A unified solution for vibration analysis of functionally gradedcylindrical, conical shells and annular plates with generalboundary conditions

Zhu Su, Guoyong Jin n, Shuangxia Shi, Tiangui Ye, Xingzhao JiaCollege of Power and Energy Engineering, Harbin Engineering University, Harbin 150001, PR China

a r t i c l e i n f o

Article history:Received 28 October 2013Received in revised form16 December 2013Accepted 2 January 2014Available online 9 January 2014

Keywords:Functionally graded materialsElastically restrained edgesFree vibrationCylindrical shellsConical shellsAnnular plates

a b s t r a c t

In this paper, a unified solution method for free vibration analysis of functionally graded cylindrical,conical shells and annular plates with general boundary conditions is presented by using the first-ordershear deformation theory and Rayleigh–Ritz procedure. The material properties of the structures areassumed to change continuously in the thickness direction according to the general four-parameterpower-law distributions in terms of volume fractions of constituents. Each of displacements androtations of those structures, regardless of boundary conditions, is expressed as a modified Fourierseries, which is constructed as the linear superposition of a standard Fourier cosine series supplementedwith auxiliary polynomial functions introduced to eliminate all the relevant discontinuities with thedisplacement and its derivatives at the edges and accelerate the convergence of series representations.The excellent accuracy and reliability of the current solutions are confirmed by comparing the presentresults with those available in the literatures, and numerous new results for functionally gradedcylindrical, conical shells and annular plates with elastic boundary conditions are presented. The effectsof boundary conditions and the material power-law distribution are also illustrated.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Functionally graded materials (FGMs) have a smooth andcontinuous variation of the material properties in the thicknessdirection. Due to the unique properties, the application of FGMshas been successfully extended to various fields. As for other shapekinds, conical, cylindrical shells and annular plates are verycommon structural elements. Recently, those structures made ofFGMs have been utilized in various engineering fields, such asaircraft, space vehicles and military industries, and in some casethey have to carry dynamic loads. Therefore, a good understandingof their vibration characteristics is necessary for designers andusers. The purpose of this paper is to study the dynamic behaviorof those structures derived from shells of revolution.

FGM shells vibration problems deal with two main concepts:shell theories and computational approaches. In the last decades, ahuge amount of research efforts have been devoted to vibrationanalysis and dynamic behaviors of the shells and a lager variety ofshell theories and computational methods have been proposedand developed by researchers.

As far as the shell deformation theories in previous studies areconcerned, there are a significant number of theories, includingtwo-dimensional (2-D) theory, three-dimensional (3-D) elasticitytheory. Most commonly used 2-D theories can be classified intothree main categories: the classical shell theory (CST), the first-order shear deformation theory (FSDT), and the higher-order sheardeformation theory (HSDT). More detailed descriptions on thedevelopment of researches on this subject may be found in severalmonographs respectively by Leissa [1], Qatu [2], Reddy [3], andCarrera [4]. Many researchers employed the CST to analyze variouscharacteristics of thin shell structures [5–16]. Since the effect ofshear deformation through the thickness is ignored, the CST isonly suitable for thin shell structures and gives proper results atlow frequencies. In order to eliminate the deficiency of the CST, theFSDT was developed, which assumes constant states of thetransverse shear stresses though the shell thickness. There exista large number of studies regarding shell structures based onFSDTs [17–34]. The shear correction factor, which is regarded as aconstant in the calculation, is introduced to adjust the transverseshear stiffness. However, the value of the shear correction factorvaries with material properties and boundary conditions in fact.The limitations of the FSDT necessitate the development of theHSDTs in which no such coefficients are required and the effects ofboth shear and normal deformations are considered. A number ofHSDTs were developed [35–41]. Compared with 2-D theories,

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International Journal of Mechanical Sciences

0020-7403/$ - see front matter & 2014 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijmecsci.2014.01.002

n Corresponding author. Tel.: þ86 451 82569458.E-mail address: guoyongjin@hrbeu.edu.cn (G. Jin).

International Journal of Mechanical Sciences 80 (2014) 62–80

since the 3-D elasticity theory does not rely on any hypotheses,such theory not only provides realistic results but also brings outphysical insights. A few investigations were carried out based on3-D elasticity theory [42–47]. Although the HSDTs and 3-Delasticity theory can provide more accurate solutions, they intro-duce mathematical and computational complexities and requiremore computational demanding compared with those FSDTs.From the existing literature, we can know that the first-ordershear deformation theory with proper shear correction factor isadequate for the prediction of the vibration behaviors of moder-ately thick shells. Therefore, in the present work, the first-ordershear deformation shell theory is just employed to formulate thetheoretical model.

The computational modeling of FGMs is an important tool tounderstanding of the structural behaviors, and has been the targetof intensive research. A huge amount of research efforts have beendevoted to the vibration analysis of FGM cylindrical shells in theliterature and a number of analytical and numerical methods havebeen proposed and developed [9–12,21–25,38–40,42,43]. Loy et al.[9] used the Ritz method to study the influence of the constituentvolume fractions and the effects of the boundary conditions on thevibration frequencies. Haddadpour [11] performed free vibrationanalysis of simply supported FGMs cylindrical shells for four setsof in-plane boundary conditions by Galerkin0s method. Iqbal et al.[12] employed the wave propagation approach to study thevibration characteristics of graded material circular cylindricalshells. Zhao et al. [21] investigated the static response and freevibration of FGM shells subjected to mechanical or thermo-mechanical loading based on Sander0s first order shear deforma-tion shell theory by using the element-free kp-Ritz method.Tornabene et al. [23,24] used generalized differential quadrature(GDQ) method to analyze the dynamic behavior of FGM conical,cylindrical shells and annular plates. A general formulation forsolving the free, steady-state and transient vibrations of FGMshells subjected to arbitrary boundary conditions is presented Quet al. [25] by means of a modified variational principle on the basisof the first-order shear deformation shell theory. Due to themechanical complexity of the structures, there are a few but notmany publications related to the vibration analysis of FGM conicalshells besides the aforementioned works of Tornabene et al.[23,24] and Qu [25]. A few methods have been developed toanalyze the dynamic behaviors of FGM conical shells[15,16,28,29,44]. Sofiyev [15,16] investigated the vibration andstability behavior of FGMs conical shells under external loads withfree and clamped boundary conditions by using the Galerkinmethod. Zhao and Liew [29] employed the element-free kp-Ritzmethod to analyze the free vibration of FGM conical shell panels.Malekzadeh [44] presented a three-dimensional free vibrationanalysis of the FGM truncated conical shells subjected to thermalenvironment, and the differential quadrature (DQ) method as anefficient method numerical tool is adopted to solve the thermaland thermo-mechanical governing equations. Compared with theanalysis of FGM cylindrical, conical shells, the literature on FGMannular plates is very limited [23,24,32–34,41,45–47]. Efraim andEisenberger [33] used exact element method to analyze thevibration of variable thickness annular isotropic and FGM platesbased on FSDT. Hosseini-Hashemi and co-authors [41] presentedan exact closed-form solution for free vibration of circular andannular moderately thick FG plates. Based on three-dimensionaltheory, dynamic analysis of multi-directional FGM annular platesis investigated by Nie and Zhong [45] using the state space-baseddifferential quadrature method. Dong [47] presented three-dimensional free vibration analysis of FGM annular plates usingChebyshev–Ritz method.

From the review of the literature, most of the previous studiesregarding the FGM cylindrical, conical shells and annular plates

are confined to the classical boundary conditions. However, thereare many non-classical boundary conditions such as elasticboundaries, which are often encountered in practical engineeringapplications, and there is a considerable lack of correspondingresearch regarding elastic boundary conditions. This work aims toprovide a unified, efficient and exact solution for free vibrationanalysis of functionally graded cylindrical, conical shells andannular plates with general boundary conditions. The presentwork can be considered as an extension of authors0 previousworks on vibration analysis of isotropic thin shells [48] andcomposite laminated shells [49,50] with general elastic bound-aries. In this paper, a unified solution method for the freevibrations of functionally graded cylindrical, conical shells andannular plates with general boundary conditions is presented. Thefirst-order shear deformation theory is adopted to formulate thetheoretical model. The material properties of the structures areassumed to change continuously in the thickness direction accord-ing to the general four-parameter power-law distributions interms of volume fractions of constituents. Each of displacementsand rotations of those structures, regardless of boundary condi-tions, is expressed as a modified Fourier series. Mathematically,such a series expansion is capable of representing any functionincluding the exact solutions. Rayleigh–Ritz procedure is used toobtain the exact solution base on the energy functions of thosestructures. The accuracy and reliability of the current solutions areconfirmed by comparing the present results with those availablein the literature. The effects of boundary conditions and thematerial power-law distribution are also illustrated.

2. Theoretical formulations

2.1. Description of the model

The geometry of shells considered hereafter is a surface ofrevolution. A differential element of such shell with its coordinatesystem is illustrated in Fig. 1. The thickness of shell element isrepresented by h respectively. The coordinate system composed ofcoordinates x, θ, and z is located on the reference surface (z¼0) ofthe shell. Rxand Rθ are the radii of curvature of the referencesurface along x and θ axes, respectively. The general boundaryconditions are represented as three of independent linear springs(ku, kv and kw) and two sets of rotational springs (Kx,Kθ) placed atthe ends, and different boundary conditions can be obtained by setting

Fig. 1. Geometry and notations of differential element of structures.

Z. Su et al. / International Journal of Mechanical Sciences 80 (2014) 62–80 63

proper spring stiffnesseses. For example a clamped boundary can besimulated by assigning the springs0 stiffness at infinity, which isrepresented by a very large number, 1015 N/m. and a free boundarycan be obtained by assigning the springs0 stiffness at zero.

Typically, FGM shells made from a mixture of two materialphases. In this paper, it is assumed that the FGM shells are made ofa mixture of ceramic and metal. Young0s modulus E(z), densityρðzÞ and Poisson0s ratio μðzÞ are assumed to vary continuouslythrough the shells thickness and can be expressed as a linearcombination:

EðzÞ ¼ ðEc�EmÞVcþEm

ρðzÞ ¼ ðρc�ρmÞVcþρm

μðzÞ ¼ ðμc�μmÞVcþμm ð1Þ

in which the subscripts c andm represent the ceramic and metallicconstituents, respectively, and the volume fraction Vc follows two

general four-parameter power-law distributions [20,23,24]:

FGMIða=b=c=pÞ : Vc ¼ 1�a12þ zh

� �þb

12þ zh

� �c� �pð2:aÞ

FGMIIða=b=c=pÞ : Vc ¼ 1�a12� zh

� �þb

12� zh

� �c� �pð2:bÞ

where p is the power-law exponent and takes only positive values.The parameters a, b and c determine the material variation profilethrough the functionally graded shell thickness. The volumefraction of all the constituent materials should add up to one, i.e.

VcþVm ¼ 1 ð3Þwhen the value of p equals to zero or infinity, the homogeneousisotropic material is obtained as a special case of the functionallygraded material. In addition, the different power-law distributionscan be obtained by setting proper value of the parameters a, b, cand p. The ordinary volume fraction profiles are presented in

0 0.2 0.4 0.6 0.8 1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

p=10

p=5

p=2

p=1

p=0.5

p=0.2

p=0.1

Vc

z/h

0 0.2 0.4 0.6 0.8 1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

p=10

p=5

p=2

p=1

p=0.5p=0.2

p=0.1

Vc

z/h

0 0.2 0.4 0.6 0.8 1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

p=10

p=5

p=2

p=1

p=0.5p=0.2

p=0.1

Vc Vc

Vc Vc

z/h

0 0.2 0.4 0.6 0.8 1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

p=10

p=5

p=2

p=1

p=0.5p=0.2

p=0.1

z/h

0 0.2 0.4 0.6 0.8 1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

p=10

p=5

p=2

p=1p=0.5

p=0.2 p=0.1

z/h

0 0.2 0.4 0.6 0.8 1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

p=10

p=5

p=2

p=1p=0.5

p=0.2 p=0.1

z/h

Fig. 2. Variations of the volume fraction Vc through the shell thickness for different values of power-law exponent p: (a) FGMI(a¼1/b¼0/c/p); (b) FGMII(a¼1/b¼0/c/p); (c) FGMI

(a¼1/b¼0.5/c¼2/p); (d) FGMII(a¼1/b¼0.5/c¼2/p); (e) FGMI(a¼0/b¼�0.5/c¼2/p); (f) FGMII(a¼0/b¼�0.5/c¼2/p).

Z. Su et al. / International Journal of Mechanical Sciences 80 (2014) 62–8064

literature as a special case of the general distribution laws bysetting a¼1 and b¼0. The variations of volume fraction Vc fordifferent values of the parameters a, b, c and p are depicted inFig. 2. More detailed descriptions on the material variation profileof FGMs can be found in Ref [20,23,24].

2.2. Kinematic relations and stress resultants

According to the FSDT, the displacement components of anarbitrary point within the FGM shell domain, designated by u, vand w, are expressed as

uðx; θ; z; tÞ ¼ uðx; θ; tÞþzψxðx; θ; tÞ ð4:aÞ

vðx; θ; z; tÞ ¼ vðx; θ; tÞþzψθðx; θ; tÞ ð4:bÞ

wðx; θ; z; tÞ ¼wðx; θ; tÞ ð4:cÞ

where u, v, and w denote the displacements of correspondingpoint on reference surface in the x, θ w, and z directions,respectively. ψxand ψθ are the rotations of normal to the referencesurface about the x and θ axes, respectively, and t is the time.When h=Rθ is negligible compared with unity, the membranestrains, denoted by ε0x , ε

0θ and ε0xθ , and curvature changes, denoted

by χ0x , χ0θ and χ0xθ , of the reference surface are given as:

ε0x ¼∂u∂x

; ε0θ ¼1A∂v∂θ

þuA∂A∂x

þwRθ; γ0xθ ¼

1A∂u∂θ

þ∂v∂x

�vA∂A∂x

;

χx ¼∂ψx

∂x; χθ ¼

1A∂ψθ

∂θþψx

A∂A∂x

; χxθ ¼1A∂ψx

∂θþ∂ψθ

∂x�ψθ

A∂A∂x

ð5Þ

where A is the parameter which is decided by the shape of the shell.According to Fig. 3, for the cylindrical shell, Rθ ¼ R, A¼ R; for theconical shell, Rθ ¼ x tan αþR1= cos α, A¼ R1þx sin α; for annularplate, Rθ ¼1, A¼ R1þx.

The linear strain–displacement relations in the shell space aredefined as:

εx ¼ ε0x þzχx; εθ ¼ ε0θ þzχθ ; γxθ ¼ γ0xθþzχxθ ;

γxz ¼ ψxþ∂w∂x

; γθz ¼ ψθ�vRθ

þ1A∂w∂θ

ð6Þ

Based on Hooke0s law, the stress–strain relations of the shellare written as:

sx

sθ

τxθ

τxz

τθz

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

¼

Q11ðzÞ Q12ðzÞ 0 0 0Q12ðzÞ Q11ðzÞ 0 0 0

0 0 Q66ðzÞ 0 00 0 0 Q66ðzÞ 00 0 0 0 Q66ðzÞ

26666664

37777775

εx

εθ

γxθγxzγθz

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

ð7Þ

where the elastic constants Qij(z) are functions of thicknesscoordinate z and are defined as:

Q11ðzÞ ¼EðzÞ

1�μ2ðzÞ; Q12ðzÞ ¼μðzÞEðzÞ1�μ2ðzÞ; Q66ðzÞ ¼

EðzÞ2½1þμðzÞ� ð8Þ

The stress and moment resultants are given as:

ðNx;Nθ ;Nxθ ;Mx;Mθ ;MxθÞ ¼Z h=2

�h=2ðsx;sθ ; τxθ ; zsx; zsθ ; zτxθÞ dz ð9Þ

ðQx;Q θÞ ¼ κ

Z h=2

�h=2ðτxz; τθzÞ dz ð10Þ

where Nx, Nθ and Nxθ are the in-place force resultants, Mx, Mθ andMxθ are moment resultants, Qx, Q θ are transverse shear forceresultants. The shear correction factor κ is computed such that thestrain energy due to transverse shear stresses in Eq. (10) equals thestrain energy due to the true transverse stresses predicted by thethree-dimensional elasticity theory [3]. Since the shear correctionfactor for a FGM shell depends on shell parameters, such as materialproperties and boundary conditions, it is difficult to obtain theaccurate value of the shear correction stresses. In this paper, theshear correction factors κ will be uniformly selected by 5/6 [20–25].

Substituting Eqs. (6)–(8) into Eqs. (9) and (10) followingconstitutive equation is obtained

Nx

Nθ

Nxθ

Mx

Mθ

Mxθ

Qx

Q θ

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;

¼

A11 A12 0 B11 B12 0 0 0A12 A11 0 B12 B11 0 0 00 0 A66 0 0 B66 0 0B11 B12 0 D11 D12 0 0 0B12 B11 0 D12 D11 0 0 00 0 B66 0 0 D66 0 00 0 0 0 0 0 κA66 00 0 0 0 0 0 0 κA66

2666666666666664

3777777777777775

ε0xε0θγ0xθχxχθχxθγxzγθz

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;

ð11Þ

Fig. 3. Schematic sketch of structures: (a) FGM cylindrical shells; (b) FGM conical shells; (c) FGM annular plates.

Z. Su et al. / International Journal of Mechanical Sciences 80 (2014) 62–80 65

where Aij, Bij and Dij (i,j¼1, 2 and 6) are the extensional,extensional-bending coupling, bending stiffness, and they arerespectively expressed as

ðAij;Bij;DijÞ ¼Z h=2

�h=2QijðzÞð1; z; z2Þdz ð12Þ

In this study, the general boundary conditions along each edgecan be described as follows:

ku0u¼Nx; kv0v¼Nθ ; kw0w¼ Qx

Kx0ψx ¼ �Mx; Kθ0ψθ ¼ �Mxθ ; at x¼ 0ð13Þ

kuau¼ �Nx; kvav¼ �Ns; kwaw¼ �Qx

Kxaψx ¼Mx; Kθaψθ ¼Mxθ ; at x¼ að14Þ

2.3. Solution procedure

To obtain the exact solutions for free vibration of FGM shellsRayleigh–Ritz method is employed in present paper. The energyfunction L is defined by Lagrangian function and can be expressed as;

L¼ Tmax�Umax ð15Þ

where Tmax and Umax are the maximum kinetic and strain energies ofthe FGM shells.

The total kinetic energy T for the FGM shell can be expressed inintegral form as:

T ¼ 12

Z Z ZVρ

∂u∂t

� �2

þ ∂v∂t

� �2

þ ∂w∂t

� �2" #

A dx dθ dz ð16Þ

Substituting Eq. (4) into Eq. (16), the kinetic energy T is written as

T ¼ 12

Z ZS

I0 ∂u∂t

� �2þ ∂v∂t

� �2þ ∂w∂t

� �2h i�

þ2I1∂u∂t

� �∂ψx

∂t

� �þ ∂v

∂t

� �∂ψθ

∂t

� �� �

þ I2∂ψx

∂t

� �2

þ ∂ψθ

∂t

� �2" #)

Adθ dx ð17Þ

where S represents the reference surface area, I0, I1 and I2 are theinertia terms, given as:

ðI0; I1; I2Þ ¼Z h=2

�h=2ρð1; z; z2Þdz ð18Þ

The total strain energy U of the FGM shell is defined as:

U ¼ΠþP ð19Þ

where Π designates the strain energy including stretching strainenergy, bending strain energy and bending-stretching couplingenergy, respectively, P represents the potential energy stored inthe boundary spring of the FGM shell.

The strain energy Π can be written as:

Π ¼ 12

Z Z ZVðsxεxþsθεθþτxθγxθþτxzγxzþτθzγθzÞAdx dθ dz ð19Þ

or

Π ¼ 12

Z ZSðNxε

0x þNθε

0θ þNxθε

0xθþNxχxþNθχθþNxθχxθ

þQxγxzþQ θγθzÞAdθ dx ð20Þ

Substituting Eqs. (5) and (11) into Eq. (20), the strain energy Π isdepicted as

Π ¼ 12

Z ZS

A11∂u∂x

� �2

þA111A∂v∂θ

þuA∂A∂x

þwRθ

� �2

þA661A∂u∂θ

þ∂v∂x

�vA∂A∂x

� �2

þκA66 ψθ�vRθ

þ1A∂w∂θ

� �2

þ2A12∂u∂x

� �1A∂v∂θ

þuA∂A∂x

þwRθ

� �þκA66 ψxþ

∂w∂x

� �2

þD11∂ψx

∂x

� �2

þD111A∂ψθ

∂θþψx

A∂A∂x

� �2

þ2D12∂ψx

∂x

� �1A∂ψθ

∂θþψx

A∂A∂x

� �

þD66ð1A∂ψx

∂θþ∂ψθ

∂x�ψθ

A∂A∂x

Þ2þ2B111A∂v∂θ

þuA∂A∂x

þwRθ

� �1A∂ψθ

∂θþψx

A∂A∂x

� �

þ2B121A∂v∂θ

þuA∂A∂x

þwRθ

� �∂ψx

∂x

� �þ2B12

∂u∂x

� �1A∂ψθ

∂θþψx

A∂A∂x

� �

þ2B661A∂u∂θ

þ∂v∂x

�vA∂A∂x

� �1A∂ψx

∂θþ∂ψθ

∂x�ψθ

A∂A∂x

� �þ2B11

∂u∂x

� �∂ψx

∂x

� �

8>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>;

Adθ dx

ð21Þ

The potential energy P can be expressed as

P ¼ZC

ðku0u2þkv0v2þkw0w2þKx0ψ2x þKθ0ψ

2θ Þx ¼ 0

þðkuLu2þkvLv2þkwLw2þKxLψ2x þKθLψ2

θ Þx ¼ L

( )Adθ ð22Þ

where C is the circumference of the FGM shell.In the Rayleigh–Ritz method, approximate solutions are

obtained by minimizing the energy function with respect to thecoefficients of the admissible functions, and improvements in theefficiency depend on the choice of the admissible functions.Therefore, it is very significant to take an appropriate form ofadmissible functions. There are many forms of admissible func-tions and most commonly used forms of admissible functions aresimple or orthogonal polynomials and trigonometric functions.However, the lower order polynomials cannot form a complete set,and the higher order polynomials trend to become numericallyunstable due to the computer round-off errors. The conventionalFourier series expression will have a convergence problem alongthe boundary conditions except for a few simple boundary edges.In order to construct appropriate admissible functions, a modifiedFourier series method previously proposed for the vibrationanalysis of elastically supported isotropic beams [51] and thinplates [52,53] is applied here. Each of displacements and rotationsfunctions of the FGM shell is expanded as a standard Fourier serieswith auxiliary functions [48-53]:

uðx; θ; tÞ ¼ Uðx; θÞejωt

¼∑M

m ¼ 0∑N

n ¼ 0Asmn cos λmx cos ðnθÞþ ∑

2

l ¼ 1∑N

n ¼ 0aslnζlðxÞ cos ðnθÞ

∑M

m ¼ 0∑N

n ¼ 0Aamn cos λmx sin ðnθÞþ ∑

2

l ¼ 1∑N

n ¼ 0aalnζlðxÞ sin ðnθÞ

8>>>><>>>>:

9>>>>=>>>>;ejωt

ð23:aÞ

vðx; θ; tÞ ¼ V ðx; θÞejωt

¼∑M

m ¼ 0∑N

n ¼ 0Bsmn cos λmx sin ðnθÞþ ∑

2

l ¼ 1∑N

n ¼ 0bslnζlðxÞ sin ðnθÞ

∑M

m ¼ 0∑N

n ¼ 0Bamn cos λmx cos ðnθÞþ ∑

2

l ¼ 1∑N

n ¼ 0balnζlðxÞ cos ðnθÞ

8>>>><>>>>:

9>>>>=>>>>;ejωt

ð23:bÞ

wðx; θ; tÞ ¼Wðx; θÞejωt

¼∑M

m ¼ 0∑N

n ¼ 0Csmn cos λmx cos ðnθÞþ ∑

2

l ¼ 1∑N

n ¼ 0cslnζlðxÞ cos ðnθÞ

∑M

m ¼ 0∑N

n ¼ 0Camn cos λmx sin ðnθÞþ ∑

2

l ¼ 1∑N

n ¼ 0calnζlðxÞ sin ðnθÞ

8>>>><>>>>:

9>>>>=>>>>;ejωt

ð23:cÞ

Z. Su et al. / International Journal of Mechanical Sciences 80 (2014) 62–8066

ψxðx; θ; tÞ ¼Ψ xðx; θÞejωt

¼∑M

m ¼ 0∑N

n ¼ 0Dsmn cos λmx cos ðnθÞþ ∑

2

l ¼ 1∑N

n ¼ 0dslnζlðxÞ cos ðnθÞ

∑M

m ¼ 0∑N

n ¼ 0Damn cos λmx sin ðnθÞþ ∑

2

l ¼ 1∑N

n ¼ 0dalnζlðxÞ sin ðnθÞ

8>>>><>>>>:

9>>>>=>>>>;ejωt

ð23:dÞ

ψθðx; θ; tÞ ¼Ψθðx; θÞejωt

¼∑M

m ¼ 0∑N

n ¼ 0Esmn cos λmx sin ðnθÞþ ∑

2

l ¼ 1∑N

n ¼ 0eslnζlðxÞ sin ðnθÞ

∑M

m ¼ 0∑N

n ¼ 0Eamn cos λmx cos ðnθÞþ ∑

2

l ¼ 1∑N

n ¼ 0ealnζlðxÞ cos ðnθÞ

8>>>><>>>>:

9>>>>=>>>>;ejωt

ð23:eÞwhere λm¼mπ/L,j¼

ffiffiffiffiffiffiffiffi�1

p. ω denotes the natural angular frequency of

the FGM shell. Non-negative integer n represents the circumferentialwave number of the corresponding mode shape. The superscripts,s and a, represent the symmetric and anti-symmetric modes. M andN are the highest degrees taken in the series. Vðx; θÞ, Wðx; θÞ, Ψ xðx; θÞand Ψθðx; θÞ express the displacements and rotations amplitudefunctions. As

mn, Aamn, B

smn, B

amn, C

smn, C

amn, D

smn,D

amn, E

smn and Eamn are

the Fourier coefficients of Fourier series expansions, respectively. asln,aaln, bsln, baln, csln, caln, dsln, daln, esln and ealn are the supplementedcoefficients of the auxiliary functions. ζlðxÞ represents a set ofclosed-form sufficiently smooth functions defined over [0, L]. Accord-ing to FSDT, it is required that at least two-order derivatives of thedisplacements and rotations functions exist and continuous at anypoint on the FGM shell. Therefore two auxiliary functions in xdirection are supplemented as demonstrated in Eqs. (23) (a)–(e).The closed-form auxiliary functions are given as follows

ζ1ðxÞ ¼ xxL�1

�2; ζ2ðxÞ ¼

x2

LxL�1

�ð24Þ

It is easy to verify that ζ01xð0Þ ¼ ζ02xðLÞ ¼ 1, and all the other firstderivatives are equal to zero at the edges. It can be provenmathematically that the series expansion given in Eq. (23) is ableto expand and uniformly converge to any function (including theexact displacement solution) whose first-order partial derivatives arecontinuous over the area of the shell. This series also can be simplydifferentiated, through term-by-term, to obtain uniformly convergentseries expansions for up to the second-order derivatives.

Substituting Eq. (23) into Eqs. (17), (21) and (22), the maximumkinetic energy Tmax and maximum strain energy Umax of the shellcan be written as

Tmax ¼ ω212

Z ZSfI0ðU2þV2þW2Þþ2I1ðUΨ xþVΨθÞþ I2ðΨ x

2

þΨθ2ÞgA dθ dx ð25Þ

and

Umax ¼ΠmaxþPmax ð26Þwhere

Πmax ¼12

Z ZS

A11∂U∂x

� �2þA111A∂V∂θþU

A∂A∂xþW

Rθ

�2þA66

1A∂U∂θþ ∂V

∂x�VA∂A∂x

� �2þκA66 Ψθ� V

Rθþ1

A∂W∂θ

�2þ2A12

∂U∂x

� �1A∂V∂θþU

A∂A∂xþW

Rθ

�þκA66 Ψ xþ ∂W

∂x

� �2þD11

∂Ψ x∂x

� �2þD111A∂Ψθ∂θ þΨ x

A∂A∂x

� �2þ2D12∂Ψ x∂x

� �1A∂Ψθ∂θ þΨ x

A∂A∂x

� �þD66

1A∂Ψ x∂θ þ ∂Ψθ

∂x �ΨθA

∂A∂x

� �2þ2B111A∂V∂θþU

A∂A∂xþW

Rθ

�1A∂Ψθ∂θ þΨ x

A∂A∂x

� �þ2B12

1A∂V∂θþU

A∂A∂xþW

Rθ

�∂Ψ x∂x

� �þ2B12∂U∂x

� �1A∂Ψθ∂θ þΨ x

A∂A∂x

� �þ2B66

1A∂U∂θþ ∂V

∂x�VA∂A∂x

� �1A∂Ψ x∂θ þ ∂Ψθ

∂x �ΨθA

∂A∂x

� �þ2B11∂U∂x

� � ∂Ψ x∂x

� �

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>;

XA dθ dx

Pmax ¼ZC

ðkuoU2þkv0V2þkw0W

2þKx0Ψ2x þKθ0Ψ

2θ Þx ¼ 0

þðkuoU2þkv0V2þkw0W

2þKx0Ψ2x þKθ0Ψ

2θ Þx ¼ a

( )Adθ

Substituting the above Eqs. (25) and (26) into Eq. (15), andminimizing the Lagrangian energy functional L with respect to theunknown coefficients

∂L∂αu

¼ 0ðαu ¼ Asmn;A

amn; a

smn; a

amn;B

smn;B

amn; b

smn; b

amn;C

smn;C

amn; c

smn; c

amn⋯

Dsmn;D

amn; d

smn; d

amn; E

smn; E

amn; e

smn; e

amn; Þ ð27Þ

which leads to the following governing eigenvalue equation inmatrix form:

f½K��ω2½M�gfXgf0g ð28Þhere [K] is the symmetric stiffness matrix obtained from the totalstrain energy, and [M] is the symmetric mass matrix obtained fromthe total kinetic energy. The general expressions for the elements ofthe stiffness matrix [K] and mass matrix [M] are given in Appendix A.{X} is the global generalized coordinate vector composed of theunknown coefficients expressed in the following form:

fXg ¼ ½fXug; fXvg; fXwg; fXxg; fXθg�Τ ð29Þin which

fXug ¼ ½As00;⋯;As

MN ; as10;⋯; as1N ; a

s20;⋯; as2N ;A

a00;⋯;Aa

MN ;

aa10;⋯; aa1N ; aa20;⋯; aa2N�

fXvg ¼ ½Bs00;⋯;Bs

MN ; bs10;⋯; bs1N ; b

s20;⋯; bs2N ;B

a00;⋯;Ba

MN ;

ba10;⋯; ba1N ; ba20;⋯; ba2N �

fXwg ¼ ½Cs00;⋯;Cs

MN ; cs10;⋯; cs1N ; c

s20;⋯; cs2N ;C

a00;⋯;Ca

MN ;

ca10;⋯; ca1N ; ca20;⋯; ca2N�

fXxg ¼ ½Ds00;⋯;Ds

MN ; ds10;⋯; ds1N ;d

s20;⋯; ds2N ;D

a00;⋯;Da

MN ;

da10;⋯; da1N ; da20;⋯; da2N �

fXθg ¼ ½Es00;⋯; EsMN ; es10;⋯; es1N ; e

s20;⋯; es2N ; E

a00;⋯; EaMN ;

ea10;⋯; ea1N ; ea20;⋯; ea2N�

All the natural frequencies and mode shapes of the FGM shellcan be determined by solving Eq. (28), which is a standardcharacteristic equation.

3. Numerical examples

In this section, a considerable number of examples about thefree vibration problems of FGM cylindrical, conical shells andannular plates are presented to verify the accuracy and reliabilityof the present method. The effects of boundary conditions, volumefractions and thickness-to-length ratios on the frequencies of theFGM shells are investigated. Besides the classical boundary condi-tions including free (F), shear-diaphragm (SD), simply-supported(SS) and clamped (C) supports of the shells, six types of elasticboundary conditions are considered. First type of elastic boundarycondition E1 only allows elastically restrained displacement in themeridional direction (i.e. ua0, v¼w¼ψx¼ψθ¼0); second type ofelastic boundary E2 is considered to be that only circumferentialdisplacements are elastically restrained (i.e. va0, u¼w¼ψx¼ψθ¼0); both meridional and circumferential displacements areelastically restrained, which is defined as third type of boundarycondition E3 (i.e. va0, ua0, w¼ψx¼ψθ¼0); fourth type of elasticboundary E4 is considered to be that all displacements androtations are elastically restrained (i.e. ua0, va0, wa0, ψxa0,ψθa0); fifth type of elastic boundary condition E5 only allowselastically restrained displacement in the normal direction(i.e. wa0, u¼v¼ψx¼ψθ¼0); Sixth type of elastic boundary E6 isconsidered to be that only rotation about x axes are elasticallyrestrained (i.e. ψxa0, u¼v¼w¼ψθ¼0). The corresponding springstiffnesses for the classical boundaries and the elastic boundariesare given in Table 1. The accuracy and reliability of the proposedmethod will be demonstrated by several numerical examples and

Z. Su et al. / International Journal of Mechanical Sciences 80 (2014) 62–80 67

numerous new results for FGM shells with elastically restrainededges will be presented in the following.

3.1. Free vibration analysis of FGM cylindrical shells

In this section, FGM cylindrical shells with different boundaryconditions and two general four-parameter power-law distributionsare investigated. First, a convergence study is checked to ensure thereliability of the present method. The convergence of frequencies ofSD–SD FGM cylindrical shells is shown in Table 2. The FGM cylindricalshells are composed of aluminum (metal) and zirconia (ceramic). Thegeometrical and material parameters of the FGM cylindrical shells aretaken to be L/R¼20, h/R¼0.002 and R¼1m, Em¼70 GPa, μm¼0.3 andρm¼2707 kg/m3, Ec¼168 GPa, μc¼0.3 and ρc¼5700 kg/m3. Thevolume fraction parameters are taken to be a¼1, b¼0, p¼1. It isobviously that the present method has an excellent convergence.

To further validate the accuracy and reliability of presentsolutions, more numerical examples are presented. In Table 3,the first eight frequencies for F–C FGM cylindrical shells madeof aluminum and zirconia with different power-law exponents p(i.e. p¼0, 0.6, 1, 5, 20, and 1) are presented. The geometrical

parameters of the FGM cylindrical shells are taken to be h/R¼0.1, L/R¼2 and R¼1m. The results are compared with other publishedsolutions reported by Tornabebe [24] using differential quadraturemethod base on FSDT and Qu et al.[25] employing a modifiedvariational approach and FSDT. It is seen that the present solutionsagree very well with the referential datum. The comparison of presentresults with those solutions obtained by Qu et al. [25] for FGMcylindrical shells with various boundaries (i.e. SD–SD, SS–SS and C–C)and different power-law exponents (i.e. p¼0.6 and 5) are shown inTable 4. The geometrical and material parameters of the FGMcylindrical shells are same with those in Table 3. An excellentagreement can be seen from the comparison, and the difference isvery small, and does not exceed 0.036% for the worst case. FromTables 3 and 4, the excellent accuracy and reliability of the currentsolutions are demonstrated.

Numerous new results are presented in Tables 5 and 6 for FGMcylindrical shells with a variety of boundary conditions includingclassical cases, elastic restraints and their combinations. The FGMcylindrical shells are fabricated from aluminum and zirconia. Thegeometrical parameters of the FGM cylindrical shells are taken tobe L/R¼2, R¼1 m and h/R¼0.02, 0.05 and 0.10. Table 5 shows the

Table 1The corresponding spring stiffnesses for the various boundary conditions.

BC Essential conditions ku kv kw Kx Kθ

C u¼v¼w¼ψx¼ψθ¼0 1e15 1e15 1e15 1e15 1e15

SS u¼v¼w¼ψθ¼0 1e15 1e15 1e15 0 1e15

SD v¼w¼0 0 1e15 1e15 0 0F No constraints 0 0 0 0 0E1 ua0, v¼w¼ψx¼ψθ¼0 1e9 1e15 1e15 1e15 1e15

E2 va0, u¼w¼ψx¼ψθ¼¼0 1e15 1e9 1e15 1e15 1e15

E3 va0, ua0, w¼ψx¼ψθ¼0 1e9 1e9 1e15 1e15 1e15

E4 ua0, va0, wa0, ψxa0,ψθa0 1e9 1e9 1e9 1e9 1e9

E5 wa0, u¼v¼ψx¼ψθ¼0 1e15 1e15 1e9 1e15 1e15

E6 ψxa0, u¼v¼w¼ψθ¼0 1e15 1e15 1e15 1e9 1e15

Table 2Convergence of frequencies of SD-SD FGM cylindrical shells (L/R¼20, h/R¼0.002 and R¼1 m).

Mode Truncated number

n m M¼6 M¼8 M¼10 M¼12 M¼14 M¼16 M¼20

FGMI(a¼1/b¼0/c/p¼1)

1 1 14.29 14.29 14.29 14.29 14.29 14.29 14.292 52.49 52.47 52.47 52.47 52.47 52.47 52.473 105.43 105.36 105.35 105.34 105.34 105.34 105.34

2 1 4.83 4.83 4.83 4.83 4.83 4.83 4.832 18.26 18.24 18.23 18.23 18.23 18.23 18.233 39.77 39.64 39.61 39.60 39.60 39.60 39.59

3 1 4.38 4.38 4.38 4.38 4.38 4.38 4.382 9.54 9.52 9.51 9.51 9.51 9.51 9.513 19.89 19.75 19.71 19.70 19.70 19.70 19.70

FGMII(a¼1/b¼0/c/p¼1)

1 1 14.29 14.29 14.29 14.29 14.29 14.29 14.292 52.49 52.47 52.47 52.47 52.47 52.47 52.473 105.43 105.36 105.35 105.34 105.34 105.34 105.34

2 1 4.83 4.83 4.83 4.83 4.83 4.83 4.832 18.26 18.23 18.23 18.23 18.23 18.23 18.233 39.77 39.64 39.61 39.60 39.60 39.59 39.59

3 1 4.38 4.38 4.38 4.38 4.38 4.38 4.382 9.53 9.52 9.51 9.51 9.51 9.51 9.513 19.89 19.75 19.71 19.70 19.70 19.70 19.70

Z. Su et al. / International Journal of Mechanical Sciences 80 (2014) 62–8068

Table 3Comparison of the first eight frequencies (Hz) for FGM cylindrical shells (h/R¼0.1, L/R¼2, R¼1 m; boundary conditions: F–C).

p Method Mode

1 2 3 4 5 6 7 8

FGMI(a¼1/b¼0.5/c¼2/p)

0 Ref [24] 152.93 152.93 220.06 220.06 253.78 253.78 383.55 383.56present 152.89 152.89 219.97 219.97 253.79 253.79 383.44 383.44

0.6 Ref [24] 152.25 152.25 219.86 219.86 252.17 252.17 383.39 383.40Ref [25] 152.02 152.02 219.54 219.54 251.91 251.91 382.87 382.87present 152.07 152.07 219.59 219.59 251.96 251.96 382.95 382.95

1 Ref [24] 151.77 151.77 219.56 219.56 251.14 251.14 382.97 382.97present 151.52 151.52 219.19 219.19 250.81 250.81 382.35 382.35

5 Ref [24] 148.97 148.97 218.87 218.88 244.40 244.40 382.46 382.47Ref [25] 148.50 148.50 218.16 218.16 243.73 243.73 381.26 381.26present 148.53 148.53 218.21 218.21 243.76 243.76 381.33 381.33

20 Ref [24] 146.46 146.46 215.90 215.90 239.84 239.84 377.34 377.34Ref [25] 146.21 146.21 215.50 215.50 239.54 239.54 376.69 376.69present 146.24 146.24 215.55 215.55 239.57 239.57 376.76 376.76

1 Ref [24] 143.25 143.25 206.12 206.12 237.71 237.71 359.26 359.27present 143.21 143.21 206.04 206.04 237.72 237.72 359.16 359.16

FGMII(a¼1/b¼0.5/c¼2/p)

0 Ref[24] 152.93 152.93 220.06 220.06 253.78 253.78 383.55 383.56present 152.89 152.89 219.97 219.97 253.79 253.79 383.44 383.44

0.6 Ref [24] 151.82 151.82 218.74 218.74 251.74 251.74 381.29 381.29Ref [25] 151.85 151.85 218.79 218.79 251.91 251.91 381.42 381.42present 151.90 151.90 218.85 218.85 251.96 251.96 381.50 381.50

1 Ref [24] 151.10 151.10 217.83 217.83 250.48 250.48 379.70 379.70present 151.25 151.25 218.03 218.03 250.82 250.82 380.08 380.08

5 Ref [24] 147.58 147.58 215.22 215.22 243.14 243.14 375.57 375.58Ref [25] 147.89 147.89 215.65 215.65 243.75 243.75 376.34 376.34present 147.92 147.92 215.69 215.69 243.79 243.79 376.42 376.42

20 Ref [24] 145.83 145.83 214.20 214.20 239.28 239.28 374.14 374.14Ref [25] 145.92 145.92 214.32 214.32 239.55 239.55 374.38 374.38present 145.95 145.95 214.36 214.36 239.58 239.58 374.45 374.45

1 Ref [24] 143.25 143.25 206.12 206.12 237.71 237.71 359.26 359.27present 143.21 143.21 206.04 206.04 237.72 237.72 359.16 359.16

Table 4Frequencies (Hz) for FGM cylindrical shells with various boundaries and power-law exponents p (h/R¼0.1, L/R¼2, R¼1 m; m¼1).

p BC Method n

1 2 3 4 5 6 7

FGMI(a¼1/b¼0.5/c¼2/p)

0.6 SD–SD Qu [25] 517.81 319.41 304.35 437.76 645.99 901.34 1194.63present 517.86 319.47 304.46 437.89 646.13 901.49 1194.79

SS–SS Qu[25] 528.23 363.83 346.90 461.40 658.82 909.22 1200.10present 528.28 363.87 346.93 461.41 658.83 909.22 1200.10

C–C Qu[25] 543.84 390.65 375.30 483.30 674.82 921.30 1209.50present 543.89 390.69 375.35 483.34 674.86 921.34 1209.54

5 SD–SD Qu[25] 500.21 309.62 299.76 434.80 642.40 895.95 1186.49present 500.27 309.70 299.88 434.94 642.54 896.10 1186.64

SS–SS Qu[25] 512.32 357.14 345.33 460.70 656.83 904.96 1192.76present 512.35 357.16 345.35 460.71 656.84 904.97 1192.77

C–C Qu[25] 526.72 380.18 369.21 479.22 670.55 915.45 1200.99present 526.75 380.22 369.25 479.25 670.59 915.48 1201.03

FGMII(a¼1/b¼0.5/c¼2/p)

0.6 SD–SD Qu[25] 517.98 319.72 304.09 436.53 643.80 898.18 1190.46present 518.02 319.75 304.17 436.65 643.94 898.33 1190.62

SS–SS Qu[25] 527.06 360.05 342.14 457.11 654.73 904.84 1195.11present 527.11 360.09 342.17 457.13 654.74 904.84 1195.12

C–C Qu[25] 543.81 390.47 374.69 481.94 672.60 918.14 1205.36present 543.86 390.51 374.74 481.99 672.64 918.18 1205.39

5 SD–SD Qu[25] 500.76 310.50 298.72 430.57 634.97 885.25 1172.42present 500.80 310.51 298.78 430.67 635.09 885.39 1172.57

SS–SS Qu[25] 508.60 345.18 330.37 447.08 643.59 890.54 1176.20present 508.63 345.20 330.38 447.09 643.60 890.54 1176.20

C–C Qu[25] 526.56 379.45 367.04 474.57 663.00 904.74 1186.98present 526.59 379.49 367.08 474.60 663.04 904.78 1187.02

Z. Su et al. / International Journal of Mechanical Sciences 80 (2014) 62–80 69

fundamental frequencies of the FGM cylindrical shells subjected toclassical boundaries (i.e. SD–SD, C–C, SD–C and SS–SS) and elasticboundaries (i.e. E1–E1, E2–E2, E3–E3 and E4–E4). The fundamentalfrequencies of FGM cylindrical shells with classical-elasticrestraints (i.e. SD–E1, SD–E2, SD–E3, SD–E4, C–E1, C–E2, C–E3 andC–E4) are presented in Table 6. It is obvious that the fundamentalfrequencies of the FGM cylindrical shells with classical boundary

conditions (i.e. SD–SD, C–C, C–SD and SS–SS) increase asthickness-to-radius ratio increases. However, the variation of thefundamental frequencies of the FGM shells subjected to elasticrestraints and classical-elastic restraints is more complex. It isinteresting that the fundamental frequencies of the FGMI cylind-rical shells are higher than those of FGMII cylindrical shells. It isalso obvious that the power-law exponent has significant effect on

Table 5Frequencies (Hz) for FGM cylindrical shell with classical and elastic boundaries (L/R¼2 and R¼1 m).

p h/R BC

SD–SD C–C SD–C SS–SS E1–E1 E2–E2 E3–E3 E4- E4

FGMI(a¼1/b¼0.5/c¼2/p)

0 0.02 142.16 189.67 168.70 184.96 155.80 187.32 154.36 151.720.05 217.05 281.87 249.05 266.98 238.03 263.87 232.24 209.730.10 305.68 376.85 339.35 346.28 208.83 324.28 208.83 194.17

0.5 0.02 141.52 188.94 167.87 184.73 155.72 186.55 154.27 151.860.05 216.03 280.51 247.85 266.78 237.36 262.84 231.62 210.900.10 304.68 375.62 338.22 346.87 219.74 325.27 219.74 202.77

5 0.02 138.20 185.37 163.52 182.39 155.24 182.39 153.73 152.170.05 210.68 273.27 241.52 262.48 233.91 257.45 228.48 214.430.10 299.88 369.25 332.57 345.35 270.75 330.02 270.75 233.10

20 0.02 136.28 183.05 161.11 179.25 154.50 180.00 152.97 151.670.05 207.71 269.15 237.98 256.34 231.55 254.14 226.23 214.340.10 296.27 364.26 328.32 337.23 291.58 326.29 291.58 239.30

FGMII(a¼1/b¼0.5/c¼2/p)

0 0.02 142.16 189.67 168.70 184.96 155.80 187.32 154.36 151.720.05 217.05 281.87 249.05 266.98 238.03 263.87 232.24 209.730.10 305.68 376.85 339.35 346.28 208.83 324.28 208.83 194.17

0.5 0.02 141.49 188.87 167.84 183.67 155.68 186.51 154.23 151.820.05 216.00 280.40 247.78 264.36 237.25 262.73 231.50 210.770.10 304.44 375.10 337.87 342.83 219.74 325.25 219.74 202.76

5 0.02 138.07 185.08 163.38 178.46 155.08 182.23 153.57 152.010.05 210.52 272.76 241.21 253.50 233.41 256.93 227.97 213.910.10 298.78 367.08 331.06 330.38 270.75 329.50 270.75 232.49

20 0.02 136.22 182.91 161.04 177.45 154.42 179.92 152.89 151.590.05 207.62 268.90 237.82 252.21 231.30 253.88 225.99 214.090.10 295.73 363.22 327.58 330.34 291.57 325.92 291.57 238.99

Table 6Frequencies (Hz) for FGM cylindrical shell with classical-elastic boundaries (L/R¼2, R¼1 m ).

p h/R BC

SD-E1 SD-E2 SD-E3 SD-E4 C-E1 C-E2 C-E3 C-E4

FGMI(a¼1/b¼0.5/c¼2/p)

0 0.02 148.90 165.21 148.26 146.97 175.72 188.65 174.70 173.080.05 202.92 240.08 202.92 202.83 259.91 272.12 256.85 242.880.10 146.24 320.71 146.24 146.18 359.32 361.99 353.97 272.89

0.5 0.02 148.54 164.49 147.89 146.72 175.18 187.95 174.16 172.690.05 212.79 239.06 212.79 212.37 258.89 270.95 255.86 243.140.10 153.70 319.32 153.70 153.63 358.29 360.47 352.46 276.91

5 0.02 146.64 160.66 145.97 145.23 172.24 184.51 171.23 170.310.05 222.07 233.73 219.51 212.11 253.54 264.75 250.68 242.430.10 188.08 311.97 188.08 187.95 353.13 352.30 344.38 291.46

20 0.02 145.31 158.46 144.63 144.02 170.42 182.25 169.42 168.650.05 219.40 230.60 216.90 210.71 250.29 261.09 247.50 240.620.10 201.82 308.00 201.82 201.67 348.69 347.51 339.67 294.67

FGMII(a¼1/b¼0.5/c¼2/p)

0 0.02 148.90 165.21 148.26 146.97 175.72 188.65 174.70 173.080.05 202.92 240.08 202.92 202.83 259.91 272.12 256.85 242.880.10 146.24 320.71 146.24 146.18 359.32 361.99 353.97 272.89

0.5 0.02 148.51 164.46 147.85 146.69 175.14 187.88 174.12 172.650.05 212.79 238.99 212.79 212.28 258.78 270.84 255.75 243.030.10 153.70 319.32 153.70 153.63 357.77 360.31 352.31 276.78

5 0.02 146.49 160.51 145.82 145.08 172.08 184.22 171.07 170.150.05 221.74 233.39 219.17 211.74 253.04 264.24 250.17 241.920.10 188.08 311.85 188.08 187.94 350.95 351.57 343.69 290.83

20 0.02 145.24 158.39 144.56 143.95 170.34 182.10 169.34 168.580.05 219.24 230.43 216.73 210.52 250.05 260.84 247.26 240.370.10 201.82 307.93 201.82 201.66 347.65 347.14 339.32 294.35

Z. Su et al. / International Journal of Mechanical Sciences 80 (2014) 62–8070

Fig. 4. The variation of frequencies for the FGM cylindrical shells with different power-law exponents (h/R¼0.1, L/R¼2 and R¼1 m): (a) FGMI(a¼1/b¼0.5/c¼2/p) cylindricalshells; (b) FGMII(a¼1/b¼0.5/c¼2/p) cylindrical shells.

Fig. 5. Mode shapes of the FGMI(a¼1/b¼0.5/c¼2/p¼1) cylindrical shells with SD-C boundary condition (h/R¼0.1, L/R¼2 and R¼1 m).

Fig. 6. Mode shapes of the FGMI(a¼1/b¼0.5/c¼2/p¼1) cylindrical shells with E1–E1 boundary condition (h/R¼0.1, L/R¼2 and R¼1 m).

Z. Su et al. / International Journal of Mechanical Sciences 80 (2014) 62–80 71

Table 7Convergence of frequencies (Hz) of FGM conical shells (R1¼0.5 m, h¼0.1 m, L1¼2 m, α¼ 401; boundary condition: SS-SS).

Mode Truncated number

n m M¼8 M¼9 M¼10 M¼11 M¼12 M¼13 M¼14 M¼15

FGMI(a¼0/b¼ -0.5/c¼2/p¼1)

1 1 330.46 330.45 330.45 330.44 330.44 330.44 330.44 330.432 451.37 451.32 451.29 451.27 451.26 451.25 451.25 451.243 610.21 610.05 609.97 609.92 609.89 609.86 609.85 609.83

2 1 232.50 232.49 232.49 232.48 232.48 232.47 232.47 232.462 392.25 392.22 392.20 392.19 392.18 392.18 392.17 392.173 569.63 569.46 569.39 569.34 569.31 569.29 569.28 569.26

3 1 217.20 217.19 217.19 217.18 217.18 217.17 217.17 217.172 387.77 387.74 387.72 387.71 387.70 387.70 387.69 387.693 591.54 591.34 591.28 591.21 591.19 591.15 591.15 591.13

FGMII(a¼0/b¼-0.5/c¼2/p¼1)

1 1 331.26 331.25 331.25 331.24 331.24 331.24 331.24 331.232 453.06 453.01 452.98 452.96 452.95 452.94 452.93 452.933 607.54 607.38 607.31 607.26 607.23 607.21 607.19 607.18

2 1 230.01 230.01 230.01 230.01 230.00 230.00 230.00 230.002 393.99 393.96 393.93 393.92 393.90 393.90 393.88 393.883 567.51 567.36 567.29 567.24 567.22 567.20 567.18 567.17

3 1 214.01 214.00 214.00 214.00 214.00 213.99 213.99 213.992 387.73 387.71 387.67 387.66 387.64 387.64 387.62 387.623 589.92 589.73 589.67 589.61 589.59 589.57 589.56 589.54

Table 8

Comparison of frequency parameters Ω¼ωR2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiρð1�μ2Þ=E

pfor isotropic conical shell with different boundary condition (α¼ 451, h/R2¼0.01, L sin α=R2 ¼ 0:5; m¼1).

BC Method n

0 1 2 3 4 5 6 7

SD–SD Shu[13] 0.2233 0.5463 0.6310 0.5062 0.3942 0.3340 0.3239 0.3514Liew[14] 0.2234 0.5462 0.6309 0.5061 0.3941 0.3337 0.3235 0.3510Qu[25] 0.2230 0.5460 0.6308 0.5061 0.3941 0.3336 0.3232 0.3504present 0.2233 0.5461 0.6308 0.5062 0.3941 0.3337 0.3233 0.3505

SD–C Shu[13] 0.8700 0.8118 0.6613 0.5246 0.4319 0.3826 0.3737 0.3987Liew[14] 0.8691 0.8113 0.6610 0.5244 0.4316 0.3822 0.3732 0.3980Qu[25] 0.8693 0.8115 0.6611 0.5245 0.4317 0.3823 0.3731 0.3977present 0.8694 0.8115 0.6611 0.5246 0.4318 0.3824 0.3732 0.3978

C–SD Shu[13] 0.7151 0.7098 0.6475 0.5201 0.4161 0.3592 0.3450 0.3648Liew[14] 0.7148 0.7095 0.6473 0.5199 0.4158 0.3589 0.3446 0.3644present 0.7147 0.7094 0.6473 0.5200 0.4159 0.3589 0.3445 0.3640

C–C Shu[13] 0.8732 0.8120 0.6696 0.5428 0.4566 0.4089 0.3963 0.4143Liew[14] 0.8732 0.8120 0.6696 0.5428 0.4565 0.4088 0.3961 0.4141Qu[25] 0.8726 0.8117 0.6694 0.5426 0.4563 0.4085 0.3956 0.4133present 0.8726 0.8117 0.6694 0.5427 0.4564 0.4086 0.3958 0.4135

Table 9Comparison of the first eight frequencies (Hz) for FGM conical shells (R1¼0.5m, h¼0.1 m, L cos α¼ 2m, α¼ 401; boundary conditions: F-C).

p Method Mode

1 2 3 4 5 6 7 8

FGMI(a¼0/b¼ -0.5/c¼2/p)

0.6 Ref [24] 208.92 208.92 230.11 230.11 284.73 284.74 321.51 321.51Ref [25] 208.75 208.75 229.96 229.96 284.60 284.60 321.33 321.33present 208.58 208.58 230.06 230.06 284.67 284.67 321.28 321.28

5 Ref[24] 204.81 204.81 223.84 223.84 275.52 275.53 316.64 316.64Ref [25] 204.25 204.25 223.36 223.36 275.02 275.02 315.94 315.94present 203.36 203.36 223.72 223.72 275.26 275.26 315.48 315.48

20 Ref [24] 204.89 204.89 227.33 227.33 282.68 282.69 312.50 312.50Ref [25] 204.27 204.27 226.78 226.78 282.10 282.10 311.71 311.71present 203.29 203.29 227.17 227.17 282.35 282.35 311.15 311.15

FGMII(a¼0/b¼ -0.5/c¼2/p)

0.6 Ref [24] 208.49 208.49 229.65 229.65 284.17 284.17 321.18 321.18Ref [25] 208.54 208.54 229.70 229.70 284.25 284.25 321.30 321.30present 208.74 208.74 229.61 229.61 284.21 284.21 321.33 321.33

5 Ref [24] 202.87 202.87 221.78 221.78 273.02 273.02 315.18 315.18Ref [25] 203.30 203.30 222.16 222.16 273.47 273.47 315.81 315.81present 204.15 204.15 221.67 221.67 273.14 273.14 315.72 315.72

20 Ref [24] 202.60 202.60 224.87 224.87 279.65 279.66 310.83 310.83Ref [25] 203.09 203.09 225.30 225.30 280.17 280.17 311.53 311.53present 203.98 203.98 224.73 224.73 279.80 279.80 311.39 311.39

Z. Su et al. / International Journal of Mechanical Sciences 80 (2014) 62–8072

the fundamental frequency of the FGM cylindrical shell. Thevariations of the fundamental frequencies of FGM cylindrical shellswith different power-law exponents and boundary conditions aredemonstrated in Fig. 4. The variation of frequencies of For theFGMI cylindrical shells and FGMII cylindrical shells are similar. Forthe case of FGM cylindrical shells with classical boundary condi-tions (i.e. C–C, SS–SS and SD–SD), the fundamental frequencies of theFGM cylindrical shells always decrease as the power-law exponent

increases. However, the variations of the frequencies of the FGMcylindrical shells with elastic restraints (i.e. E1–E1 and E2–E2) aredifferent. The fundamental frequencies of the E1–E1 FGM cylindricalshells increase as the power-law exponent increases. The funda-mental frequencies of the E2–E2 FGM cylindrical shell first increaseand then decrease as the power-law exponent increases. Some modeshapes for SD–C and E1–E1 FGM cylindrical shells are depicted inFigs. 5 and 6.

Table 10Frequencies (Hz) for FGM conical shells with classical and elastic boundaries (R1¼0.5 m, L cos α¼ 2m, and α¼ 401).

p h/R1 BC

SD-SD C-C C-SD SS-SS E1- E1 E2- E2 E3- E3 E4- E4

FGMI(a¼0/b¼ -0.5/c¼2/p)

0 0.02 65.78 89.97 68.70 88.90 76.31 88.36 74.84 74.750.10 103.03 179.28 147.00 171.05 152.83 164.50 143.98 138.670.20 122.04 238.10 200.86 218.32 217.06 213.31 195.64 158.70

0.5 0.02 65.34 89.42 68.33 88.48 76.21 87.86 74.76 74.680.10 102.54 178.44 146.27 170.79 152.24 163.83 143.47 138.410.20 121.46 236.54 199.49 217.74 215.60 212.57 195.06 159.56

5 0.02 63.38 86.99 66.64 86.51 75.92 85.66 74.62 74.540.10 100.21 174.48 142.83 168.48 149.71 160.87 141.42 137.380.20 118.69 229.75 193.32 215.04 209.43 209.34 192.68 163.05

20 0.02 64.09 87.75 67.09 87.29 77.56 86.53 76.36 76.300.10 99.88 174.33 143.17 168.92 151.49 161.98 143.76 140.280.20 118.34 232.80 195.88 217.52 213.80 211.02 195.00 168.68

FGMII(a¼0/b¼ -0.5/c¼2/p)

0 0.02 65.78 89.97 68.70 88.90 76.31 88.36 74.84 74.750.10 103.03 179.28 147.00 171.05 152.83 164.50 143.98 138.670.20 122.04 238.10 200.86 218.32 217.06 213.31 195.64 158.70

0.5 0.02 65.32 89.35 68.23 88.18 76.12 87.79 74.67 74.590.10 102.54 178.30 146.09 169.68 152.04 163.64 143.23 138.220.20 121.46 236.27 199.33 216.03 215.27 212.31 194.78 159.41

5 0.02 63.27 86.60 66.11 84.90 75.44 85.27 74.12 74.060.10 100.18 173.60 141.83 162.59 148.66 159.87 140.19 136.400.20 118.63 228.29 192.44 205.95 207.70 207.96 191.23 162.64

20 0.02 63.95 87.32 66.50 85.45 77.02 86.11 75.81 75.760.10 99.85 173.44 142.06 162.27 150.30 160.84 142.39 139.180.20 118.26 231.06 194.91 207.24 211.78 209.44 193.35 168.29

Table 11Frequencies (Hz) for FGM conical shells with classical-elastic boundaries (R1¼0.5 m, L cos α¼ 2 m, and α¼ 401).

p h/R1 BC

SD–E1 SD–E2 SD–E3 SD–E4 C–E1 C–E2 C–E3 C–E4

FGMI(a¼0/b¼ -0.5/c¼2/p)

0 0.02 74.97 87.47 73.79 73.74 77.69 88.37 76.56 76.500.10 148.82 160.48 140.64 137.14 155.24 164.72 146.18 142.280.20 191.55 207.44 190.77 165.69 218.19 214.78 199.35 172.44

0.5 0.02 74.82 87.08 73.66 73.62 77.39 87.87 76.48 76.430.10 148.20 159.81 140.12 136.89 154.65 164.06 145.67 142.050.20 192.92 206.74 190.23 166.83 216.74 214.04 198.76 173.59

5 0.02 74.29 85.07 73.28 73.25 76.27 85.66 75.54 75.510.10 145.50 156.84 137.99 135.83 152.10 161.10 143.60 141.110.20 200.77 203.61 187.90 171.54 210.59 210.80 196.26 178.40

20 0.02 76.00 85.67 75.06 75.04 78.19 86.54 77.49 77.460.10 147.38 158.02 140.40 138.64 153.69 162.20 145.79 143.750.20 209.76 205.09 189.88 176.65 214.86 212.45 198.37 183.80

FGMII(a¼0/b¼ -0.5/c¼2/p)

0 0.02 74.97 87.47 73.79 73.74 77.69 88.37 76.56 76.500.10 148.82 160.48 140.64 137.14 155.24 164.72 146.18 142.280.20 191.55 207.44 190.77 165.69 218.19 214.78 199.35 172.44

0.5 0.02 74.78 87.00 73.61 73.57 77.31 87.80 76.39 76.340.10 148.07 159.61 139.91 136.62 154.45 163.87 145.44 141.800.20 192.91 206.50 189.99 166.30 216.42 213.78 198.49 173.14

5 0.02 74.11 84.70 73.02 72.99 75.84 85.28 75.12 75.080.10 144.82 155.78 136.93 134.49 151.06 160.11 142.40 139.790.20 200.68 202.36 186.67 168.88 208.86 209.44 194.88 176.12

20 0.02 75.77 85.22 74.76 74.73 77.72 86.11 77.02 76.990.10 146.55 156.81 139.18 137.16 152.51 161.06 144.43 142.290.20 208.08 203.64 188.45 173.79 212.85 210.88 196.79 181.33

Z. Su et al. / International Journal of Mechanical Sciences 80 (2014) 62–80 73

3.2. Free vibration analysis of FGM conical shells

In this section, studies on the free vibration of FGM conicalshells are carried out. The convergence of the current method forFGM conical shells also is examined. Table 7 shows the frequenciesof the SS–SS FGM conical shells made of aluminum and zirconia.The geometrical parameters of the FGM conical shells are taken tobe R1¼0.5 m, h¼0.1 m, L1¼2 m, and α¼ 401. The volume fractionparameters are taken to be a¼0, b¼-0.5, c¼2, and p¼1. Theconvergence trends of the frequencies for the FGM conical shellsare obvious.

For the isotropic conical shells having α¼ 451, L sin α=R2 ¼ 0:5and h/R2¼0.01, comparison of frequency parametersΩ¼ ωR2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiρð1�μ2Þ=E

pwith those CST solutions obtained by Shu

[13] using the GDQ method and Liew et al. [14] using the element-free kp-Ritz method, and FSDT solutions obtained by Qu et al. [25]employing a modified variational approach is presented in Table 8.Four combinations of boundary conditions (i.e. SD–SD, SD–C, C–SD, and C–C) are considered. An excellent agreement is observed.

Table 9 shows the first eight frequencies for FGM conical shellswith F–C boundary condition, which made of aluminum andzirconia and have R1¼0.5 m, h¼0.1 m, L cos α¼ 2 m, andα¼ 401. It is seen that the present results are in good agreementwith those FSDT solutions obtained by Tornabene [24] and Quet al. [25]. The slight discrepancies may be due to the differentsolution methods used in the three studies. From the competitionstudies presented in Tables 8 and 9. It is concluded that thepresent method for FGM conical shells is numerically accurate.

There are also numerous new results which are given inTables 10 and 11. The free vibrations of aluminum/zirconia conicalshells with different thickness and boundary condition are stu-died. The boundary conditions including classical cases (i.e. SD-SD,C-C, C-SD, and SS-SS), elastic restraints (i.e. E1–E1, E2–E2, E3–E3 andE4–E4) and classical-elastic boundaries (i.e. SD–E1, SD–E2, SD–E3,SD–E4, C–E1, C–E2, C–E3 and C–E4) are considered, here. Thegeometrical parameters of the FGM conical shells are taken to beR1¼0.5 m, L cos α¼ 2m, and α¼ 401 and h/R1¼0.02, 0.10, and0.20. The fundamental frequencies of the FGM cylindrical shells

Fig. 7. The variation of frequencies for the FGM conical shells with different power-law exponents (R1¼0.5 m, h¼0.1 m, L cos α¼ 2m and α¼ 401): (a) FGMI(a¼0/b¼�0.5/c¼2/p)

conical shells; (b) FGMII(a¼0/b¼�0.5/c¼2/p) conical shells.

Fig. 8. Mode shapes of the FGMI(a¼0/b¼�0.5/c¼2/p¼1) conical shells with C-SD boundary condition (R1¼0.5 m, h¼0.1 m, L cos α¼ 2m and α¼ 401).

Z. Su et al. / International Journal of Mechanical Sciences 80 (2014) 62–8074

Fig. 9. Mode shapes of the FGMI(a¼0/b¼�0.5/c¼2/p¼1) conical shells with E4-E4 boundary condition (R1¼0.5 m, h¼0.1 m, L cos α¼ 2m and α¼ 401).

Table 12Comparison of frequencies (Hz) for isotropic annular plates with different boundary conditions (R1¼0.5 m, h¼0.1 m, R2–R1¼1.5 m, α¼ 901; ρ¼ 7800 kg=m3,

E¼ 2:1� 1011 Pa, and μ¼ 0:3).

Mode C–C SD–SD SD–C F–C

Ref [23] present Ref [23] present Ref [23] present Ref [23] present

1 238.05 238.06 115.42 115.42 182.68 182.68 67.03 67.032 246.01 246.02 129.88 128.41 196.69 195.19 121.86 121.913 246.01 246.02 129.88 128.41 196.69 195.19 121.86 121.914 275.54 275.56 175.58 172.28 242.03 238.58 203.17 203.185 275.54 275.56 175.58 172.28 242.03 238.58 203.17 203.186 335.89 335.91 250.65 247.21 319.28 316.20 283.85 283.857 335.89 335.91 250.65 247.21 319.28 316.20 302.85 302.868 427.31 427.33 347.67 344.40 421.25 419.66 302.85 302.869 427.31 427.33 347.67 344.40 421.25 419.66 340.65 340.84

10 542.01 542.03 432.75 432.76 540.29 539.74 340.65 340.84

Table 13The first ten frequencies (Hz) for FGM annular plate with different power-law exponent p (R1¼0.5 m, h¼0.1 m, R2-R1¼1.5 m, and α¼ 901and FGMI–II(a¼0/b¼ -0.5/c¼2/p);boundary condition F–C).

Mode p¼0.6 p¼1 p¼5 p¼20

Ref[24] present Ref[24] present Ref[24] present Ref[24] present

1 69.24 69.24 68.71 68.71 66.36 66.36 68.84 68.842 125.92 125.99 124.95 125.04 120.75 121.05 125.11 125.463 125.92 125.99 124.95 125.04 120.75 121.05 125.11 125.464 209.95 209.96 208.35 208.37 201.35 201.50 208.58 208.775 209.95 209.96 208.35 208.37 201.35 201.50 208.58 208.776 293.35 293.35 291.13 291.14 281.41 281.41 291.33 291.337 312.99 313.00 310.64 310.65 300.29 300.34 310.82 310.878 312.99 313.00 310.64 310.65 300.29 300.34 310.82 310.879 352.12 352.33 349.50 349.73 337.98 338.35 349.49 349.86

10 352.12 352.33 349.50 349.73 337.98 338.35 349.49 349.86

Z. Su et al. / International Journal of Mechanical Sciences 80 (2014) 62–80 75

also increase as thickness-to-radius ratio increases. It is obviousthat the frequencies of the FGMI conical shells are also alwayshigher than those of FGMII conical shells. And the power-lawexponent has a significant effect on the fundamental frequency ofthe FGM conical shell. Fig. 7 shows the variation of the funda-mental frequencies for FGM conical shells with various power-lawexponents. The variations of the fundamental frequencies of theFGMI and FGMII conical shells are also similar. Six types of theboundary conditions including C–C, SS–SS, E1–E1, E2–E2, C–E1 andC–E2 are considered. The characters of the fundamental frequen-cies for different restraints are similar. The first eleven of modeshapes for C–SD and E4–E4 FGM conical shells are given inFigs. 8 and 9.

3.3. Free vibration analysis of FGM annular plates

The annular plates can be considered as particular conical shellshaving α¼ 901. In this section, the free vibrations of the FGMannular plates are analyzed. First, comparison of the frequenciesfor isotropic annular plates with different boundary conditions (i.e.C–C, SD–SD, SD–C and F–C) are presented in Table 12. Thegeometrical and material parameters of the FGM annular platesare taken to be R1¼0.5 m, h¼0.1 m, R2–R1¼1.5 m, ρ¼ 7800 kg=m3,E¼ 2:1� 1011 Pa and μ¼ 0:3. It is seen that the present results arein excellent agreement with those obtained by Tornabene et al.[23]. Table 13 shows the first ten frequencies for aluminum/zirconia annular plates with different power-law exponents

(i.e. p¼0.6, 1, 5, and 20). The geometrical parameters of the FGMannular plates are same with those in Table 12. The F–C boundarycondition is considered. The present results are compared withthose obtained by Tornabene [24] employing differential quadra-ture method base on FSDT. A good agreement is presented.

The FGM annular plates with various boundary conditions arestudied. Table 14 shows the fundamental frequencies of the FGMannular plates subjected to classic restraints (i.e. SD–SD, C–C,

Table 14Frequencies (Hz) for FGM annular plate with classical and elastic boundary condition (R1¼0.5 m, R2–R1¼1.5 m, α¼ 901and FGMI� II(a¼0/b¼ -0.5/c¼2/p).

p h/R1 BC

SD–SD C–C C–SD SD–C SS–SS E5–E5 E6–E6 E5–E6

0 0.02 12.20 25.80 16.99 19.50 12.20 25.78 25.79 25.790.10 60.83 127.79 84.46 97.02 60.83 120.43 127.15 122.400.20 120.76 249.08 166.19 191.14 120.76 172.89 240.15 186.52

0.5 0.02 12.07 25.52 16.81 19.29 12.08 25.51 25.52 25.510.10 60.18 126.44 83.56 95.99 60.24 119.62 125.85 121.460.20 119.49 246.54 164.47 189.16 119.60 174.35 238.26 187.52

5 0.02 11.54 24.40 16.07 18.45 11.87 24.39 24.40 24.390.10 57.55 120.95 79.91 91.80 59.20 116.20 120.54 117.500.20 114.30 236.14 157.42 181.04 117.54 180.77 230.32 191.88

20 0.02 11.98 25.33 16.68 19.15 12.40 25.32 25.33 25.320.10 59.72 125.44 82.91 95.25 61.81 121.03 125.06 122.250.20 118.53 244.28 163.06 187.55 122.62 191.85 238.91 202.73

Table 15Frequencies (Hz) for FGM annular plate with classical-elastic boundary condition (R1¼0.5 m, R2–R1¼1.5 m,α¼ 901, FGMI-II(a¼0/b¼�0.5/c¼2/p)).

p h/R1 BC

SD–E5 SD–E6 SS–E5 SS–E6 C–-E5 C–E6 F–E5 F–E6

0 0.02 19.50 19.50 19.50 19.50 25.79 25.80 7.06 7.060.10 95.59 96.79 95.59 96.79 125.17 127.47 35.12 35.160.20 168.76 187.65 168.76 187.65 209.60 244.48 68.23 68.97

0.5 0.02 19.29 19.29 19.30 19.30 25.52 25.52 6.98 6.980.10 94.67 95.77 94.69 95.80 124.03 126.14 34.75 34.790.20 168.46 185.93 168.50 185.97 209.93 242.29 67.63 68.32

5 0.02 18.45 18.45 18.58 18.58 24.40 24.40 6.68 6.680.10 90.89 91.65 91.54 92.31 119.30 120.74 33.26 33.280.20 166.75 178.79 167.84 180.08 210.63 233.17 65.16 65.62

20 0.02 19.15 19.15 19.32 19.32 25.33 25.33 6.93 6.930.10 94.41 95.11 95.23 95.95 123.91 125.25 34.53 34.550.20 174.39 185.47 175.79 187.09 220.87 241.54 67.73 68.16

Fig. 10. The variation of frequencies for the FGM(a¼0/b¼�0.5/c¼2/p) conical shellswith different power-law exponents (R1¼0.5 m, h¼0.1 m and R2¼2 m).

Z. Su et al. / International Journal of Mechanical Sciences 80 (2014) 62–8076

C–SD, SD–C and SS–SS) and elastic restraints (i.e. E5–E5, E6–E6 andE5–E6). The fundamental frequencies of FGM annular plates withclassical-elastic restraints (i.e. SD–E5, SD–E6, SS–E5, SS–E6, C–E5,C–E6, F–E5 and F–E6) are presented in Table 15. The FGM annularplates are made of aluminum and zirconia. The geometricalparameters of the FGM annular plates are taken to be R1¼0.5 m,R2–R1¼1.5 m and h/R1¼0.02, 0.10, and 0.20. It is obviously thatboundary conditions and power-law exponents have a greatinfluence on the vibration frequencies of the FGM annularplates. Similarly, the thickness-to-radius ratio also has a conspic-uous effect on the fundamental frequencies which increaseas the thickness-to-radius ratio increases. The variations of thefundamental frequencies for FGM annular plates with variouspower-law exponents are presented in Fig. 10. Some modeshapes for C–SD and E6–E6 FGM annular plates are given inFigs. 11 and 12.

4. Conclusions

A unified solution method for the free vibrations of FGMcylindrical, conical shells and annular plates with general bound-ary conditions has been presented. The first-order shear

deformation theory was adopted to formulate the theoreticalmodel. The material properties of the structures are assumed tochange continuously in the thickness direction according to thegeneral four-parameter power-law distributions in terms ofvolume fractions of constituents. Each of displacements androtations of those structures, regardless of boundary conditions,is expressed as a modified Fourier series, which is constructed asthe linear superposition of a standard Fourier cosine seriessupplemented with auxiliary polynomial functions introduced toeliminate all the relevant discontinuities with the displacementand its derivatives at the edges and accelerate the convergence ofseries representations. Mathematically, such a series expansion iscapable of representing any function including the exact solutions.Rayleigh–Ritz procedure is used to obtain the exact solution baseon the energy functions of those structures. The excellent accuracyand reliability of the current solutions are confirmed by comparingthe present results with those available in the literatures, andnumerous new results for functionally graded cylindrical, conicalshells and annular plates with elastic boundary conditions arepresented, which can serve as the benchmark solution for othercomputational techniques in the future research. The effects ofboundary conditions and the material power-law distribution arealso illustrated.

Fig. 11. Mode shapes of the FGMI-II(a¼0/b¼�0.5/c¼2/p¼1) annular plates with C-SD boundary condition (R1¼0.5 m, h¼0.1 m and R2¼2 m).

Fig. 12. Mode shapes of the FGMI-II(a¼0/b¼�0.5/c¼2/p¼1) annular plates with E5–E5 boundary condition (R1¼0.5 m, h¼0.1 m and R2¼2 m).

Z. Su et al. / International Journal of Mechanical Sciences 80 (2014) 62–80 77

Acknowledgments

The authors would like to thank the reviewers for their Constructive comments. The authors gratefully acknowledge the financialsupport from the National Natural Science Foundation of China (Nos. 51175098 and 51279035).

Appendix A. Detailed expressions for the stiffness matrix and mass matrix

The detailed expressions of the mass matrix [M] and stiffness matrix [K] in Eq. (28) are given as follows. To make the expressionssimple and clear, some indexes are pre-defined.

s¼mþnðMþ1Þþ1; q¼m1þn1ðMþ1Þþ1s1 ¼ nþðl�1ÞðMþ1Þþ1; q1 ¼ n1þðl1�1ÞðMþ1Þþ1

The mass matrix [M] can be expressed as:

½M� ¼

½Muu� 0 0 ½Mux� 00 ½Mvv� 0 0 ½Mvθ�0 0 ½Mww� 0 0

½Mux� 0 0 ½Mxx� 00 ½Mvθ � 0 0 ½Mθθ�

26666664

37777775

ðA:1Þ

The first row elements of the mass matrices can be written as:

fM11uugsq ¼ I0

Z L

0Ccc cos λmx cos λm1xA dx ðA:2Þ

fM12uugsq1 ¼ I0

Z L

0cos λmxζl1 ðxÞA dx ðA:3Þ

fM11ux gsq ¼ I1

Z L

0Ccc cos λmx cos λm1xA dx ðA:4Þ

fM12ux gsq1 ¼ I1

Z L

0Ccc cos λmxζl1 ðxÞA dx ðA:5Þ

fM13uugsq ¼ 0; fM14

uugsq1 ¼ 0; fM13ux gsq ¼ 0; fM14

ux gsq1 ¼ 0

The mass matrix [K] can be expressed as:

½K� ¼

½Kuu� ½Kuv� ½Kuw� ½Kux� ½Kuθ�½Kvv� ½Kvw� ½Kvx� ½Kvθ�

½Kww� ½Kwx� ½Kwθ�sym ½Kxx� ½Kxθ�

½Kθθ�

26666664

37777775

ðA:6Þ

The first row elements of the stiffness matrix are given as

fK11uugsq ¼

Z L

0

A11Cccλmλm1 sin λmx sin λm1xþA66=Rθnn1Css cos λmx cos λm1x

þS2A11Ccc cos λmx cos λm1xþA12=SCccð�λm1 Þ cos λmx sin λm1x

þA12=SCccð�λmÞ sin λmx cos λm1xþCcc=A½kux0þkuxað�1Þmþm1 �

8><>:

9>=>;A dx ðA:7Þ

fK12uugsq1 ¼

Z L

0

A11Cccð�λmÞ sin λmxζ0l1 ðxÞþA66=Rθnn1Css cos λmxζl1 ðxÞþS2A11Ccc cos λmxζl1 ðxÞþA12=SCcc cos λmxζ0l1 ðxÞþA12=SCccð�λmÞ sin λmxζl1 ðxÞ

8>><>>:

9>>=>>;A dx ðA:8Þ

fK11uvgsq ¼

Z L

0

SA11=An1Ccc cos λmx cos λm1xþA12=Að�λmÞn1Ccc sin λmx cos λm1x

þA66=Að�λm1Css cos λmx sin λm1x�SCss cos λmx cos λm1xÞ

( )A dx ðA:9Þ

fK12uvgsq1 ¼

Z L

0

SA11=An1Ccc cos λmxζl1 ðxÞþA12=Að�λmÞn1Ccc sin λmxζl1 ðxÞþA66=AðCss cos λmxζ0l1 ðxÞ�SCss cos λmxζl1 ðxÞÞ

( )Adx ðA:10Þ

fK11uwgsq ¼

Z L

0fA12=RθCccð�λmÞ sin λmx cos λm1xþSA11=RθCcc cos λmx cos λm1xgA dx ðA:11Þ

fK12uwgsq1 ¼

Z L

0fA12=RθCccð�λmÞ sin λmxζl1 ðxÞþSA11=Ccc cos λmxζl1 ðxÞgA dx ðA:12Þ

Z. Su et al. / International Journal of Mechanical Sciences 80 (2014) 62–8078

fK11ux gsq ¼

Z L

0

B11Cccλmλm1 sin λmx sin λm1xþSB12Cccð�λmÞ sin λmx cos λm1x

þSB12Cccð�λm1 Þ cos λmx sin λm1xþS2B11Ccc cos λmx cos λm1x

þB66=A2nn1Css cos λmx cos λm1x

8>><>>:

9>>=>>;Adx ðA:13Þ

fK12ux gsq1 ¼

Z L

0

B11Cccð�λmÞ sin λmxζ0l1 ðxÞþSB12Cccð�λmÞ sin λmxζl1 ðxÞþSB12Ccc cos λmxζ0l1 ðxÞþS2B11Ccc cos λmxζl1 ðxÞþB66A

2nn1Css cos λmxζl1 ðxÞ

8>><>>:

9>>=>>;Adx ðA:14Þ

fK11uθ gsq1 ¼

Z L

0

B66=Anλm1Css cos λmx sin λm1x�SB66=Að�nÞCss cos λmx cos λm1x

þB12=An1ð�λmÞCcc sin λmx cos λm1xþSB11=ACcc cos λmx cos λm1x

( )Adx ðA:15Þ

fK12uθ gsq1 ¼

Z L

0

B66=Að�nÞCss cos λmxζ0l1 ðxÞ�SB66=Að�nÞCss cos λmxζl1 ðxÞþB12=An1ð�λmÞCcc sin λmxζl1 ðxÞþSB11=ACcc cos λmxζl1 ðxÞ

( )Adx ðA:16Þ

fK13uugsq ¼ 0; fK14

uugsq1 ¼ 0; fK13uvgsq ¼ 0; fK14

uvgsq1 ¼ 0; fK13uwgsq ¼ 0; fK14

uwgsq1 ¼ 0;

fK13ux gsq ¼ 0; fK14

ux gsq1 ¼ 0; fK13uygsq ¼ 0; fK14

uygsq1 ¼ 0

where

S¼ 1A∂A∂x

Ccc ¼Z 2π

0cos ðnθÞ cos ðn1θÞdθ Css ¼

Z 2π

0sin ðnθÞ sin ðn1θÞdθ

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