuresuch 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 istoroidal shell segment. Stein and McElman [6] carried out thehomogenous and isotropic toroidal shell segments about thebuckling problem. Moreover, the initial post-buckling behavior oftoroidal shell segments subjected to several loading conditionsbased on the basic of Koiter's general theory was performed byHutchinson [7]. Parnell [8] gave a simple technique for the analysisof 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 andnonlinear buckling of functionally graded toroidal shell segmentunder lateral pressure based on the classical thin shell theory, thesmeared 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 theanalytical approach are investigated by Ninh et al. [12,13].

Today, sandwich FGM structures have received mentionableattention in structural applications. The smooth and continuouschange in material properties enables sandwich FGMs to avoidinterface problems and unexpected thermal stress concentrations.Furthermore, the sandwich structures also have the remarkableproperties, especially thermal and sound insulation. Sofiyev andKuruoglu [14] investigated the parametric instability of simply-supported sandwich cylindrical shell with a FGM core under staticand time dependent periodic axial compressive loads. The gov-erning equations of sandwich cylindrical shell with an FGM corewere derived to reduce the second order differential equation withthe time-dependent periodic coefficient or Mathieu-type equationby using the Garlerkin's method and the equation was solved byBolotin's method. Moreover, the dynamic instability of three-layered cylindrical shells containing a FG interlayer under staticand time dependent periodic axial compressive loads was studiedby Sofiyev and Kuruoglu [15]. The expressions for boundaries ofunstable regions of three-layered cylindrical shell with an FG in-terlayer were found. The bending response of the sandwich panelwith FG skins using Fourier conduction equation and obtainingtemperature distribution under thermal mechanical load based onhigher order sandwich plate theory was studied by Sadighi et al.[16]. Taibi et al. [17] analyzed the deformation behavior of sheardeformable FG sandwich plates resting on Pasternak foundationunder thermo-mechanical loads. The influences of shear stressesand rotary inertia on the vibration of FG coated sandwich cylind-rical shells resting on Pasternak elastic foundation based on the

D.G. Ninh, D.H. Bich / Thin-Walled Structures 104 (2016) 198–210 199

modification of Donnell type equations of motion were examinedby Sofiyev et al. [18]. The basic equations were reduced to an al-gebraic equation of the sixth order and numerically solving thisalgebraic 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 sandwichplates with FGM face sheets in thermal environment. The for-mulations were based on a higher-order shear deformation platetheory and a general von-Karman-type equation that includes athermal effect and the equations of motion were solved by animproved perturbation technique. The refined hierarchical kine-matics quasi-3D Ritz models for free vibration analysis of doublycurved FGM shells and sandwich shells with FGM core were in-vestigated by Fazzolari and Carrera [20]. Sburlati [21] presented ananalytical solution in the framework of the elasticity theory todescribe the elastic bending response of axisymmetric circularsandwich panels with functionally graded material cores andhomogeneous face-sheets. The elastic solution was obtained usinga Plevako representation, which reduced the problem to thesearch of potential functions satisfying linear fourth-order partialdifferential equations. The effect of continuously grading fiber or-ientation face sheets on free vibration of sandwich panels withfunctionally graded using generalized power-law distribution wasinvestigated by Aragh and Yas [22]. Woodward and Kashtalyan[23] analyzed a three-dimensional elasticity for a sandwich panelwith stiffness of the core graded in the thickness direction, on thebasic of the developed 3D elasticity solution subjected to dis-tributed and concentrated loadings.

The vibration problem about the shell structures have beenattracted a large number of studies. The free vibration analysis offunction analysis of FGM cylindrical shell with holes was studiedwith the variational equation and the unified displacement mode-shape function of the shells with various boundary conditions byCao and Wang [24]. Sofiyev and Kuruoglu [25] carried out the vi-bration and stability of FGM orthotropic cylindrical shells underexternal pressures using the shear deformation shell theory. Thebasic equations of shear deformable FG shell were derived usingDonnell shell theory and solved using the Galerkin method. Bichet al. [26] investigated the characteristics of free vibration andnonlinear responses, using the governing equations of motion ofeccentrically stiffened functionally graded cylindrical panels withgeometrically imperfections based on the classical shell theorywith the geometrical nonlinearity in von Karman-Donnell senseand smeared stiffeners technique. Moreover, Bich and Nguyen [27]presented the study of the nonlinear vibration of a functionallygraded 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 usingthe Lagrangian theory and multiple scale method was presentedby Du et al. [28]. The energy approach was applied to derive thereduced 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 deformationtheory. The vibration of FGM cylindrical shells under variousboundary conditions with the strain displacement relations formLove's shell theory were studied by Pradhan et al. [30]. Based onthe Rayleigh method, the governing equations were derived andthe 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 governingequation 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 bymeans of double mixed series based on Chebyshev orthogonal

polynomials for the longitudinal variable and harmonic functionsfor the circumferential variable. Sofiyev [32] studied the dynamicbehavior of FGM structures such as dynamic response of an FGMcylindrical shell subjected to combined action of the axial tension,internal compressive load and ring-shaped compressive pressurewith constant velocity based on the von Karman-Donnell typenonlinear kinematics using the superposition and Galerkinmethods. The nonlinear vibration of simply supported FGM cy-lindrical shells with embedded piezoelectric layers using a semianalytical approach was addressed by Jafari et al. [33]. The gov-erning differential equations of motion of the FG cylindrical shellwere derived using the Lagrange equations under the assumptionof 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 thenonlinear Sanders-Koiter shell theory to derive strain-displace-ment. The governing equations in linear state were solved by theRayleigh–Ritz procedure. Based on strain-displacement relationsfrom the Love's shell theory and the eigenvalue governing equa-tion using Rayleigh–Ritz method, Loy et al. [35] gave the study atvibration filed of functionally cylindrical shells. The dynamic be-havior of FGM truncated conical shells subjected to asymmetricinternal ring-shaped moving loads using Hamilton's principlebased on the first order shear deformation theory was studied byMalekzadeh and Daraie [36]. The vibration of FGM cylindricalshells on elastic foundations using wave propagation to solve dy-namical equations were analyzed by Abdul et al. [37]. The shellwas assumed to be simply supported with movable edges and theequations of motion were reduced using Galerkin method to asystem of infinite nonlinear ordinary differential equations withquadratic and cubic nonlinearities. Haddadpour et al. [38] per-formed free vibration analysis of functionally graded cylindricalshells using the equations of motion based on Love's shell theoryand the von Karman-Donnell type of kinematic nonlinearity forthe thermal effects investigated by specifying arbitrary high tem-perature on the outer surface and ambient temperature on theinner surface of the cylindrical. Based on the first-order sheardeformation theory of shells, the influences of centrifugal andCoriolis forces in combination with the other geometrical andmaterial parameters on the free vibration behavior of rotatingfunctionally graded truncated conical shells subjected to differentboundary conditions were investigated by Malekzadeh and Hey-darpour [39].

Noda [40], Praveen et al. [41] firstly realized the heat-resistantFGM structures and studied material properties dependent ontemperature 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 Karmannonlinear theory. The coupled nonlinear partial differential equa-tions are discretized based on a series expansion of linear modesand a multiterm Galerkin's method. Furthermore, Shen [43] tookinto account the nonlinear vibration of shear deformable FGMcylindrical shells of finite length embedded in a large outer elasticmedium and in thermal environments. The motion equations werebased on a higher order shear deformation shell theory that in-cluded shell-foundation interaction. The transient thermoelasticanalysis of functionally graded cylindrical shells under movingboundary pressure and heat flux was presented by Malekzadehand Heydarpour [44]. The hyperbolic heat conduction equationswere used to include the influence of finite heat wave speed. Theresulting system of differential equations was solved using New-mark's time integration scheme in the temporal domain. Kianiet al. [45] investigated the thermoelastic dynamic behavior of anFGM doubly curved panel under the action of thermal and me-chanical loads based on the first-order shear deformation theoryof modified Sanders assumptions applying the Laplace

D.G. Ninh, D.H. Bich / Thin-Walled Structures 104 (2016) 198–210200

transformation. Furthermore, Malekzadeh et al. [46,47] studied theinfluences of thermal environment on the free vibration char-acteristics of functionally graded shells and panels based on thefirst-order shear deformation theory. By taking into account boththe temperature dependence of material properties, which wereassumed to be graded in the thickness direction, and the initialthermal 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 anystudy to the nonlinear thermal vibration of eccentrically stiffenedCeramic-FGM-Metal layer toroidal shell segments surrounded byan elastic foundation.

In the present paper, the nonlinear vibration of eccentricallystiffened FGM sandwich toroidal shell segments on elastic mediumin thermal environment are investigated. Based on the classicalshell theory with the nonlinear strain-displacement relation of largedeflection, the Galerkin method, Stein and McElman assumption,Volmir's assumption and the numerical method using fourth-orderRunge-Kutta are performed for dynamic analysis of shells to giveexpression of natural frequencies and nonlinear dynamic responses.

Fig. 1. The coordinate system of ES-C

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, x1 and x2 is the axial and circumferential direc-tions, respectively and z is the normal to the two axes. Considerthe sandwich toroidal shell segment of thickness h, length L, whichis formed by rotation of a plane circular arc of radius R about anaxis in the plane of the curve as shown in Fig. 1 in a coordinatesystem (x1, x2, z) consists of ceramic, FGM and metal described inFig. 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 shellvaries smoothly along the thickness in such a way that the innersurface 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 h�hc�hm. Thesubscripts m and c are referred to the metal and ceramic

-FGM-M toroidal shell segment.

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

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

D.G. Ninh, D.H. Bich / Thin-Walled Structures 104 (2016) 198–210 201

constituents respectively. Denote Vm and Vc as volume - fractionsof metal and ceramic phases respectively, where VmþVc¼1. Ac-cording to the mentioned law, the volume fraction is expressed as

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

( ) = − ≤ ≤ − −

( ) =+ −− −

− − ≤ ≤ −

( ) = − ≤ ≤

≥

( )

V zh

zh

h

V zz h hh h h

hh z

hh

V zh

h zh

k

0,2 2

/2,

2 2

1,2 2

, 0

1

c m

cm

c m

k

m c

c c

According to the mentioned law, the Young modulus andthermal expansion coefficient of C-FGM-M shell are expressed ofthe form

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

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

the Poisson's ratio ν is assumed to be constant.

2.2. Constitutive relations and governing equations

For the middle surface of a toroidal shell segment, from theFig. 1 we have:

φ= − ( − )r a R 1 sin ,

where a is the equator radius and φ is the angle between the axisof revolution and the normal to the shell surface. For a sufficientlyshallow toroidal shell in the region of the equator of the torus, theangle φ is approximately equal to π/2, thus φ ≈sin 1; φ ≈cos 0 andr¼a [6]. The form of governing equation is simplified by putting:

φ θ= =dx Rd dx ad,1 2

The radius of arc R is positive with convex toroidal shell seg-ment and negative with concave toroidal shell segment. The shellis surrounded by an elastic foundation with Winkler foundationmodulus K1 (N/m3) and the shear layer foundation stiffness ofPasternak 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. Inorder to provide continuity within the shell and stiffeners andeasier manufacture, the homogeneous stiffeners can be used. Be-cause pure ceramic ones are brittleness we used metal stiffenerand put them at metal side of the shell. With the law indicated in(1) the outer surface is metal, so the external metal stiffeners areput at outer side of the shell. Fig. 3 depicts the geometry and co-ordinate system of stiffened C-FGM-M shell on elastic foundation.

The von Karman type nonlinear kinematic relation for thestrain component across the shell thickness at a distance z from

the middle surface are of the form [48]:

ε ε χ ε ε χ γ γ χ= − = − = − ( )z z z; ; 2 , 21 10

1 2 20

2 12 120

12

where ε10 and ε2

0 are normal strains, γ120 is the shear strain at the

middle surface of the shell and χ1 and χ2 are the curvatures and χ12is the twist.

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

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟ε ε

γ χ χ χ

= ∂∂

− + ∂∂

= ∂∂

− + ∂∂

= ∂∂

+ ∂∂

+ ∂∂

∂∂

= ∂∂

= ∂∂

= ∂∂ ∂ ( )

ux

wR

wx

vx

wa

wx

ux

vx

wx

wx

wx

wx

wx x

12

;12

;

; ; ; .3

10

1 1

2

20

2 2

2

120

2 1 1 21

2

12 2

2

22 12

2

1 2

Hooke's law for toroidal shell segment is defined as

σν

ε νε αν

Δ Δ

σν

ε νε αν

Δ

σν

γ

= ( )−

( + ) − ( ) ( )−

= −

= ( )−

( + ) − ( ) ( )−

= ( )( + ) ( )

E z E z zT T T T

E z E z zT

E z

1.

1, ,

1.

1,

2 1 4

sh

sh

sh

1 2 1 2 0

2 2 2 1

12 12

and for metal stiffeners

σ ε α Δ σ ε α Δ= − = −E E T E E T; .stm m m

stm m m1 1 2 2

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

By integrating the stress–strain equations and their momentsthrough the thickness of the shell and using the smeared stiffenerstechnique; the expressions for force and moment resultants of aC-FGM-M toroidal shell segment can be obtained as [13,48]:

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

ε ε χ χ Φ Φ

ε ε χ χ Φ Φ

γ χ

= + + − ( + ) − − − *

= + + − − ( + ) − − **

= − ( )

N AE A

sA B C B

N A AE A

sB B C

N A B

,

,

2 , 5

ma a

ma a

1 111

110

12 20

11 1 1 12 2

2 12 10

222

220

12 1 22 2 2

12 66 120

66 12

D.G. Ninh, D.H. Bich / Thin-Walled Structures 104 (2016) 198–210202

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

ε ε χ χ Φ Φ

ε ε χ χ Φ Φ

γ χ

= ( + ) + − + − − − *

= + ( + ) − − + − − **

= − ( )

M B C B DE Is

D

M B B C D DE Is

M B D

,

,

2 , 6

mm m

mm m

1 11 1 10

12 20

111

11 12 2

2 12 10

22 2 20

12 1 222

22

12 66 120

66 12

where A B D, ,ij ij ij (i, j¼1, 2, 6) are extensional, coupling andbending stiffnesses of the shell without stiffeners.

ννν ν

ννν ν

ννν ν

= =−

=−

=( + )

= =−

=−

=( + )

= =−

=−

=( + ) ( )

A AE

AE

AE

B BE

BE

BE

D DE

DE

DE

1,

1,

2 1,

1,

1,

2 1,

1,

1,

2 1,

7

11 221

2 121

2 661

11 222

2 122

2 662

11 223

2 123

2 663

where

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟⎤⎦⎥

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

∫

∫

∫

= ( ) = + +( − − )

+

= ( ) = − ++

− ( − − )

−( + )( + )

( − − )

= ( ) =+

− ( − − ) −( + )( + )

− ( − − )

+( + )( + )( + )

( − − ) +

+ − + + −

+ ( − − ) − − − ( − − )

−

−

−

E E z dz E h E hE h h h

k

E E z zdzE h h E h E

kh

h h h h

Ek k

h h h

E E z z dzE

kh

h h h hE

k kh

h h h h

Ek k k

h h hE h

E hh hh

Eh

hh hh

Eh h h

hh

hh h h h

1,

2 2 1 2

1 2,

1 221 2 2

21 2 3 3

2 2 33

2 2

33

2 2

h

hm cm c

cm c m

h

h cm c cm c cmc c m

cmc m

h

h cmc c m

cmc c m

cmc m

c c

c cc

mm

mm

mc m m c c m

1 /2

/2

2 /2

/2 2

2

3 /2

/2 22

2

33

3

3

in which Ecm¼Ec�Em

= − = −CE A z

sC

E A zs

, ,m m1

1 1

12

2 2

2

= = ( )A h d A h d, , 81 1 1 2 2 2

= + = +Id h

A z Id h

A z12

,12

.11 1

3

1 12

22 2

3

2 22

and

∫ ∫

∫ ∫

∫ ∫ ( )

**

**

Φν

α Φ α

Φ α Φν

α

Φ α Φ α

=−

( ) ( )Δ = Δ

= Δ =−

( ) ( )Δ

= Δ = Δ

*

*

− − −

−

− −

−

−

− −

−

− −

−

9

E z z Tdzds

E Tdz

ds

E Tdz E z z Tzdz

ds

E Tzdzds

E Tzdz

11

; ;

;1

1;

; .

a h

ha h h

hm m

a h h

hm m m h

h

m h h

hm m m h h

hm m

/2

/2 1

1 /2 1

/2

2

2 /2 2

/2

/2

/2

1

1 /2 1

/2 2

2 /2 2

/2

If ΔT¼const.

**

Φν

α α αα

α α

Φ α Φ α

=−

Δ

= + + ( − ) +( − − )

+

+( − − )

++

( − − )+

* = Δ = Δ

P T

P E h E h E h hE h h h

kE h h h

kE h h h

kd h

sE T

d hs

E T

11

;

1

1 2 1

; .

a

m m c c c m m cm cm c m

cm m c m cm cm c m

a m m a m m1 1

1

2 2

2

in which

α α α= − ( )10cm c m

The spacings of the stringer and ring stiffeners are denoted bys1 and s2 respectively. The quantities A1, A2 are the cross sectionareas of the stiffeners and I1, I2, z1, z2 are the second moments ofcross 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 surroundedby an elastic foundation based on the classical shell theory aregiven by [48]:

⎛⎝⎜⎜

⎞⎠⎟⎟

( )

ρ

ρ

ρ ρ ε

∂∂

+∂∂

= ∂∂

∂∂

+∂∂

= ∂∂

∂∂

+∂∂ ∂

+∂∂

+ ∂∂

+ ∂∂ ∂

+ ∂∂

− ∂∂

+

+ + − + ∂∂

+ ∂∂

= ∂∂

+ ∂∂ 11

Nx

Nx

u

t

Nx

Nx

v

t

M

x

Mx x

M

xN

w

xN

wx x

Nw

xph

w

x

NR

Na

q K w Kw

x

w

x

w

t

wt

,

,

2 2

2 ,

1

1

12

21

2

2

12

1

2

21

2

2

21

12

212

1 2

22

22 1

2

12 12

2

1 22

2

22

2

12

1

21 2

2

12

2

22 1

2

2 1

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

⎛⎝⎜

⎞⎠⎟ρ ρ ρ

ρρ= + +

( − − )+

+ +( )

h hh h h

kAs

As1 12m cm c

cm c mm1

1

1

2

2

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

ρ

ρ

Φ Φ Φ Φ ρ ρ ε

( ) + ( ) + ( ) + ( ) = ∂∂

( ) + ( ) + ( ) + ( ) = ∂∂

( ) + ( ) + ( ) + ( ) + ( ) + ( ) − ∂∂

+ − ( + *) − ( + **) = ∂∂

+ ∂∂ ( )

Y u Y v Y w P wu

t

Y u Y v Y w P wv

t

Y u Y v Y w P w Q u w R v w phw

x

qR a

wt

wt

,

,

, ,

1 12 ,

13a a a a

11 12 13 1 1

2

2

21 22 23 2 1

2

2

31 32 33 3 3 3

2

12

1

2

2 1

where the linear operators ( )( = )Y i j, 1, 2, 3ij and the nonlinear

operators ( )( = )P i 1, 2, 3i , 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 thermalenvironment.

3. Nonlinear analysis

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

= = = = = ( )w v M at x and x L0, 0, 0. 0 141 1 1

The approximate solutions of the system of Eq. (13) satisfyingthe conditions (14) can be expressed as:

π π

π

= ( ) = ( )

= ( ) ( )

u U tm x

Lnx

av V t

m xL

nxa

w W tm x

Lnx

a

cos sin2

; sin cos2

;

sin sin2

, 15

mn mn

mn

1 2 1 2

1 2

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

Substituting Eq. (15) into Eq. (13) and then applying the Ga-lerkin method leads to:

D.G. Ninh, D.H. Bich / Thin-Walled Structures 104 (2016) 198–210 203

⎡⎣⎢

⎤⎦⎥

( )

**

ρ

ρ

π

δ δπ

Φ Φ Φ Φ

ρ ρ ε

+ + + =

+ + + =

+ + + + + +

+ + − ( + *) − ( + )

= +16

y U y V y W n Wd U

dt

y U y V y W n Wd V

dt

y U y V y W ph m

LW n W n W n U W

n V Wmn

qR a

d W

dt

dWdt

,

,

4 1 1

2 ,

mn mn mn mnmn

mn mn mn mnmn

mn mn mn mn mn mn mn mn

mn mn a a a a

mn mn

11 12 13 12

1

2

2

21 22 23 22

1

2

2

31 32 33

2 2

2 32

43

5

61 2

2

1

2

2 1

where yij; ni are given in Appendix B.Otherwise, the Volmir's assumption [49] can be used in the

dynamic analysis. Taking the inertia forces ρ ( ) →d U dt/ 012 2 and

ρ ( ) →d V dt/ 012 2 into consideration because of < < < <u w v w, , Eq.

(16) can be rewritten as follows:

⎡⎣⎢

⎤⎦⎥

( )

**

π

δ δ

πΦ Φ Φ Φ

ρ ρ

+ + + =

+ + + =

+ + + +

+ + + + − ( + ) − ( + )

= + ϵ

*

17

y U y V y W n W

y U y V y W n W

y U y V y W ph m

LW n W

n W n U W n V Wmn

qR a

d W

dt

dWdt

0,

0,

4 1 1

2

mn mn mn mn

mn mn mn mn

mn mn mn mn mn

mn mn mn mn mn a a a a

mn mn

11 12 13 12

21 22 23 22

31 32 33

2 2

2 32

43

5 61 2

2

1

2

2 1

Solving the first and the second obtained equations with re-spect to Umn and Vmn and then substituting the results into thethird equation yields

⎡⎣⎢

⎤⎦⎥

ρ ρ ε

δ δπ

Φ Φ Φ Φ

+ + − +

= − ( + *) − ( + **)( )

d W

dt

dWdt

g W g W g W

mnq

R a

2

4 1 118

mn mnmn mn mn

a a a a

1

2

2 1 1 22

33

1 22

where

( )

π= − −( − )

−−

( − )−

−

= +( − )

−+

( − )−

+( − )

−

+( − )

−

= − −( − )

−−

( − )− 19

g yy y y y y

y y y y

y y y y y

y y y yp

h m

L

g ny y n y n

y y y y

y y n y n

y y y y

n y y y y

y y y y

n y y y y

y y y y

g n ny n y n

y y y y

n y n y n

y y y y

,

,

.

1 3331 12 23 22 13

11 22 12 21

32 21 13 11 23

11 22 12 21

2 2

2

2 331 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 4 512 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 of Eq. (16) and putting p¼0, q¼0and ε¼0 the natural frequencies of the shell can be directly cal-culated by solving determinant

ρ ω

ρ ω

ρ ω

+

+

+

=

( )

y y y

y y y

y y y

0

20

11 12

12 13

21 22 12

23

31 32 33 12

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 beapproximately determined by explicit expression in Eq. (18)

ωρ

=( )

g

21mn

1

1

.Solving Eq. (20) leads to exact solution but implicit expressionwhile Eq. (21) performs approximate frequencies but explicit ex-pression and simpler.

3.2. Frequency amplitude curve

Consider nonlinear vibration of a toroidal shell segment undera uniformly distributed transverse load Ω=q Q tsin includingthermal effects. Assuming pre-loaded compression p, Eq. (18)takes the form.

⎡⎣⎢

⎤⎦⎥

ρ ρ ε

δ δπ

Ω Φ Φ Φ Φ

+ + − +

= − ( + *) − ( + **)( )

d W

dt

dWdt

g W g W g W

mnQ t

R a

2

4sin

1 1.

22

mn mnmn mn mn

a a a a

1

2

2 1 1 22

33

1 22

Eq. (22) can be rewritten as

( )ε ω Ω+ + ( − + ) − + =23

d W

dt

dWdt

W HW KW F t G2 sin 0,mn mnmn mn mn mn

2

22 2 3

where ω =ρmng1

1is the fundamental frequency of linear vi-

bration of the toroidal shell segment and H¼g2/g1, K¼g3/g1,

= δ δ

π ρF Q

mn

4 1 22

1,

⎡⎣⎢

⎤⎦⎥Φ Φ Φ Φ= ( + *) + ( + **)δ δ

π ρG

mn R a a a a a4 1 11 2

21

.

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

Ω= ( )W A tsin , 24mn

into Eq. (23) to give

ω Ω Ω ε Ω Ω ω Ω

ω Ω Ω

= ( − ) + −

+ − + = ( )

Y A t A t HA t

K A t F t G

sin 2 cos sin

sin sin 0 25

mn mn

mn

2 2 2 2 2

2 3 3

Integrating over a quarter of vibration period

∫ Ω =π Ω

Y tdtsin 0,0

/2

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

⎛⎝⎜

⎞⎠⎟Ω ε

πΩ ω

π π− = − + − +

( )HA

KA

FA

GA

41

83

34

426mn

2 2 2

By denoting α Ω ω= / mn2 2 2 Eq. (26) is rewritten as

α επω

απ ω πω

− = − + − +( )

HAK

AF

AG

A4

18

334

4

27mn mn mn

2 22 2

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

απ ω πω

= − + − +( )

HAK

AF

AG

A1

83

34

4

28mn mn

2 22 2

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

⎛⎝⎜

⎞⎠⎟ω ω

π= − +

( )HA

KA1

83

34 29NL mn

2 2 2

where ωNL is the nonlinear vibration frequency.

D.G. Ninh, D.H. Bich / Thin-Walled Structures 104 (2016) 198–210204

3.3. Nonlinear vibration responses

Consider an eccentrically stiffened functionally graded toroidalshell segment acted on by a uniformly distributed transverse load

Ω( ) =q t Q tsin and a pre-loaded axial compression p, the set ofmotion Eq. (16) has of the form

⎡⎣⎢

⎤⎦⎥

( )

**

ρ

ρ

π

δ δ

πΩ Φ Φ Φ Φ

ρ ρ ε

+ + + =

+ + + =

+ + + + + +

+ + − ( + ) − ( + )

= +

*

30

y U y V y W n Wd U

dt

y U y V y W n Wd V

dt

y U y V y W ph m

LW n W n W n U W

n V Wmn

Q tR a

d W

dt

dWdt

,

,

4sin

1 1

2

mn mn mn mnmn

mn mn mn mnmn

mn mn mn mn mn mn mn mn

mn mn a a a a

mn mn

11 12 13 12

1

2

2

21 22 23 22

1

2

2

31 32 33

2 2

2 32

43

5

61 2

2

1

2

2 1

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

⎡⎣⎢

⎤⎦⎥

ρ ρ ε

δ δπ

Ω Φ Φ Φ Φ

+ + − +

= − ( + *) − ( + **)( )

d W

dt

dWdt

g W g W g W

mnQ t

R a

2

4sin

1 131

mn mnmn mn mn

a a a a

1

2

2 1 1 22

33

1 22

Using the fourth-order Runge-Kutta method into Eq. (30) or Eq.(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 nopublication about nonlinear vibration of ES-C-FGM-M toroidalshell segment, that is reason to compare the results in this paperwith homogenous and FGM cylindrical shell (i.e. a toroidal shellsegment with R-1 and hc¼hm¼0).

Firstly, the results of natural frequencies in present will becompared with results for the un-stiffened isotropic cylindricalshell studied by Lam and Loy [50], Li [51] and Shen [43] and can beseen in Table 1.

As can be seen in Table 1 that good agreements are obtained inthis comparison. Moreover the frequencies calculated by Eq. (20)(the full order equation ODE) and Eq. (21) (the Volmir's assump-tion) are quite close to each other.

Secondly, the natural frequencies of FGM cylindrical shell illu-strated in Table 2 are computed and compared with the results ofLoy et al. [35] using Rayleigh–Ritz method and Shen [43] with twokinds of micromechanics models: Voigt model and Mori–Tanakamodel based on a higher order shear deformation shell theory. AFGM cylindrical shell is made of stainless steel and nickel materialin initial temperature T0¼300 K is considered with the followingmaterial properties.

Table 1

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

cylindrical shell (a/L¼2; h/a¼0.06, E¼210 GPa, υ¼0.3, ρ = 7850 kg/m3).

(m, n) Lam and Loy[50]

Li [51] Shen [43] Present(Eq. 21)

Present(Eq. 20)

(1, 1) 0.03748 0.03739 0.03712 0.03781 0.03755(1, 2) 0.03671 0.03666 0.03648 0.03761 0.03751(1, 3) 0.03635 0.03634 0.03620 0.03733 0.03717(1, 4) 0.03720 0.03723 0.03700 0.03705 0.03684

ENi¼205.09 GPa; υNi¼0.31; ρNi¼8900 kg/m3; ESS¼207.7877GPa; 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 differencewith those of Ref. [43] because the author used other theories.

In the following sections, the materials consist of Aluminumand Alumina with = ×E N m70 10 /m

9 2; ρ = 2702m kg/m3;

α = × −−C23 10m

61

0 ; = ×E N m380 10 /c9 2; ρ = 3800c kg/m3;

α = × −−C5.4 10c

61

0 and Poisson's ratio is chosen to be 0.3. Theelastic foundation parameters are taken as K1¼2.5�108 N/m3,K2¼5�105 N/m with Pasternak foundation. The parametersn1¼50 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 segmentin three cases using Eq. (20) are illustrated in Table 3. The datum ofproblem: h¼0.01 m; hc¼0.3 h; hm¼0.1 h; a¼300 h; R¼500 h;L¼2a; d1¼d2¼h/2; h1¼h2¼h/2; n1¼n2¼50; k¼1. It can be seenthat the natural frequencies of the shell on Pasternak foundationare the highest while the natural frequencies with pre-loaded axialcompression (p¼1 GPa) are the lowest. It means that when theshell is subjected to pre-loaded axial compression, the naturalfrequencies 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). Thefrequency-amplitude curve of nonlinear free vibration of the shelland the effects of pre-loaded axial compression, elastic foundationare indicated in Fig. 4. It can be observed that the lowest frequencywill increase when the shell is on elastic foundation. Whereas, itwill 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 in Fig. 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-loadedaxial compression (p¼2.5 GPa) and without elastic foundationunder excited forces with F¼5�105 and F¼8�105, respectively.Finally, the lines 4 and 5 correspond to the free vibration and theforced vibration cases of the shell on Pasternak foundation, re-spectively. It can be seen, the frequency-amplitude curve trendfurther from the curve of the free vibration case when the am-plitude of external force increases. The frequency-amplitudecurves of the shell on elastic foundation move ahead in the in-creasing frequency direction in comparison with those curves ofthe shell without elastic foundation.

4.4. Nonlinear vibration responses

The comparison of the nonlinear response of the shell calcu-lated by the approximate Eq. (31) (Volmir's assumption) and thefull order system Eq. (30) is shown in Fig. 6.

From Tables 1 and 2 and Fig. 6, we conclude that the Volmir'sassumption can be used to investigate nonlinear dynamical ana-lysis with an acceptable accuracy.

In the next sections, the full order system Eq. (30) is used toinvestigate nonlinear vibration responses. As following the effectsof the characteristics of functionally graded materials, the pre-loaded axial compression, the dimensional ratios, the elasticfoundation and thermal loads on the nonlinear dynamic responses

Table 2

Comparisons of natural frequencies = Ωπ

f2(Hz) for FGM cylindrical shells (L/a¼20, a/h¼20, h¼0.05 m, T¼300 K).

(m, n) Source k

0.0 0.5 1.0 2.0 5.0 15.0

SUS304/Ni(1, 7) Loy et al. [35] 580.78 570.25 565.46 560.93 556.45 553.37

Shen [43] 585.788 575.266 570.471 565.925 561.399 558.274Present (Eq. 21) 601.267 591.025 586.279 581.741 577.242 574.228Present (Eq. 20) 578.011 568.187 563.630 559.267 554.934 552.027

(1, 8) Loy et al. [35] 763.98 750.12 743.82 737.86 731.97 737.92Shen [43] 759.914 746.278 740.065 734.176 728.311 724.262Present (Eq. 21) 784.929 771.557 765.362 759.437 753.565 749.631Present (Eq. 20) 761.367 748.427 742.425 736.677 730.971 727.140

Ni/SUS304(1, 7) Loy et al. [35] 551.22 560.94 565.63 570.25 575.03 578.40

Shen [43] 556.073 565.780 570.478 575.119 579.936 583.355Present (Eq. 21) 572.189 581.633 586.279 590.913 595.704 599.008Present (Eq. 20) 550.057 559.113 563.574 568.028 572.642 575.829

(1, 8) Loy et al. [35] 725.08 737.87 744.04 750.13 756.41 760.84Shen [43] 721.406 733.988 740.074 746.088 752.327 756.757Present (Eq. 21) 746.968 759.297 765.362 771.411 777.666 781.979Present (Eq. 20) 724.546 736.474 742.349 748.217 754.294 758.492

Table 3The fundamental frequencies (s�1) using Eq. (20) in various cases of ES-C-FGM-Mtoroidal shell segment.

Cases ω1 (1, 1)* ω2 (1, 2) ω3 (1, 3) ω4 (2, 1) ω5 (2, 2)

Natural frequencies 2362.216 2214.457 2096.708 2805.679 2727.048Natural frequencies ofshell on Pasternakfoundation

2932.162 3037.544 3116.408 3846.529 3788.811

Natural frequencies ofshell 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 orderequation system).

D.G. Ninh, D.H. Bich / Thin-Walled Structures 104 (2016) 198–210 205

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 toroidalshell are considered in Figs. 7–21.

Figs. 7 and 8 depict the effect of R/h ratio on nonlinear vibrationof convex and concave stiffened FGM sandwich toroidal shellsegment, respectively. It can be seen that when increasing R/hratio, the amplitudes of nonlinear vibration of both stiffened

C-FGM-M shell also increase and the frequency does not modifymuch. Furthermore, the amplitudes of nonlinear vibration ofconvex ES-FGM sandwich shell are smaller than ones of concaveES-FGM sandwich shell.

The effect of L/R ratio is described in Figs. 9 and 10. It can beseen that when L/R ratios increase, the amplitudes of nonlinearvibration of convex ES-FGM core shell also go up while those ofconcave ES-C-FGM-M shell decrease. It means that the amplitudes

Fig. 7. Effect of R/h ratio on nonlinear vibration response of convex ES-FGMsandwich toroidal shell segment on elastic medium.

Fig. 8. Effect of R/h ratio on nonlinear vibration response of concave ES-FGMsandwich toroidal shell segment on elastic medium.

Fig. 9. Effect of L/R ratio on nonlinear vibration response of convex ES-FGMsandwich toroidal shell segment on elastic medium.

Fig. 10. Effect of L/R ratio on nonlinear vibration response of concave ES-FGMsandwich toroidal shell segment on elastic medium.

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

Fig. 12. Effect of L/a on nonlinear vibration response of concave ES-FGM sandwichtoroidal 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.

D.G. Ninh, D.H. Bich / Thin-Walled Structures 104 (2016) 198–210206

Fig. 15. Effect of volume-fraction k on nonlinear vibration response of ES-FGMsandwich toroidal shell segment on elastic medium.

Fig. 16. Effect of elastic foundation on nonlinear vibration response of ES-FGMsandwich toroidal shell segment.

Fig. 17. Effect of pre-loaded axial compression on nonlinear vibration response ofES-FGM sandwich toroidal shell segment.

Fig. 18. Effect of thermal environment on nonlinear vibration response of ES-FGMsandwich convex toroidal shell segment.

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

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

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

D.G. Ninh, D.H. Bich / Thin-Walled Structures 104 (2016) 198–210 207

of the more convex shells are greater than that of the less convexones whereas this feature of concave shells is completely on thecontrary. On the other hand, the amplitudes of nonlinear vibrationof convex shell are lower than ones of concave shell.

As can be observed in Figs. 11 and 12, the influence of ratio L/aon 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 ofnonlinear vibration of both ES-FGM core shells decrease when thethickness of ceramic layer increases. It means that the sandwichstructures will be stiffer than FGM structures with the same geo-metry parameters. Thus, the amplitudes of nonlinear vibration ofsandwich structure will be lower than those of FGM structure.

D.G. Ninh, D.H. Bich / Thin-Walled Structures 104 (2016) 198–210208

Furthermore, the amplitudes of nonlinear vibration of convex shellare lower than those of concave shell.

The effect of volume-fraction k on nonlinear vibration of ES-FGM core shell is illustrated in Fig. 15. The amplitudes of nonlinearvibration of ES-FGM toroidal shell segment increase when thevalue of volume fraction index increases.

This property evidently appropriates to the real characteristicof material, because the higher value of k corresponds to a metal-richer shell which has less stiffness than a ceramic-richer one.

Based on Fig. 16, the amplitudes of nonlinear vibration of shellwithout elastic foundation are the highest and those on Pasternakfoundation are lowest. In addition, the amplitudes of nonlinearvibration of shell without elastic foundation in this example areabout 3 times more than ones on Pasternak foundation.

The effect of pre-loaded axial compression on the nonlinearresponses of ES-FGM core shell is indicated in Fig. 17. As can beseen, the amplitude of nonlinear vibration of the shell increaseswhen the value of axial compression load increases. The pre-loa-ded axial compression makes the load bearing capacity of the shellreduce under dynamical loads.

Based on Figs. 18 and 19, we can see that the temperature in-creases, the stiffness of the structure will reduce and the ampli-tude will rise. Therefore, the load bearing capacity of the structurewill mitigate.

The shell is preheated, the temperature field makes the shell bedeflected outward (negative deflection) before it is affected bylateral load. It means that the amplitude of structure will benegative.

Furthermore, the amplitude of nonlinear vibration of convexshell is about 3 times higher than one of concave shell. This can beexplained as follows: The geometry of the shell is concave, thuswhen temperature field affects, it is expanded and has trend ascylindrical shell. In contrarily, the shape of convex shell is convexthus it is expanded so much.

In the previous sections nonlinear dynamic responses of ES-FGM core toroidal shell segments are investigated when the fre-quency of excited force is far from the natural frequency of theshell. The picture of nonlinear dynamic response of the shell isquite different when the frequency of the excitation is near to thenatural frequency of the shell.

Fig. 20 describes the nonlinear response of ES-C-FGM-M con-vex toroidal shell segment with the natural frequencyω = 3239.20941 rad/s under excitation q¼1000sin (3150t).

Similarly, Fig. 18 demonstrates the nonlinear response of ES-FGM sandwich concave shell with the same datum of problemexcepted only R/h¼�500 and in this case the natural frequency ofthe shell ω = 2748.40631 rad/s and the external frequencyΩ¼2650 rad/s.

From obtained figures, the interesting phenomenon is observedlike harmonic beat of linear vibration, where the number of thebeats of ES-FGM sandwich convex shell is smaller than those ofconcave shell.

5. Concluding remarks

The nonlinear thermal vibration of eccentrically stiffenedCeramic-FGM-Metal layer toroidal shell segment subjected tomechanical loads using the displacement method is investigated.Based on the classical shell theory with the geometrical nonlinearin von Karman-Donnell sense and the smeared stiffeners techni-que, the governing equations of motion of eccentrically stiffenedfunctionally graded toroidal shell segments are derived in thispaper. Furthermore, the Galerkin method is used for the vibration

analysis of shells to perform natural frequencies, nonlinear fre-quency-amplitude relation and nonlinear dynamic responses.Some remarkable conclusions are obtained as follows:

– The obtained results of natural frequencies are compared withthe results of the other authors to validate. Moreover, elasticfoundation and pre-loaded axial compression also remarkablyinfluence to the fundamental frequencies.

– The lowest frequency in frequency-amplitude curve increasewhen the toroidal shell segment is on elastic foundation while itwill decrease if the shell bears the pre-loaded axial compression.

– The effects of characteristic of functionally graded materials,geometrical ratios, elastic foundation, pre-loaded axial com-pression, ceramic and metal layer and thermal environment onthe nonlinear vibration behavior of shells are indicated. Thetrend of the nonlinear vibration responses of ES-FGM sandwichconcave shell is contrast with those of ES-FGM sandwich convexshell when changing L/R and L/a ratios. Furthermore, the ther-mal environment significantly impacts on the nonlinear vibra-tion response to make the amplitude of structural shell negative.

– The beating vibration phenomenon is shown when the fre-quency of the external force is near to the natural frequency ofthe shell. Moreover, the number of the beats of ES-FGM sand-wich concave shell is more than those of ES-FGM sandwichconvex shell.

Appendix A

Operators in Eq. (13)

⎛⎝⎜

⎞⎠⎟( ) = + ∂

∂+ ∂

∂Y u A

E As

ux

Au

x,11 11

0 1

1

2

12 66

2

22

( )( ) = + ∂∂ ∂

Y v A Av

x x,12 12 66

2

1 2

⎛⎝⎜

⎞⎠⎟( ) = −

++ ∂

∂− ( + ) ∂

∂− ( + ) ∂

∂ ∂Y w

A E A sR

Aa

wx

B Cw

xB B

w

x x

/2 ,13

11 0 1 1 12

111 1

3

13 12 66

3

1 22

⎛⎝⎜

⎞⎠⎟ ( )( ) = + ∂

∂∂∂

+ + ∂∂

∂∂ ∂

+ ∂∂

∂∂

P w AE A

swx

wx

A Awx

wx x

Awx

wx

,1 110 1

1 1

2

12 12 66

2

2

1 266

1

2

22

( )( ) = + ∂∂ ∂

Y u A Au

x x,21 66 12

2

1 2

⎛⎝⎜

⎞⎠⎟( ) = ∂

∂+ + ∂

∂Y v A

vx

AE A

sv

x,22 66

2

122

0 2

2

3

22

⎛⎝⎜

⎞⎠⎟

( ) ( )( ) = − + ∂∂

− + ∂∂ ∂

− ++ ∂

∂

Y w B Cw

xB B

wx x

AR

A E A sa

wx

2

/,

23 22 2

3

23 12 66

3

12

2

12 22 0 2 2

2

⎛⎝⎜

⎞⎠⎟( )( ) = + ∂

∂∂

∂ ∂+ ∂

∂∂∂

+ + ∂∂

∂∂

P w A Awx

wx x

Awx

wx

AE A

swx

wx

,2 66 121

2

1 266

2

2

12 22

0 2

2 2

2

22

⎛⎝⎜

⎞⎠⎟

( ) ( )( ) = + ∂∂

+ + ∂∂ ∂

+ ++ ∂

∂

Y u B Cu

xB B

ux x

Aa

A E A sR

ux

2

/,

31 11 1

3

13 66 12

3

1 22

12 11 0 1 1

1

D.G. Ninh, D.H. Bich / Thin-Walled Structures 104 (2016) 198–210 209

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎤

⎦⎥⎥

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

( ) ( )

( )

( )

( )

( )

( ) ( )

( )

Φ Φ

Φ Φ

( ) = + ∂∂ ∂

+ + ∂∂

+ ++ ∂

∂

( ) = − + ∂∂

− + ∂∂

−+

+ − + ( + ) ∂∂

−+

+ − + ( + ) ∂∂

− ( + ) ∂∂ ∂

−+

+ ++

−

( ) = − ∂∂ ∂

++

+ ∂∂

++

+ ∂∂

+ − ∂∂

∂∂

−+

+ ∂∂

−+

+ ∂∂

+ + ∂∂

∂∂

+ + ∂∂

∂∂ ∂

+ ∂∂

∂∂ ∂

+ + ∂∂

∂∂

++ ∂

∂∂∂

+ ∂∂

∂∂

+ ∂∂

∂∂

++ ∂

∂∂∂

+ ∂∂

∂∂

∂∂ ∂

( ) = + ∂∂

∂∂

+ ∂∂

∂∂ ∂

+ ∂∂

∂∂

⁎

⁎⁎

Y v B Bv

x xB C

v

x

AR

A E A sa

vx

Y w DE Is

w

xD

E Is

w

x

B C

RBa

Kw

x

B C

aBR

Kw

xD D

w

x x

A E A s

R

ARa

A E A s

aK w

P w B Bw

x xA E A s

RA

awx

A E A sa

AR

wx

B Bw

x

w

x

A E A sR

Aa

ww

x

A E A sa

AR

ww

x

B Cwx

w

xB B

wx

w

x x

wx

w

x x

B Cwx

w

x

A E A s w

x

wx

A w

x

wx

w

x

wx

A E A s w

x

wx

Awx

wx

wx x

Q u w AE A

sux

w

xA

ux

wx x

Aux

w

x

2/

,

2 2

2 22 4

/ 2 /,

2 2/

2 2

/2 2

2 2/

/

2

/2 2

/2

2 ,

, 2 ,

a a

a a

32 12 66

3

12

222 2

3

23

12 22 0 2 2

2

33 110 1

1

4

14 22

0 2

2

4

24

11 1 122

2

12

22 2 122

2

22 12 66

4

12

22

11 0 1 12

12 22 0 2 22 1

3 12 66

2

1 2

211 0 1 1 12

1

2

22 0 2 2 12

2

2

66 12

2

12

2

22

11 0 1 1 122

12

22 0 2 2 122

22

22 22

3

23 12 66

1

3

1 22

2

3

12

2

11 11

3

13

11 0 1 12

12

1

212

2

12

2

2 2

22

1

2

22 0 2 22

22

2

2

661 2

2

1 2

3 110 1

1 1

2

12 66

2

2

1 212

1

2

22

⎛⎝⎜

⎞⎠⎟( ) = ∂

∂∂∂

+ ∂∂

∂∂ ∂

+ + ∂∂

∂∂

R v w Avx

wx

Avx

wx x

AE A

svx

wx

, 21

.3 122

2

12 66

2

1 222

0 2

2 2

2

22

Appendix B

Coefficients in system of Eq. (16)

⎛⎝⎜

⎞⎠⎟

π= − + −y AE A

sm

LA

na4

,11 110 1

1

2 2

66

2

2

( ) π= − +y A Amn

La2,12 12 66

⎛⎝⎜

⎞⎠⎟

π π π= −+

+ + ( + ) + ( + )yA E A s

RAa

mL

B Cm

LB B

mL

n

a

/2

4,13

11 0 1 1 1211 1

3 3

3 12 66

2

2

( ) π= − +y A Amn

La2,21 66 12

⎛⎝⎜

⎞⎠⎟

π= − − +y Am

LA

E As

na4

,22 66

2 2

2 220 2

2

2

2

⎛⎝⎜

⎞⎠⎟

( ) ( ) π= + + +

− ++

y B Cn

aB B

m nL a

AR

A E A sa

na

82

2/

2,

23 22 2

3

3 12 66

2 2

2

12 22 0 2 2

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

( ) ( )

( )

( )

π π

π π

= − + + +

− ++

= +

+ + − ++

y B Cm

LB B

mL

na

Aa

A E A sR

mL

y B Bm

L

na

B Cn

a

AR

A E A sa

na

24

/, 2

2 8

/2

,

31 11 1

3 3

3 66 12

2

2

12 11 0 1 132 12 66

2 2

2

22 2

3

312 22 0 2 2

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

( )

( )

( )

π

Φ Φ π

Φ Φ π

πδ δ π π

πδ δ π

= − + − +

++

+ − + ( + )

++

+ − + ( + ) − ( + )

−+

+ ++

+

= + +

+ +

⁎

⁎⁎

y DE Is

m

LD

E Is

n

a

B C

RBa

Km

L

B C

aBR

Kn

aD D

m

L

n

a

A E A s

R

ARa

A E A s

aK

nmn

AE A

sm

LA

m n

La

mnA A

m n

La

16

2 2

2 2

42 4

4

/ 2 /,

8

9 4

4

9 4

a a

a a

33 110 1

1

4 4

4 220 2

2

4

4

11 1 122

2 2

2

22 2 122

2

2 12 66

2 2

2

2

2

11 0 1 12

12 22 0 2 22 1

1 2 1 2 110 1

1

3 3

3 66

2

2

2 1 2 12 66

2

2

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

( )π

δ δ π

πδ δ π

= +

− + +

nmn

A Am n

L a

mnA

m nL a

AE A

sn

a

49 2

89 2 8

2 2 1 2 66 12

2 2

2

2 1 2 66

2 2

2 220 2

2

3

3

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

⎛⎝⎜⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟⎟

( )

( ) ( ) ( )

( )

πδ δ π

πδ δ π

π π

π

π π

ππ π

= −

++

+ ++

+

− + − + + +

+ − ++

+

++

+

= −+

+ − ++

naLmn

B Bm n

L a

aLmn

A E A sR

Aa

m

L

A E A sa

AR

n

a

B Cn

aB B

m n

L aB C

m

L

aLmnB B

m n

L a

A E A sR

Aa

m

L

A E A sa

AR

n

a

naL

A E A s m

LA A

m n

L a

A E A s n

a

8

92 2

4

16

9

/2 2

/2 2 4

164 2

4

128

92 2

4

/

/

4,

38

/2

23 4

/2 16

,

3 3 1 2 12 66

2 2 2

2 2

3 1 211 0 1 1 12

2 2

222 0 2 2 12

2

2

22 2

4

4 66 12

2 2 2

2 2 11 1

4 4

4

3 66 12

2 2 2

2 211 0 1 1 12

2 2

2

22 0 2 2 122

2

411 0 1 1

4 4

4 12 66

2 2 2

2 222 0 2 2

4

4

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥π

π ππ

δ δ π= + + +nmn

AE A

sm

LA

m nLa

Lamn

Am n

La64

9 4169 4

,5 2 110 1

1

3 3

3 12

2

2 1 2 66

2

2

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥π

ππ

δ δ π= + + +nmn

AE A

sn

aA

m nLa

Lamn

Am n

L a64

9 8 2169 2

.6 2 220 2

2

3

3 12

2 2

1 2 66

2 2

2

where δ = ( − ) −1 1m1 and δ = ( − ) −1 1n

2

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