Free Vibrations of Four-Parameter Functionally Graded
Parabolic Panels and Shells of Revolution
Francesco Tornabene, Erasmo Viola
To cite this version:
Francesco Tornabene, Erasmo Viola. Free Vibrations of Four-Parameter Functionally Graded
Parabolic Panels and Shells of Revolution. European Journal of Mechanics - A/Solids, Elsevier,
2009, 28 (5), pp.991. .
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Accepted Manuscript
Title: Free Vibrations of Four-Parameter Functionally Graded Parabolic Panels and
Shells of Revolution
Authors: Francesco Tornabene, Erasmo Viola
PII:
S0997-7538(09)00054-0
DOI:
10.1016/j.euromechsol.2009.04.005
Reference:
EJMSOL 2522
To appear in:
European Journal of Mechanics / A Solids
Received Date: 19 May 2008
Revised Date:
4 February 2009
Accepted Date: 16 April 2009
Please cite this article as: Tornabene, F., Viola, E. Free Vibrations of Four-Parameter Functionally
Graded Parabolic Panels and Shells of Revolution, European Journal of Mechanics / A Solids (2009),
doi: 10.1016/j.euromechsol.2009.04.005
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ARTICLE IN PRESS
FREE VIBRATIONS OF FOUR-PARAMETER FUNCTIONALLY
GRADED PARABOLIC PANELS AND SHELLS OF REVOLUTION
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Francesco Tornabene1*, Erasmo Viola1
DISTART - Department, Faculty of Engineering, University of Bologna
Keywords: Functionally Graded Materials, Doubly Curved Shells, FSD Theory, Free Vibrations, Generalized
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Differential Quadrature.
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Abstract. Basing on the First-order Shear Deformation Theory (FSDT), this paper focuses on the dynamic behaviour
of moderately thick functionally graded parabolic panels and shells of revolution. A generalization of the power-law
distribution presented in literature is proposed. Two different four parameter power-law distributions are considered
for the ceramic volume fraction. Some symmetric and asymmetric material profiles through the functionally graded
shell thickness are illustrated by varying the four parameters of power-law distributions. The governing equations of
motion are expressed as functions of five kinematic parameters. For the discretization of the system equations the
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Generalized Differential Quadrature (GDQ) method has been used. Numerical results concerning four types of
parabolic shell structures illustrate the influence of the parameters of the power-law distribution on the mechanical
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behaviour of shell structures considered.
* Corresponding author, Tel.: +39-051-209-3379; Fax: +39-051-209-3495
E-mail address: [email protected]
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1.
Introduction
Functionally graded materials (FGM) are a class of composites that have a smooth and continuous variation of
material properties from one surface to another and thus can alleviate the stress concentrations found in laminated
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composites. Typically, these materials consist of a mixture of ceramic and metal, or a combination of different
materials. Extensive research work has been carried out on this new class of composites since the concept itself was
first introduced and proposed in the late 1980s in Japan. One of the advantages of using functionally graded
materials is that they can survive environments with high temperature gradients, while maintaining structural
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integrity. Furthermore, the continuous change in the compositions leads to a smooth change in the mechanical
properties, which has many advantages over the laminated composites, where the delamination and cracks are most
likely to initiate at the interfaces due to the abrupt variation in mechanical properties between laminae.
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In the last years, some researchers have analyzed various characteristics of functionally graded structures (Ng et al.,
2000; Yang et al., 2003; Della Croce and Venini, 2004; Liew et al., 2004; Wu and Tsai, 2004; Elishakoff et al.,
2005; Patel et al., 2005; Abrate, 2006; Pelletier and Vel, 2006; Zenkour, 2006; Arciniega and Reddy, 2007; Nie and
Zhong, 2007; Roque et al., 2007; Yang and Shen, 2007).
The aim of this paper is to study the dynamic behaviour of functionally graded parabolic shell structures derived
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from shells of revolution, which are very common structural elements. As a matter of fact, the approach for studying
the vibration of isotropic shell (Tornabene and Viola, 2008) is now extended to shells made of four parametric
functionally graded materials.
The work is based on the First-order Shear Deformation Theory (FSDT) (Reddy, 2003). The geometric model refers
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to a moderately thick shell, and the effects of transverse shear deformation as well as rotary inertia are taken into
account. Several studies have been presented earlier for the vibration analysis of such revolution shells and the most
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popular numerical tool in carrying out these analyses was the finite element method (Reddy, 2003). The generalized
collocation method based on the ring element method has also been applied. With regard to the latter method, each
static and kinematic variable is transformed into a theoretically infinite Fourier series of harmonic components, with
respect to the circumferential co-ordinate (Viola and Artioli, 2004; Artioli et al., 2005; Artioli and Viola, 2005,
2006). In other words, when dealing with a completely closed shell, the 2-D problem can be reduced using standard
Fourier decomposition. In a panel, however, it is not possible to perform such a reduction operation, and the twodimensional field must be directly dealt with. In this paper, the governing equations of motion are a set of five twodimensional partial differential equations with variable coefficients. These fundamental equations are expressed in
terms of kinematic parameters and can be obtained by combining the three basic sets of equations, namely
equilibrium, kinematic and constitutive equations.
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This paper is motivated by the lack of studies in the technical literature about the free vibration analysis of
functionally graded parabolic panels and shells and the effect of the power-law distribution on their mechanical
behaviour.
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Firstly, two different power-law variations of the volume fraction of the constituents in the thickness direction are
proposed. The effect of the power-law exponent and of the power-law distribution choice on the mechanical
behaviour of functionally graded parabolic panels and shells is investigated.
Symmetric and asymmetric volume fraction profiles are presented in this paper. Classical volume fraction profiles
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can be obtained as special cases of the general distribution laws presented in this work. For the first power-law
distribution, the bottom surface of the structure is ceramic rich, whereas the top surface can be metal rich, ceramic
rich or made of a mixture of the two constituents and, on the contrary, for the second power-law distribution. The
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homogeneous isotropic material can be regarded as a special case of functionally graded materials, too. From this
point of view, the present work generalizes the paper by Tornabene and Viola (2008).
Secondly, another aim of the present paper is to demonstrate an efficient application of the differential quadrature
approach (Viola and Tornabene, 2005; Tornabene, 2007; Tornabene and Viola, 2007, 2008; Viola et al., 2007;
Marzani et al. 2008), by solving the equations of motion governing the free vibration of functionally graded thick
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parabolic panels and shells of revolution.
The system of second-order linear partial differential equations is solved without resorting to the one-dimensional
formulation of the dynamic equilibrium of the shell. The discretization of the system by means of the Generalized
Differential Quadrature method (GDQ) leads to a standard linear eigenvalue problem, where two independent
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variables are involved. In this way, it is possible to compute the complete assessment of the modal shapes
corresponding to natural frequencies of panel structures. It should be noted that there is comparatively little
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literature available for these structures, compared to literature on the free vibration analysis of complete shells of
revolution. In this study, complete revolution shells are obtained as special cases of shell panels by satisfying the
kinematic and physical compatibility at the common meridian with ϑ = 0, 2π .
2.
Fundamental System for Functionally Graded Panels and Shells of Revolution
The basic configuration of the problem considered here is a doubly curved shell as shown in Fig. 1. The co-ordinates
along the meridian and circumferential directions of the reference surface are ϕ and s , respectively. The distance of
each point from the shell mid-surface along the normal is ζ . The shells considered are assumed to be single-layer
shells of uniform thickness h . It is worth noting that, differently from the work by Tornabene and Viola (2008), the
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co-ordinates of the shell reference surface are changed
( (ϕ ,ϑ ) → (ϕ , s) )
in order to simplify the fundamental
equations. Now, all the geometric relations and fundamental equations are re-written following the new co-ordinate
system.
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The angle formed by the external normal n to the reference surface and the axis of rotation x3 , or the geometric
axis x3′ of the meridian curve, is defined as the meridian angle ϕ and the angle between the radius of the parallel
circle R 0 (ϕ ) and the x1 axis is designated as the circumferential angle ϑ as shown in Fig. 2. The parametric coordinates (ϕ , s ) define the meridian curves and the parallel circles upon the middle surface of the shell, respectively.
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The curvilinear abscissa s (ϕ ) of a generic parallel is related to the circumferential angle ϑ by the relation
s
( 0 ≤ s ≤ s0 )
(ϕ
0
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s = ϑ R 0 . The position of an arbitrary point within the shell material is known by the co-ordinates ϕ
≤ ϕ ≤ ϕ1 ) ,
upon the middle surface, and ζ directed along the outward normal n and measured from the
reference surface ( −h 2 ≤ ζ ≤ h 2 ) .
The geometry of shells considered is a surface of revolution with a parabolic curved meridian. The parabolic
meridian can be described with the following equation:
(
where k = s12 − d 2
)
0
− Rb
)
2
− k x3′ = 0
(1)
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(R
S is a characteristic parameter of the parabolic curve. The horizontal radius R 0 (ϕ ) of a
generic parallel of the shell represents the distance of each point from the axis of revolution and for a shell with
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parabolic meridian assumes the form R 0 (ϕ ) = k tan ϕ 2 + R b . Rb is the shift of the geometric axis of the meridian
x3′ with reference to the axis of revolution x3 .
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The radii of curvature R ϕ (ϕ ) , R s (ϕ ) in the meridian and circumferential directions and the first derivative of
R ϕ (ϕ ) with respect to ϕ can be expressed according to the well known differential geometry formulae.
As regards shell theory, this work is based on the following assumptions: (1) the transverse normal is inextensible so
that the normal strain is equal to zero: ε n = ε n (ϕ , s, ζ , t ) = 0 ; (2) the transverse shear deformation is supposed to
influence the governing equations so that normal lines to the reference surface of the shell before deformation
remain straight, but not necessarily normal after deformation (a relaxed Kirchhoff-Love hypothesis); (3) the shell
deflections are small and the strains are infinitesimal; (4) the shell is moderately thick and therefore it is possible to
assume that the thickness-direction normal stress is negligible so that the plane assumption can be invoked:
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σ n = σ n (ϕ , s, ζ , t ) = 0 ; (5) the linear elastic behaviour of anisotropic materials is assumed; (6) the rotary inertia is
also taken into account.
Consistent with the assumptions of a moderately thick shell theory reported above, the displacement field considered
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in this study is that of the First-order Shear Deformation Theory and can be expressed in the following, well known,
form:
Uϕ (ϕ , s, ζ , t ) = uϕ (ϕ , s, t ) + ζβϕ (ϕ , s, t )
U s (ϕ , s, ζ , t ) = u s (ϕ , s, t ) + ζβ s (ϕ , s, t )
(2)
W ( ϕ , s , ζ , t ) = w (ϕ , s , t )
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where uϕ , us , w are the displacement components of points lying on the middle surface (ζ = 0 ) of the shell, along
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meridian, circumferential and normal directions, respectively, while t is the time variable. βϕ and β s represent
normal-to-mid-surface rotations. The kinematic hypothesis expressed by equations (2) should be supplemented by
the statement that the shell deflections are small and strains are infinitesimal, that is w (ϕ , s, t )
h . It is worth
noting that in-plane displacements U ϕ and U s vary linearly through the thickness, while W remains independent
of ζ .
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Relationships between strains and displacements along the shell reference (middle) surface (ζ = 0 ) and the five
equations of dynamic equilibrium in terms of internal actions can be written as in Tornabene and Viola (2008).
As far as the constitutive relations for a functionally graded linear elastic material are concerned, they relate internal
stress resultants and internal couples with generalized strain components on the middle surface:
A12
A11
0
B12
0
0
A66
0
B11
B12
0
D11
B12
B11
0
D12
0
0
B 66
0
B11
0
0
0
0
B 66
0
0
D12
0
0
0
D11
0
0
0
0
0
0
D 66
0 κ A66
0
0
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⎡ Nϕ ⎤ ⎡ A11
⎢ N ⎥ ⎢A
⎢ s ⎥ ⎢ 12
⎢ Nϕ s ⎥ ⎢ 0
⎢
⎥ ⎢
⎢ M ϕ ⎥ = ⎢ B11
⎢ M ⎥ ⎢ B12
⎢ s⎥ ⎢
⎢Mϕs ⎥ ⎢ 0
⎢T ⎥ ⎢ 0
⎢ ϕ ⎥ ⎢
⎢⎣ Ts ⎥⎦ ⎢⎣ 0
0
0
0
0
⎤ ⎡ ε ϕ0 ⎤
⎥⎢ 0 ⎥
⎥ ⎢ εs ⎥
⎥ ⎢ γ ϕ0s ⎥
⎥⎢ ⎥
⎥ ⎢ χϕ ⎥
0 ⎥ ⎢ χs ⎥
⎥⎢ ⎥
0 ⎥ ⎢ χϕ s ⎥
0 ⎥ ⎢γ ϕn ⎥
⎥⎢ ⎥
κ A66 ⎥⎦ ⎢⎣ γ sn ⎥⎦
0
0
0
0
(3)
where κ is the shear correction factor, which is usually taken as κ = 5 6 , such as in the present work. In particular,
it is worth noting that the determination of shear correction factors for composite laminated structures is still an
unsolved issue, because these factors depend on various parameters (Alfano et al., 2001; Auricchio and Sacco, 1999,
2003).
In equations (3) the three components Nϕ , N s , Nϕ s are the in-plane meridian, circumferential and shearing force
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resultants, M ϕ , M s , M ϕ s are the analogous couples, while Tϕ , Ts are the transverse shear force resultants. We notice
that, in the above definitions (3) the symmetry of shearing force resultants Nϕ s , N sϕ and torsional couples M ϕ s , M sϕ
is assumed as a further hypothesis, even if it is satisfied only in the case of spherical shells and flat plates (Toorani
respect to unity ( ζ R ϕ , ζ R s
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and Lakis, 2000). This assumption is derived from the consideration that ratios ζ R ϕ , ζ R s can be neglected with
1 ). Moreover, the first three strains ε ϕ0 , ε s0 , γ ϕ0s
are in-plane meridian,
circumferential and shearing components and χϕ , χ s , χϕ s are the corresponding analogous curvature changes. The
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last two components γ ϕ n , γ sn are the transverse shearing strains.
The extensional stiffnesses Aij , the bending stiffnesses Dij and the bending-extensional coupling stiffnesses Bij are
h
2
∫ Q (ζ ) d ζ ,
Aij =
ij
h
−
2
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defined as:
Bij =
h
2
∫ Q (ζ ) ζ d ζ ,
ij
Dij =
h
−
2
h
2
∫ Q (ζ ) ζ
ij
2
dζ
(4)
h
−
2
where the elastic constants Qij = Qij (ζ ) are functions of thickness coordinate ζ and are represented by the
following:
E (ζ )
1 − (ν (ζ ) )
, Q12 =
ν (ζ ) E (ζ )
1 − (ν (ζ ) )
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Q11 =
2
2
, Q66 =
E (ζ )
2 (1 + ν (ζ ) )
(5)
Typically, the functionally graded materials are made of a mixture of two constituents. In this work, it is assumed
that the functionally graded material is made of a mixture of a ceramic and metal constituent. The material
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properties of the functionally graded shell vary continuously and smoothly along the thickness direction ζ and are
functions of the volume fractions and properties of the constituent materials. The Young’s modulus E (ζ ) ,
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Poisson’s ratio ν (ζ ) and mass density ρ (ζ ) of the functionally graded shell can be expressed as a linear
combination:
ρ (ζ ) = ( ρC − ρ M )VC + ρ M
E (ζ ) = ( EC − EM )VC + EM
(6)
ν (ζ ) = (ν C −ν M )VC + ν M
where ρC , EC ,ν C , VC and ρ M , EM ,ν M , VM represent mass density, Young’s modulus, Poisson’s ratio and volume
fraction of the ceramic and metal constituent materials, respectively.
In this paper, the ceramic volume fraction VC follows two simple four parameter power-law distributions:
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p
c
⎛
⎛1 ζ ⎞ ⎛1 ζ ⎞ ⎞
FGM 2 ( a / b / c / p ) : VC = ⎜ 1 − a ⎜ − ⎟ + b ⎜ − ⎟ ⎟
⎜
⎝ 2 h ⎠ ⎝ 2 h ⎠ ⎟⎠
⎝
p
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c
⎛
⎛1 ζ ⎞ ⎛1 ζ ⎞ ⎞
FGM 1 ( a / b / c / p ) : VC = ⎜ 1 − a ⎜ + ⎟ + b ⎜ + ⎟ ⎟
⎜
⎝ 2 h ⎠ ⎝ 2 h ⎠ ⎟⎠
⎝
(7)
where the volume fraction index p ( 0 ≤ p ≤ ∞ ) and the parameters a, b, c dictate the material variation profile
through the functionally graded shell thickness. It is worth noting that the volume fractions of all the constituent
VC + VM = 1
(8)
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materials should add up to unity:
In order to choose the three parameters a, b, c in a suitable way, the relation (8) must be always satisfied for every
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volume fraction index p . By considering the relations (7), when the power-law exponent is set equal to zero ( p = 0 )
or equal to infinity ( p = ∞ ), the homogeneous isotropic material is obtained (Tornabene and Viola, 2008) as a
special case of the functionally graded material. In fact, from equations (8), (7) and (6) it is possible to obtain:
p = 0 → VC = 1, VM = 0 → ρ (ζ ) = ρC , E (ζ ) = EC , ν (ζ ) = ν C
p = ∞ → VC = 0, VM = 1 → ρ (ζ ) = ρ M , E (ζ ) = EM , ν (ζ ) = ν M
(9)
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Some material profiles through the functionally graded shell thickness are illustrated in Figs. 3, 4 and 5. In Fig. 3
the classical volume fraction profiles, such as reported in literature, are presented as special cases of the general
distribution laws (7) by setting a = 1 and b = 0 . As can be seen from Fig. 3(a), for the first distribution
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FGM 1 ( a =1/ b = 0 / c / p ) (7) the material composition is continuously varied such that the bottom surface ( ζ h = − 0.5 ) of
the shell is ceramic rich, whereas the top surface ( ζ h = 0.5 ) is metal rich. Fig. 3(b) shows that conversely, for the
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second distribution FGM 2 ( a =1/ b = 0 / c / p ) (7) the top surface ( ζ h = 0.5 ) of the shell is ceramic rich, whereas the bottom
surface ( ζ h = − 0.5 ) is metal rich. Choosing other values for the parameters a, b, c , it is possible to obtain symmetric
and asymmetric volume fraction profiles as shown in Figs. 4 and 5. In Figs. 4(a) and 4(b), by setting a = 1 , b = 1 and
c = 2 the two distributions FGM 1 ( a =1/ b =1/ c = 2 / p ) , FGM 2 ( a =1/ b =1/ c = 2 / p ) (7) present the same profiles by varying the
volume fraction index p and are symmetric respect to the reference surface ( ζ h = 0 ) of the shell. Furthermore,
these distributions are characterized by the fact that both the top ( ζ h = 0.5 ) and bottom surface ( ζ h = − 0.5 ) are
ceramic rich, while there is a mixture of two constituents through the thickness. Figs. 4(c) and 4(d) illustrate
asymmetric profiles FGM 1 ( a =1/ b =1/ c = 4 / p ) , FGM 2 ( a =1/ b =1/ c = 4 / p ) (7) obtained by setting a = 1 , b = 1 and c = 4 . As
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shown by the figures being reviewed, we have the same constituent at the top and bottom surface but, unlike the
previous cases (Figs. 4(a) and 4(b)), profiles are not symmetric with respect to the reference surface ( ζ h = 0 ) of the
shell. Fig. 5 shows other cases obtained by varying the parameters a, b, c . These profiles are characterized by the fact
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that one of the two shell surfaces (the top or bottom surface) presents a mixture of two constituents. For example, by
considering the FGM 1 ( a =1/ b = 0.5 / c = 4 / p =1) distribution (Fig. 5(a)), at the top surface we have a mixture of ceramic and
metal made up by fifty per cent of ceramic and fifty per cent of metal, while the bottom surface ( ζ h = − 0.5 ) is
ceramic rich. Figs. 5(b), (c), (d), (e) and (f) illustrate various power-law distribution cases obtained by modifying the
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parameters a, b, c, p . From a design point of view it is important to know if the top surface of the shell ( ζ h = 0.5 ) is
ceramic or metal rich, if the bottom surface ( ζ h = − 0.5 ) is metal or ceramic rich, or if one of these surfaces presents
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a mixture of two constituents. One of the aim of this work is to propose a generalization of the power-law
distribution presented in literature and to illustrate the influence of the power-law exponent p and of the choice of
the parameters a, b, c on the mechanical behaviour of shell structures.
The three basic sets of equations, namely the kinematic, constitutive and equilibrium equations may be combined to
give the fundamental system of equations, also known as the governing system of equations. By substituting the
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kinematic equations into the constitutive equations and the result of this substitution into the equilibrium equations,
the complete equations of motion in terms of displacements can be written as:
L12
L 22
L13
L 23
L 32
L 33
L 34
L 43
L 44
L 52
L 53
L15 ⎤ ⎡ uϕ ⎤ ⎡ I 0
L 25 ⎥⎥ ⎢ us ⎥ ⎢ 0
⎢ ⎥ ⎢
L 35 ⎥ ⎢ w ⎥ = ⎢ 0
⎥⎢ ⎥ ⎢
L 45 ⎥ ⎢ βϕ ⎥ ⎢ I1
L 55 ⎥⎦ ⎢⎣ β s ⎥⎦ ⎢⎣ 0
L14
L 24
L 42
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⎡ L11
⎢L
⎢ 21
⎢ L 31
⎢
⎢ L 41
⎢L
⎣ 51
L 54
0
0
I1
I0
0
0
I0
0
0
0
0
I2
I1
0
0
0 ⎤ ⎡ u&&ϕ ⎤
⎢ ⎥
I1 ⎥⎥ ⎢ u&&s ⎥
&& ⎥
0⎥⎢ w
⎥ ⎢ && ⎥
0 ⎥ ⎢ βϕ ⎥
I 2 ⎥⎦ ⎢⎣ β&&s ⎦⎥
(10)
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where the explicit forms of the equilibrium operators L ij , i, j = 1,...,5 are listed in Appendix and the mass inertias
assumed the following form:
Ii =
h
2
∫ ρ (ζ ) ζ
h
−
2
i
⎛
ζ ⎞⎛ ζ ⎞
d ζ , i = 0,1, 2
⎜1 +
⎟ 1+
⎜ R ϕ ⎟ ⎜ Rs ⎟
⎠
⎝
⎠⎝
(11)
Three kinds of boundary conditions are considered, namely the fully clamped edge boundary condition (C), the
simply supported edge boundary condition (S) and the free edge boundary condition (F). The equations describing
the boundary conditions can be written as follows:
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Clamped edge boundary condition (C)
uϕ = us = w = βϕ = β s = 0
at ϕ = ϕ 0 or ϕ = ϕ 1 , 0 ≤ s ≤ s0
(12)
uϕ = us = w = βϕ = β s = 0
at s = 0 or s = s 0 , ϕ 0 ≤ ϕ ≤ ϕ 1
(13)
uϕ = us = w = β s = 0, M ϕ = 0
at ϕ = ϕ 0 or ϕ = ϕ 1 , 0 ≤ s ≤ s 0
uϕ = us = w = βϕ = 0, M s = 0
at s = 0 or s = s 0 , ϕ 0 ≤ ϕ ≤ ϕ 1
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Simply supported edge boundary condition (S)
(14)
(15)
at ϕ = ϕ 0 or ϕ = ϕ 1 , 0 ≤ s ≤ s 0
Nϕ = Nϕ s = Tϕ = M ϕ = M ϕ s = 0
N s = Nϕ s = Ts = M s = M ϕ s = 0
at s = 0 or s = s 0 , ϕ 0 ≤ ϕ ≤ ϕ 1
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Free edge boundary condition (F)
(16)
(17)
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If we wish to consider a complete shell of revolution, the kinematic and physical compatibility should be satisfied at
the common meridian with s = 0, 2π R 0 in addition to the external boundary conditions. The kinematic compatibility
conditions include the continuity of displacements. The physical compatibility conditions can only be the five
continuous conditions for the generalized stress resultants. To consider complete revolute shells characterized by
s0 = 2π R 0 , it is necessary to implement the kinematic and physical compatibility conditions between the two
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meridians with s = 0 and with s0 = 2π R 0 :
Kinematic compatibility conditions
uϕ (ϕ , 0, t ) = uϕ (ϕ , s0 , t ), us (ϕ , 0, t ) = us (ϕ , s0 , t ), w(ϕ , 0, t ) = w(ϕ , s0 , t ),
ϕ0 ≤ϕ ≤ϕ 1
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βϕ (ϕ , 0, t ) = βϕ (ϕ , s0 , t ), β s (ϕ , 0, t ) = β s (ϕ , s0 , t )
(18)
Physical compatibility conditions
N s (ϕ , 0, t ) = N s (ϕ , s0 , t ), Nϕ s (ϕ , 0, t ) = Nϕ s (ϕ , s0 , t ), Ts (ϕ , 0, t ) = Ts (ϕ , s0 , t ),
3.
ϕ0 ≤ ϕ ≤ ϕ 1
(19)
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C
M s (ϕ , 0, t ) = M s (ϕ , s0 , t ), M ϕ s (ϕ , 0, t ) = M ϕ s (ϕ , s0 , t )
Generalized Differential Quadrature Method Review
The Generalized Differential Quadrature method will be used to discretize the derivatives in the governing equations
in terms of displacements and the boundary and compatibility conditions.
The essence of GDQ method is that the n-th order derivative of a smooth one-dimensional function f ( x) defined
over an interval [ 0, L ] at the i-th point of abscissa xi , can be approximated as:
d n f ( x)
dx n
N
= ∑ ς ij( n ) f ( x j ),
x = xi
i = 1, 2,..., N
(20)
j =1
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ARTICLE IN PRESS
where ς ij( n ) are the weighting coefficients at the i-th point calculated for the j-th sampling point of the domain. In
equation (20) N is the total number of the sampling points of the grid distribution and f ( x j ) is the value of the
function at the j-th point.
RI
PT
Some simple recursive formulas are available for calculating n-th order derivative weighting coefficients ς ij( n ) by
means of Lagrange polynomial interpolation functions (Shu, 2000). The weighting coefficients for the first order
derivative are:
ς ii(1) = −
N
∑
j =1, j ≠ i
ς ij(1) , i, j = 1, 2,..., N ,
i≠ j
SC
L (1) ( xi )
, i, j = 1, 2,..., N ,
( xi − x j ) L (1) ( x j )
i= j
M
AN
U
ς ij(1) =
(21)
(22)
In equation (21) the first derivative of Lagrange interpolating polynomials at each point xk , k = 1, 2,..., N , is:
L (1) ( xk ) =
N
∏
l =1, l ≠ k
( xk − xl ),
k = 1, 2,..., N
(23)
For higher order derivatives ( n = 2,3,..., N − 1 ), one gets iteratively:
⎛
⎜
⎝
ς ij( n −1) ⎞
⎟ , i, j = 1, 2,..., N , i ≠ j
xi − x j ⎟⎠
TE
D
ς ij( n ) = n ⎜ ς ii( n −1)ς ij(1) −
ς ii( n ) = −
N
∑
j =1, j ≠ i
ς ij( n ) , i, j = 1, 2,..., N , i = j
(24)
(25)
From the above equations that the weighting coefficients of the second and higher order derivatives can be
EP
determined from those of the first order derivative. Furthermore, it is interesting to note that the preceding
coefficients ς ij( n ) are dependent on the derivative order n , on the grid point distribution x j , j = 1, 2,..., N , and on
AC
C
the specific point xi , where the derivative is computed. It is worth noting that, this set of expressions for the
determination of the weighting coefficients is so compact and simple that it is very easy to implement them in
formulating and programming because of the recurrence feature.
Throughout the paper, the Chebyshev-Gauss-Lobatto (C-G-L) grid distribution is assumed, for which the coordinates of grid points (ϕi , s j ) along the reference surface are:
⎞ ⎞ (ϕ 1 − ϕ 0 )
+ ϕ 0 , i = 1, 2,..., N , for ϕ ∈ ⎡⎣ϕ 0 , ϕ 1 ⎤⎦
2
⎠⎠
⎝
⎛
⎛ j −1 ⎞ ⎞ s 0
π ⎟ ⎟ , j = 1, 2,..., M ,
s j = ⎜ 1 − cos⎜
for s ∈ ⎡⎣ 0, s 0 ⎤⎦ (with s ≤ ϑ R 0 )
⎝ M −1 ⎠ ⎠ 2
⎝
⎛
⎛ i −1
⎝
ϕ i = ⎜1 − cos⎜
π⎟⎟
N −1
(26)
10
ARTICLE IN PRESS
where N , M are the total number of sampling points used to discretize the domain in ϕ and s directions,
respectively, of the double curved shell (Fig. 6). It has been proven that for the Lagrange interpolating polynomials,
the Chebyshev-Gauss-Lobatto sampling point rule guarantees convergence and efficiency to the GDQ technique
4.
Discrete Gorverning Equations and Numerical Implementation
RI
PT
(Tornabene, 2007; Tornabene and Viola, 2007, 2008).
This section analyses the free vibration of functionally graded shells. Using the method of separation of variables, it
rotations can be written as follows:
uϕ (ϕ , s, t ) = U ϕ (ϕ , s ) eiωt
M
AN
U
us (ϕ , s, t ) = U s (ϕ , s ) eiωt
SC
is possible to seek solutions that are harmonic in time and whose frequency is ω . The displacements and the
w (ϕ , s, t ) = W (ϕ , s ) eiωt
β ϕ (ϕ , s , t ) = B (ϕ , s ) e
ϕ
(27)
iω t
β s (ϕ , s, t ) = B s (ϕ , s ) eiωt
where the vibration spatial amplitude values U ϕ , U s , W , Bϕ , B s fulfil the fundamental differential system.
The GDQ procedure allows writing of the equations of motion in discrete form, transforming each space derivative
TE
D
into a weighted sum of node values of dependent variables. Each approximate equation is valid in a single sampling
point. The governing equations can be discretized and for the domain points, i = 2,3,..., N − 1 , j = 2,3,..., M − 1 we
have:
A11
Rϕ2i
N
EP
1) Translational equilibrium along the meridian direction ϕ
⎛ cos ϕi
1 dRϕ
−
⎜ Rϕ i R 0i Rϕ3i dϕ
⎝
ϕ 2
∑ ς ik ( )U kjϕ + A11 ⎜
k =1
M
⎞ N ϕ (1) ϕ
⎛ A12 sin ϕi A11 cos 2 ϕi
A66 ⎞
s 2
ϕ
−⎜
+
+ κ 2 ⎟U ijϕ +
⎟ ∑ ς ik U kj + A 66 ∑ ς jm( )U im
2
⎟ k =1
⎜
R 0i
Rϕ i ⎟⎠
m =1
i⎠
⎝ Rϕ i R 0i
⎞ M s (1) s ⎛ A12 A66 ⎞ N ϕ (1) M s (1) s ⎛ A11 A12 sin ϕi
A 66 ⎞ N ϕ 1
+
+ κ 2 ⎟ ∑ ς ik ( )Wkj +
⎟ ∑ ς jm U im + ⎜
⎟ ∑ ς ik ∑ ς jm U km + ⎜ 2 +
⎜ Rϕ i Rϕ i ⎟ k =1
⎟ m =1
⎜ Rϕ i
Rϕ i R 0i
Rϕ i ⎟⎠ k =1
m =1
⎝
⎠
⎠
⎝
⎛
⎛ cos ϕi
1 dRϕ
+ ⎜ A11 ⎜
− 3
⎜
⎜
R
R
R
i
i
ϕ
0
ϕ i dϕ
⎝
⎝
⎞ A11 sin ϕi cos ϕi
⎟−
⎟
R 20i
i⎠
AC
C
⎛ A11 cos ϕi A 66 cos ϕi
−⎜
+
⎜ R 0i
R 0i
⎝
⎞
⎛ cos ϕi
B N ϕ ( 2) ϕ
1 dRϕ
⎟Wij + 11
− 3
ς Bkj + B 11 ⎜
2 ∑ ik
⎜
⎟
Rϕ i k =1
R
R
R
i
i
ϕ
0
ϕ i dϕ
⎝
⎠
M
⎛ B 12 sin ϕi B 11 cos 2 ϕi
A66 ⎞ ϕ ⎛ B11 cos ϕi B 66 cos ϕi
s 2
ϕ
+ B 66 ∑ ς jm( ) Bim
−⎜
+
−κ
+
⎟ Bij − ⎜
2
⎜
⎜ R 0i
R 0i
Rϕ i ⎟⎠
R 0i
m =1
⎝ Rϕ i R 0i
⎝
⎛ B 12 B 66 ⎞ N ϕ (1) M s (1) s
+⎜
+
⎟ ς ik ∑ ς jm Bkm = −ω 2 I 0U ijϕ + I1Bijϕ
⎜ Rϕ i Rϕ i ⎟ ∑
m =1
⎝
⎠ k =1
(
)
⎞ N ϕ1
⎟ ∑ ς ik ( ) Bkjϕ +
⎟ k =1
i⎠
⎞ M s (1) s
⎟ ∑ ς jm Bim +
⎟ m =1
⎠
(28)
11
ARTICLE IN PRESS
2) Translational equilibrium along the circumferential direction s
⎞ M s (1) ϕ ⎛ A12 A 66 ⎞ N ϕ (1) M s (1) ϕ
A 66
+
⎟ ∑ ς jm U im + ⎜
⎟ ς ik ∑ ς jm U km + 2
⎜ Rϕ i Rϕ i ⎟ ∑
⎟ m =1
Rϕ i
m =1
⎝
⎠ k =1
⎠
⎛ B 11 cos ϕi B 66 cos ϕi
+⎜
+
⎜ R 0i
R 0i
⎝
⎞
A 66 sin 2 ϕi
⎟+κ
⎟
R 20i
⎠
⎞ M s 1 ϕ ⎛ B 12 B 66
+⎜
+
⎟ ∑ ς jm( ) Bim
⎜ Rϕ i Rϕ i
⎟ m =1
⎝
⎠
M
⎛
⎛ cos 2 ϕi sin ϕi
s 2
+ B 11 ∑ ς jm( ) Bims − ⎜ B 66 ⎜
−
⎜ R 20i
⎜
Rϕ i R 0 i
m =1
⎝
⎝
⎞
⎛A
A sin ϕi
A 66 sin ϕi
⎟U ijs + ⎜ 12 + 11
+κ
⎜ Rϕ i
⎟
R
R 0i
i
0
⎝
⎠
⎞ N ϕ 1 M s 1 ϕ B 66
+ 2
⎟ ∑ ς ik ( ) ∑ ς jm( ) Bkm
⎟ k =1
Rϕ i
m =1
⎠
⎞
A 66 sin ϕi
⎟ −κ
⎟
R 0i
⎠
k =1
⎛ cos ϕi
1 dRϕ
−
U kjs + A 66 ⎜
⎜ Rϕ i R 0i Rϕ3i dϕ
⎝
ϕ ( 2)
ik
N
∑ς
k =1
⎞
⎟ Bijs = −ω 2 I 0U ijs + I1Bijs
⎟
⎠
(
3) Translational equilibrium along the normal direction ζ
ϕ ( 2)
ik
⎞ M s (1)
⎟ ∑ ς jm Wim +
⎟ m =1
⎠
⎛ cos ϕi
1 dRϕ
Bkjs + B 66 ⎜
−
⎜ Rϕ i R 0i Rϕ3i dϕ
⎝
)
N
∑ς
k =1
⎛ cos ϕi
1 dRϕ
−
Wkj + κ A 66 ⎜
⎜ Rϕ i R 0i Rϕ3i dϕ
⎝
ϕ ( 2)
ik
⎞⎞
⎟ ⎟U ijϕ +
⎟⎟
i ⎠⎠
⎞ N ϕ (1)
⎟ ∑ ς ik Wkj +
⎟
i ⎠ k =1
⎞
⎛ B 11 B 12 sin ϕi
A 66 ⎞ N ϕ (1) ϕ
−κ
⎟Wij − ⎜ 2 +
⎟ ∑ ς ik Bkj +
⎟
⎜ Rϕ i
Rϕ i R 0 i
Rϕ i ⎟⎠ k =1
⎠
⎝
TE
D
M
⎛ A11 2 A12 sin ϕi A11 sin 2 ϕi
s 2
+κ A 66 ∑ ς jm( )Wim − ⎜ 2 +
+
⎜ Rϕ i
Rϕ i R 0i
R 20 i
m =1
⎝
M
AN
U
⎞ M s (1) s
A 66
⎟ ∑ ς jm U im + κ 2
⎟ m =1
Rϕ i
⎠
⎛ B 12 cos ϕi B 11 sin ϕi cos ϕi
A 66 cos ϕi
−⎜
+
−κ
2
⎜ Rϕ i R 0i
R 0i
R 0i
⎝
⎞ N ϕ1
⎟ ∑ ς ik ( ) Bkjs +
⎟ k =1
i⎠
(29)
⎛ A12 cos ϕi A 66 sin ϕi cos ϕi
⎛ A11 A12 sin ϕi
⎛ cos ϕi
A 66 ⎞ N ϕ 1
1 dRϕ
−⎜ 2 +
+ κ 2 ⎟ ∑ ς ik ( )U kjϕ − ⎜
+
+ κ A 66 ⎜
− 3
2
⎜ Rϕ i
⎟
⎜
⎜
R
R
R
R
R
R
R
R
R
k
=
1
ϕ i 0i
ϕi ⎠
0i
ϕ i dϕ
⎝
⎝ ϕi 0i
⎝ ϕ i 0i
⎛ A12 A11 sin ϕi
A 66 sin ϕi
−⎜
+
+κ
⎜ Rϕ i
R 0i
R 0i
⎝
⎞ N ϕ (1) s
⎟ ∑ ς ik U kj +
⎟
i ⎠ k =1
RI
PT
M
⎛
⎛ cos 2 ϕi sin ϕi
s 2
+ A11 ∑ ς jm( )U ims − ⎜ A 66 ⎜
−
⎜ R 20i
⎜
Rϕ i R 0i
m =1
⎝
⎝
N
∑ς
SC
⎛ A11 cos ϕi A 66 cos ϕi
+
⎜
⎜ R 0i
R 0i
⎝
⎞ ϕ ⎛ B 12 B 11 sin ϕi
⎞ M s1
+
− κ A 66 ⎟ ∑ ς jm( ) Bims = −ω 2 I 0Wij
⎟ Bij − ⎜
⎟
⎜ Rϕ i
⎟ m =1
R 0i
⎠
⎝
⎠
(30)
4) Rotational equilibrium about the circumferential direction s
N
∑ς
2
ϕ i k =1
R
⎛ cos ϕi
1 dRϕ
−
U kjϕ + B 11 ⎜
⎜ Rϕ i R 0 i Rϕ3i dϕ
⎝
ϕ ( 2)
ik
M
⎞ N ϕ (1) ϕ
⎛ B 12 sin ϕi B 11 cos 2 ϕi
A 66 ⎞ ϕ
s 2
ϕ
−⎜
+
−κ
⎟ ∑ ς ik U kj + B 66 ∑ ς jm( )U im
⎟U ij +
2
⎟ k =1
⎜
R 0i
Rϕ i ⎟⎠
m =1
i⎠
⎝ Rϕ i R 0i
EP
B 11
⎞ M s (1) s ⎛ B 12 B 66 ⎞ N ϕ (1) M s (1) s ⎛ B 11 B 12 sin ϕi
A 66 ⎞ N ϕ (1)
+
−κ
⎟ ∑ ς jm U im + ⎜
⎟ ∑ ς ik Wkj +
⎟ ∑ ς ik ∑ ς jm U km + ⎜ 2 +
⎜ Rϕ i Rϕ i ⎟ k =1
⎟ m =1
⎜ Rϕ i
Rϕ i R 0i
Rϕ i ⎟⎠ k =1
m =1
⎝
⎠
⎠
⎝
⎛
⎛ cos ϕi
1 dRϕ
+ ⎜ B 11 ⎜
−
⎜ Rϕ i R 0 i Rϕ3i dϕ
⎜
⎝
⎝
⎞ B 11 sin ϕi cos ϕi
⎟−
⎟
R 20 i
i⎠
AC
C
⎛ B 11 cos ϕi B 66 cos ϕi
−⎜
+
⎜ R 0i
R 0i
⎝
⎞
⎛ cos ϕi
D N ϕ ( 2) ϕ
1 dRϕ
⎟Wij + 11
−
ς Bkj + D11 ⎜
2 ∑ ik
⎜ Rϕ i R 0i Rϕ3i dϕ
⎟
Rϕ i k =1
⎝
⎠
M
⎛ D12 sin ϕi D11 cos 2 ϕi
⎞
⎛ D 11 cos ϕi D 66 cos ϕi
s 2
ϕ
+ D 66 ∑ ς jm( ) Bim
−⎜
+
+ κ A 66 ⎟ Bijϕ − ⎜
+
2
⎜
⎟
⎜
R 0i
R 0i
R 0i
m =1
⎝ Rϕ i R 0 i
⎠
⎝
⎛ D 12 D 66 ⎞ N ϕ (1) M s (1) s
+⎜
+
⎟ ∑ ς ik ∑ ς jm Bkm = −ω 2 I1U ijϕ + I 2 Bijϕ
⎜ Rϕ i
⎟ k =1
R
m =1
i
ϕ
⎝
⎠
(
)
⎞ N ϕ (1) ϕ
⎟ ∑ ς ik Bkj +
⎟
i ⎠ k =1
⎞ M s (1) s
⎟ ∑ ς jm Bim +
⎟ m =1
⎠
(31)
12
ARTICLE IN PRESS
5) Rotational equilibrium about the meridian direction ϕ
⎞ M s (1) ϕ ⎛ B 12 B 66 ⎞ N ϕ (1) M s (1) ϕ B 66
+
⎟ ∑ ς jm U im + ⎜
⎟ ς ik ∑ ς jm U km + 2
⎜ Rϕ i Rϕ i ⎟ ∑
⎟ m =1
Rϕ i
m =1
⎝
⎠ k =1
⎠
⎛ D 11 cos ϕi D 66 cos ϕi
+⎜
+
⎜
R 0i
R 0i
⎝
⎞
A 66 sin ϕi
⎟ −κ
⎟
R 0i
⎠
k =1
⎞ N ϕ (1) s
⎟ ∑ ς ik U kj +
⎟
i ⎠ k =1
⎞
⎛B
⎞ M s1
B sin ϕi
⎟U ijs + ⎜ 12 + 11
− κ A 66 ⎟ ∑ ς jm( )Wim +
⎜ Rϕ i
⎟ m =1
⎟
R 0i
⎝
⎠
⎠
⎞ M s (1) ϕ ⎛ D12 D 66 ⎞ N ϕ (1) M s (1) ϕ D 66
+
⎟ ∑ ς jm Bim + ⎜
⎟ ∑ ς ik ∑ ς jm Bkm + 2
⎜ Rϕ i
⎟ m =1
Rϕ i ⎟⎠ k =1
Rϕ i
m =1
⎝
⎠
M
⎛
⎛ cos 2 ϕi sin ϕi
s 2
+ D 11 ∑ ς jm( ) Bims − ⎜ D 66 ⎜
−
⎜ R 20i
⎜
Rϕ i R 0i
m =1
⎝
⎝
⎛ cos ϕi
1 dRϕ
U kjs + B 66 ⎜
−
⎜ Rϕ i R 0 i Rϕ3i dϕ
⎝
ϕ ( 2)
ik
⎞
⎞
⎟ + κ A 66 ⎟ Bijs = −ω 2 I1U ijs + I 2 Bijs
⎟
⎟
⎠
⎠
(
N
∑ς
k =1
ϕ ( 2)
ik
RI
PT
M
⎛
⎛ cos 2 ϕi sin ϕi
s 2
+ B 11 ∑ ς jm( )U ims − ⎜ B 66 ⎜
−
⎜ R 20 i
⎜
Rϕ i R 0i
m =1
⎝
⎝
N
∑ς
⎛ cos ϕi
1 dRϕ
−
Bkjs + D 66 ⎜
⎜ Rϕ i R 0i Rϕ3i dϕ
⎝
)
⎞ N ϕ (1) s
⎟ ∑ ς ik Bkj +
⎟
i ⎠ k =1
(32)
SC
⎛ B 11 cos ϕi B 66 cos ϕi
+
⎜
⎜ R 0i
R 0i
⎝
In equations (28)-(32), ς ikϕ (1) , ς sjm(1) , ς ikϕ ( 2) and ς sjm( 2) are the weighting coefficients of the first and second derivatives in
directions.
M
AN
U
ϕ and s directions, respectively. Furthermore, N and M are the total number of grid points in ϕ and s
Applying the GDQ methodology, the discretized forms of the boundary and compatibility conditions are given as
follows:
Clamped edge boundary condition (C)
U ajϕ = U ajs = Waj = Bajϕ = Bajs = 0
ϕ
U ib = U = Wib = Bib = B = 0
s
ib
s
ib
for a = 1, N and j = 1, 2,..., M
TE
D
ϕ
for b = 1, M and i = 1, 2,..., N
(33)
Simply supported edge boundary condition (S)
for a = 1, N and j = 1, 2,..., M
⎧U iϕb = U ibs = Wib = Bibϕ = 0
⎪
N
M
s (1) s
⎪ B12 ς ϕ (1)U ϕ + B11 cos ϕi U ϕ + B
ib
11 ∑ ς bm U im +
⎪ R ∑ ik kb
R
m =1
ϕ i k =1
0i
⎪
⎪
D 12 N ϕ (1) ϕ
⎨ ⎛ B 12 B11 sin ϕi ⎞
+
⎟Wib +
∑ ς ik Bkb +
⎪+ ⎜⎜
⎟
R 0i ⎠
Rϕ i k =1
⎪ ⎝ Rϕ i
⎪ D cos ϕ
M
s1
i
⎪+ 11
Bibϕ + D 11 ∑ ς bm( ) Bims = 0
⎪⎩
R 0i
m =1
for b = 1, M and i = 1,2,..., N
AC
C
EP
⎧U ajϕ = U ajs = Waj = Bajs = 0
⎪
M
⎪ B11 N ϕ (1) ϕ B12 cos ϕa ϕ
s1
s
+
U aj + B 12 ∑ ς jm( )U am
⎪ R ∑ ς ak U kj +
R 0a
m =1
⎪ ϕ a k =1
⎪
D11 N ϕ (1) ϕ
⎨ ⎛ B11 B 12 sin ϕa ⎞
+
⎟Waj +
⎪+ ⎜⎜
∑ ς ak Bkj +
⎟
R 0a ⎠
Rϕ a k =1
⎪ ⎝ Rϕ a
⎪ D cos ϕ
M
s1
a
s
⎪+ 12
=0
Bajϕ + D 12 ∑ ς jm( ) Bam
⎪
R
=
1
m
0
a
⎩
(34)
13
ARTICLE IN PRESS
Free edge boundary condition (F)
RI
PT
⎧A N
M
M
⎛ A11 A12 sin ϕa ⎞
A12 cos ϕ a ϕ
B 11 N ϕ (1) ϕ B 12 cos ϕa ϕ
s1
s1
ϕ (1) ϕ
s
s
⎪ 11 ∑ ς ak
+⎜
+
=0
U kj +
U aj + A12 ∑ ς jm( )U am
ς ak Bkj +
Baj + B 12 ∑ ς jm( ) Bam
⎟Waj +
∑
⎜ Rϕ a
⎟
⎪ Rϕ a k =1
R 0a
R
R
R
m =1
k
m
1
1
=
=
0a
0a
ϕa
⎝
⎠
⎪
M
M
B 66 N ϕ (1) s B 66 cos ϕ a s
A 66 N ϕ (1) s A 66 cos ϕ a s
⎪
s (1) ϕ
s1 ϕ
+
Baj = 0
ς ak U kj −
U aj + B 66 ∑ ς jm( ) Bam
∑ ς ak Bkj − R
∑
⎪ A 66 ∑ ς jm U am +
Rϕ a k =1
Rϕ a k =1
R 0a
m =1
0a
⎪ m =1
⎪⎪
A 66 ϕ
A 66 N ϕ (1)
U aj + κ
(35)
⎨−κ
∑ ς ak Wkj + κ A66 Bajϕ + κ A66 Bajs = 0
Rϕ a k =1
⎪ Rϕ a
⎪
N
M
M
⎛ B 11 B 12 sin ϕ a ⎞
D 11 N ϕ (1) ϕ D 12 cos ϕ a ϕ
s (1) s
s1
⎪ B 11 ς ϕ (1)U ϕ + B 12 cos ϕ a U ϕ + B
s
+
=0
Baj + D 12 ∑ ς jm( ) Bam
ς ak Bkj +
⎟Waj +
∑
ak
kj
aj
12 ∑ ς jm U am + ⎜
⎪R ∑
⎜
⎟
R
R 0a
R
R
R
m =1
k
m
1
1
=
=
a
a
a
a
ϕ a k =1
ϕ
0
ϕ
0
⎝
⎠
⎪
M
M
⎪
B 66 N ϕ (1) s B 66 cos ϕ a s
D 66 N ϕ (1) s D 66 cos ϕ a s
s1 ϕ
ϕ
⎪ B 66 ∑ ς sjm(1)U am
ς ak U kj −
U aj + D 66 ∑ ς jm( ) Bam
Baj = 0
+
+
∑
∑ ς ak Bkj − R
Rϕ a k =1
R 0a
Rϕ a k =1
⎪⎩ m =1
m =1
0a
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for a = 1, N and j = 1, 2,..., M
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M
M
⎧
A 66 N ϕ (1) s A 66 cos ϕi s
B 66 N ϕ (1) s B 66 cos ϕi s
s (1) ϕ
s1
ϕ
+
ς ik U kb −
U ib + B 66 ∑ ς bm( ) Bim
⎪ A 66 ∑ ς bm U im +
∑
∑ ς ik Bkb − R Bib = 0
R
R
Rϕ i k =1
m =1
⎪ m =1
ϕ i k =1
0i
0i
⎪
N
M
N
M
⎛
⎞
B cos ϕi ϕ
A
A
cos
ϕ
A
A
sin
ϕ
B
⎪ 12
s1
ϕ (1) ϕ
s1
i
i
11
12
11
12
+
ς ikϕ (1) Bkbϕ + 11
Bib + B 11 ∑ ς bm( ) Bims = 0
U ibϕ + A11 ∑ ς bm( )U ims + ⎜
⎟Wib +
∑
⎪ R ∑ ς ik U kb +
⎜
⎟
R
R
R
R
R
m =1
m =1
0i
0i
ϕ i k =1
0i
⎝ ϕi
⎠
⎪ ϕ i k =1
⎪⎪ A N
66
⎨κ
∑ ς ikϕ (1)Wkb + κ A66 Bibϕ + κ A66 Bibs = 0
⎪ Rϕ i k =1
⎪
M
M
B 66 N ϕ (1) s B 66 cos ϕi s
D 66 N ϕ (1) s D 66 cos ϕi s
s (1) ϕ
s (1) ϕ
⎪ B 66 ∑ ς bm
U im +
ς bm
Bim +
∑ ς ik U kb − R U ib + D 66 ∑
∑ ς ik Bkb − R Bib = 0
⎪ m =1
Rϕ i k =1
Rϕ i k =1
m =1
0i
0i
⎪
M
N
M
⎛
⎞
⎪ B 12 N ϕ (1) ϕ B 11 cos ϕi ϕ
B
B
ϕ
D
D cos ϕi ϕ
sin
s 1
s1
i
12
11
12
ς ik U kb +
U ib + B11 ∑ ς bm( )U ims + ⎜
ς ikϕ (1) Bkbϕ + 11
Bib + D 11 ∑ ς bm( ) Bims = 0
+
⎟Wib +
⎪
∑
∑
⎜
⎟
R
R
R
R
R
R
k
m
k
m
1
1
1
=
=
=
=1
0i
0i
ϕi
0i
⎝ ϕi
⎠
⎩⎪ ϕ i
for b = 1, M and i = 1, 2,..., N
(36)
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Kinematic and physical compatibility conditions
ϕ
s
ϕ
s
U iϕ1 = U iM
,U is1 = U iM
,Wi1 = WiM , Biϕ1 = BiM
, Bis1 = BiM
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M
M
⎧
A 66 N ϕ (1) s A 66 cos ϕi s
B 66 N ϕ (1) s B 66 cos ϕi s
s (1) ϕ
s1 ϕ
+
ς ik U k 1 −
U i1 + B 66 ∑ ς 1m( ) Bim
⎪ A 66 ∑ ς 1m U im +
∑
∑ ς ik Bk1 − R Bi1 =
Rϕ i k =1
R 0i
Rϕ i k =1
m =1
⎪ m =1
0i
⎪
M
M
B 66 N ϕ (1) s
B 66 cos ϕi s
A 66 N ϕ (1) s
A 66 cos ϕi s
s (1) ϕ
s (1) ϕ
⎪= A
ς Mm Bim +
∑ ς ik U kM − R U iM + B 66 ∑
∑ ς ik BkM − R BiM
66 ∑ ς Mm U im +
⎪
Rϕ i k =1
Rϕ i k =1
m =1
m =1
0i
0i
⎪
M
M
N
⎪ A12 N ϕ (1)
⎛
⎞
A
cos
A
A
sin
B
B
cos ϕi ϕ
ϕ
ϕ
s1
s1
12
11
12
i
i
⎪
+
U iϕ1 + A11 ∑ ς 1m( )U ims + ⎜
Bi1 + B 11 ∑ ς 1m( ) Bims =
ς ik U kϕ1 + 11
ς ikϕ (1) Bkϕ1 + 11
⎟Wi1 +
∑
∑
⎜
⎟
R
R
R
R
R
R
⎪ ϕ i k =1
m =1
m =1
ϕ i k =1
0i
0i
0i
⎝ ϕi
⎠
⎪
M
M
⎛ A12 A11 sin ϕi ⎞
⎪ A12 N ϕ (1) ϕ
A cos ϕi ϕ
B 12 N ϕ (1) ϕ
B cos ϕi ϕ
s (1) s
s (1) s
+
ς ik U kM + 11
U iM + A11 ∑ ς Mm
U im + ⎜
ς ik BkM + 11
BiM + B 11 ∑ ς Mm
Bim
⎟WiM +
⎪=
∑
∑
⎜
⎟
R 0i
R 0i ⎠
Rϕ i k =1
R 0i
m =1
m =1
⎪ Rϕ i k =1
⎝ Rϕ i
⎪
A N
⎪ A 66 N ϕ (1)
ϕ
s
ς ik Wk1 + κ A 66 Biϕ1 + κ A 66 Bis1 = κ 66 ∑ ς ikϕ (1)WkM + κ A 66 BiM
+ κ A 66 BiM
⎨κ
∑
R
Rϕ i k =1
ϕ i k =1
⎪
⎪
M
M
B 66 N ϕ (1) s B 66 cos ϕi s
D 66 N ϕ (1) s D 66 cos ϕi s
s1 ϕ
ϕ
⎪ B 66 ∑ ς 1sm(1)U im
+
ς ik U k 1 −
U i1 + D 66 ∑ ς 1m( ) Bim
+
∑
∑ ς ik Bk1 − R Bi1 =
⎪ m =1
Rϕ i k =1
R 0i
Rϕ i k =1
m =1
0i
⎪
M
M
B 66 N ϕ (1) s
B 66 cos ϕi s
D 66 N ϕ (1) s
D 66 cos ϕi s
⎪
s (1) ϕ
s (1) ϕ
U iM + D 66 ∑ ς Mm Bim +
∑ ς ik BkM − R BiM
⎪= B 66 ∑ ς Mm U im + R ∑ ς ik U kM −
R 0i
Rϕ i k =1
m =1
m =1
ϕ i k =1
0i
⎪
⎪B N
M
N
M
⎛ B 12 B 11 sin ϕi ⎞
D 12
D cos ϕi ϕ
s (1) s
s1
⎪ 12 ς ϕ (1)U ϕ + B 11 cos ϕi U ϕ + B
+
ς ikϕ (1) Bkϕ1 + 11
Bi1 + D 11 ∑ ς 1m( ) Bims =
⎟Wi1 +
∑
∑
11 ∑ ς 1m U im + ⎜
ik
k1
i1
⎜ Rϕ i
⎟
⎪ Rϕ i k =1
R 0i
R
R
R
1
1
m =1
k
m
=
=
0
ϕ
0
i
i
i
⎝
⎠
⎪
⎪ B N
M
M
⎛ B 12 B 11 sin ϕi ⎞
B 11 cos ϕi ϕ
D 12 N ϕ (1) ϕ
D cos ϕi ϕ
ϕ (1) ϕ
s (1) s
s (1) s
12
⎪=
+
U iM + B 11 ∑ ς Mm U im + ⎜
ς ik BkM + 11
BiM + D11 ∑ ς Mm
Bim
ς ik U kM +
⎟WiM +
∑
∑
⎜ Rϕ i
⎟
(37)
R 0i
R
R
R
⎪ Rϕ i k =1
=
=
1
1
m =1
k
m
ϕi
0i
0i
⎝
⎠
⎩
for i = 2,..., N − 1
Thus, the whole system of differential equations has been discretized and the global assembling leads to the
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following set of linear algebraic equations:
⎡ K bb
⎢K
⎢⎣ db
K bd ⎤ ⎡ δb ⎤
0 ⎤ ⎡ δb ⎤
⎡0
= ω2 ⎢
⎥
⎢
⎥
⎥⎢ ⎥
K dd ⎥⎦ ⎢⎣δ d ⎥⎦
⎢⎣ 0 M dd ⎥⎦ ⎢⎣δ d ⎥⎦
(38)
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In the above matrices and vectors, the partitioning is set forth by subscripts b and d , referring to the system
degrees of freedom and standing for boundary and domain, respectively. In this sense, the b -equations represent the
discrete boundary and compatibility conditions, which are valid only for the points lying on constrained edges of the
shell; while the d -equations are the equilibrium equations, assigned on interior nodes. In order to make the
)
− K db ( K bb ) K bd δ d = ω 2 M dd δ d
-1
dd
(39)
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(K
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computation more efficient, kinematic condensation of non-domain degrees of freedom is performed:
The natural frequencies of the structure considered can be determined by solving the standard eigenvalue problem
(39). In particular, the solution procedure by means of GDQ technique has been implemented in a MATLAB code.
Finally, the results in terms of frequencies are obtained using the eigs function of MATLAB program.
It is worth noting that, with the present approach, differing from the finite element method, no integration occurs
prior to the global assembly of the linear system, and this leads to a further computational cost saving in favour of
5.
Results and Discussion
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the differential quadrature technique.
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This section introduces some results and considerations about the free vibration problem of functionally graded
parabolic panels and shells of revolution. The analysis has been carried out by means of numerical procedures
illustrated above.
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No literature is available about the results of the GDQ solution for free vibrations of FGM shells with parabolic
meridian. Several attempts to validate the present formulations have been made for the isotropic and anisotropic
cases and can be found in the PhD Thesis by Tornabene (2007) and in articles by Tornabene and Viola (2007, 2008).
In this work, the frequency parameters from the present formulations are in good agreement with the results
presented in the literature and obtained with the finite element method.
Regarding the functionally graded materials, their two constituents are taken to be zirconia (ceramic) and aluminum
(metal). Young’s modulus, Poisson’s ratio and mass density for the zirconia are EC = 168GPa , ν C = 0.3 ,
ρC = 5700 Kg m3 , and for the aluminum are EM = 70 GPa , ν M = 0.3 , ρ M = 2707 Kg m3 , respectively. The
details regarding the geometry of the structures considered are indicated below:
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Toro-parabolic panel (C-F-C-F)/(S-F-S-F): k = 9, Rb = 9 m, h = 0.4 m, s1 = 3m, s0 = −3m, S = 2 m, ϑ0 = 120° ;
2.
Parabolic panel (C-F-F-F): k = 8, Rb = 0 m, h = 0.2 m, s1 = 4 m, s0 = 1m, S = 2 m, ϑ0 = 120° ;
3.
Parabolic toroid (F-C): k = 4.5, Rb = 6 m, h = 0.3m, s1 = 0 m, s0 = −3m, S = 2 m, ϑ0 = 360° ;
4.
Parabolic dome (C-F): k = 8, Rb = 0 m, h = 0.2 m, s1 = 4 m, s0 = 1m, S = 2 m, ϑ0 = 360° .
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1.
The geometrical boundary conditions for the shell panel are identified by the following convention. For example, the
symbolism C-S-C-F indicates that the edges ϕ = ϕ 1 , s = s0 , ϕ = ϕ 0 , s = 0 are clamped, simply supported, clamped
and free, respectively (Fig. 6). For the complete shell of revolution, for example, the symbolism C-F indicates that
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the edges ϕ = ϕ 1 and ϕ = ϕ 0 are clamped and free, respectively. In this case, the missing boundary conditions are
the kinematic and physical compatibility conditions that are applied at the same meridian for s = 0 and s0 = 2π R 0 .
FGM 1 ( a =1/ b =1/ c = 3 / p =1)
toro-parabolic panel (C-F-C-F), the
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In Figs. 7-10, the first mode shapes for the FGM 1 ( a =1/ b = 0.5 / c = 2 / p =1)
parabolic panel (C-F-F-F), the
FGM 1 ( a = 0 / b = −0.5 / c = 2 / p =1)
parabolic toroid (F-C) and the
FGM 1 ( a = 0.8 / b = 0.2 / c = 3 / p =1) parabolic dome (C-F) are illustrated. In particular, for the parabolic toroid and dome, there are
some symmetrical mode shapes due to the symmetry of the problem considered in 3D space. In these cases, the
symmetrical mode shapes have been summarized in one figure. The mode shapes of all the structures under
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discussion have been evaluated by the authors. By using the authors’ MATLAB code, these mode shapes have been
reconstructed in three-dimensional view by means of considering the displacement field (2) after solving the
eigenvalue problem (39).
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Tables 1-6 illustrate the first ten frequencies of the four structures under consideration. These tables show how by
varying only the power-law index p of the volume fraction VC it is possible to modify natural frequencies of FGM
panels and shells of revolution. For the GDQ results presented in Tables 1-6, the grid distributions (26) with
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N = M = 21 have been taking into account. The results are obtained using various values of the power-law exponent
p (i.e. p = 0 ceramic rich and p = ∞ metal rich) for the two power-law distributions FGM 1 ( a / b / c / p ) and
FGM 2 ( a / b / c / p ) . For each power-law distribution the same values of the three parameters a, b, c are chosen.
Tables 1-6 show that by considering the two power-law distributions FGM 1 ( a / b / c / p ) and FGM 2 ( a / b / c / p ) with the
same values of the four parameters a, b, c, p the natural frequencies are different. In fact, for curved shells it is
important, from the dynamic vibration point of view, to know if the top surface of the shell (ζ = h 2 ) is ceramic or
metal rich and, inversely, if the bottom surface (ζ = − h 2 ) is metal or ceramic rich, respectively. Using one of the
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two power-law distributions FGM 1 ( a / b / c / p ) and FGM 2 ( a / b / c / p ) changes the dynamic behaviour of the shell structure.
Furthermore, Tables 1-6 show the effect of the power-law distribution choice on the frequency parameters. It is
worth noting that the difference between FGM 1 ( a / b / c / p ) and FGM 2 ( a / b / c / p ) shell frequencies increases with
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increasing values of the shell thickness because of the curvature of the shell. From tables 1-6 it is evident that
functionally graded shell and panel structures behave like the corresponding homogeneous structures ( p = 0 ), for
p = 0.6 and p = 1 . A similar behaviour was pointed out before by Abrate (2006) for the functionally graded plate.
The influence of the index p on the vibration frequencies is also shown in Figs. 11-16. As it can be seen from
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figures, natural frequencies of FGM panels and shells of revolution often present an intermediate value between the
natural frequencies of the limit cases of homogeneous shells of zirconia ( p = 0 ) and of aluminium ( p = ∞ ), as
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expected. However, natural frequencies sometimes exceed the limit cases, as evident in Figs. 11, 15 and 16. This
fact can depend on various parameters, such as the geometry of the shell, the boundary conditions, the power-law
distribution profile, etc. In particular, for specific values of the four parameters a, b, c, p it is possible to exceed or
approach the ceramic limit case as shown in Figs. 11-16, even if the ceramic content is not high. Increasing the
values of the parameter index p up to infinity reduces the ceramic content and at the same time increases the metal
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percentage. Therefore, it is possible to obtain dynamic characteristics similar or better than the isotropic ceramic
limit case by choosing suitable values of the four parameters a, b, c, p .
Each of Figs. 11-16 is divided into two parts. On the left, figures (a) show the first four frequencies versus the
power-law index p obtained using the FGM 1 ( a / b / c / p ) distribution, while on the right, figures (b) illustrate the first
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four frequencies versus the power-law index p obtained using the FGM 2 ( a / b / c / p ) distribution.
Fig. 11 shows the first four natural frequencies of the toro-parabolic panel (C-F-C-F) versus the power-law index p
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for various values of the parameter b , when a = 1, c = 2 . Figs. 11(a) illustrate the variation of the first four
frequencies obtained using the FGM 1 ( a =1/ 0≤b ≤1/ c = 2/ p ) distribution, while in figures 11(b) the first four frequencies for
the FGM 2 ( a =1/0≤ b ≤1/ c = 2/ p ) distribution are reported. It is interesting to note that, for the structures under consideration,
frequencies attain a minimum value for a shell made only of metal, due to the fact that aluminium has a much
smaller Young’s modulus than zirconia. In particular, it is evident that in Fig. 11 for low values of the parameter b
most frequencies exhibit a fastly decreasing behaviour from the ceramic limit case ( p = 0 ) varying the power-law
index from p = 0 to p ≈ 1 . For values of p greater than unity frequencies increase until a maximum value. After
this maximum, frequencies slowly decrease by increasing the power-law exponent p and tend to the metal limit
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ARTICLE IN PRESS
case ( p = ∞ ). This is expected because the more p increases the more the ceramic content is low and the FGM
shell approaches the case of the fully metal shell. Otherwise, in Fig. 11 for values of the parameter b approaching
the unity the frequency curves present a fastly rising behaviour up to a maximum value by increasing the power-law
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index p and exceed the ceramic limit case. After this maximum, frequencies gradually decrease.
In the same way, by setting different values of parameters a, c , Figs. 12-13 ( a = 1, c = 1 ) and 14 ( a = 0, c = 2 ) show
the influence of the parameter b on the dynamic vibration of the toro-parabolic panel (S-F-S-F), of the parabolic
panel (C-F-F-F) and of the parabolic toroid (F-C), respectively. In Figs. 12 and 13 the parameter b varies from 0 to
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1 , as in Fig. 11, while in Fig. 14 from −1 to 0 . Moreover, the influence of the parameter a on the dynamic
vibration of parabolic dome (C-F) is investigated in Fig. 15 by considering b = 0.2, c = 3 . In this case the parameter
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a varies from 0.2 to 1.2 . It can be noted that in Fig. 15, for low values of the parameter a , most frequencies
exhibit a fastly decreasing behaviour from the ceramic limit case ( p = 0 ) varying the power-law index from p = 0
to p ≈ 1 , while for values of p greater than unity frequencies increase until a maximum value. After this maximum,
frequencies slowly decrease by increasing the power-law exponent p and tend to the metal limit case ( p = ∞ ).
Otherwise, approaching the value a = 1.2 the frequency curves do not present a knee as previously described, but
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present a fast rising behaviour up to a maximum value by increasing the power-law index p and exceed the ceramic
limit case. After this maximum, frequencies gradually decrease. This behaviour depends on the type of vibration
mode. It is worth noting that some frequencies do not present a knee or a maximum value as described above, but
decrease gradually from the ceramic limit case ( p = 0 ) to the metal limit case ( p = ∞ ) by increasing the power-law
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exponent p . In particular, the types of vibration mode that can present a monotone gradually decrease of frequency
are torsional, bending and axisymmetric mode shapes, while the circumferential and radial mode shapes are
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characterized by a knee or a maximum value, as it can be seen by comparing the mode shapes with variations of
frequencies as functions of the power-law exponent p . For examples, the frequencies
f 3,4
and
f7
of the parabolic
dome of Fig. 15 correspond to bending and axisymmetric mode shape as can be observed from Fig. 10, respectively.
However, this behaviour depends on the geometry of the shell and on its boundary conditions.
Finally, in Fig. 16 the influence of the parameter c on the dynamic vibration of parabolic panel (C-F-F-F) is
investigated by considering a = 1, b = 1 . The parameter c varies from 1 to 11 . It can be noted that due to the effect of
the parameter c frequencies exhibit a fastly rising behaviour up to a maximum value exceeding the ceramic limit
case ( p = 0 ) by varying the power-law index from p = 0 to p ≈ 10 . After this maximum, frequencies gradually
decrease by increasing the power-law exponent p and tend to the metal limit case ( p = ∞ ). The value of the
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maximum depends on the value of c . By increasing the value of the parameter c ( c > 1 ) the maximum gradually
decreases.
In order to investigate the GDQ procedure convergence, the first ten frequencies of the functionally graded toro-
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parabolic panel (C-F-C-F) for the FGM 1 ( a =1/ b = 0 / c = 0 / p = 5) power-law distribution are examinated by varying the
number of grid points. Results are collected in Table 7 when the number of points of the Chebyshev-Gauss-Lobatto
grid distribution (26) is increased from N = M = 11 up to N = M = 31 . It can be seen that the proposed GDQ
formulation well captures the dynamic behaviour of the system by using only 21 points in two co-ordinate
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directions. It can also be seen that for the system under investigation, the formulation is stable while increasing the
number of points and that the use of 21 points guarantees convergence of the procedure. Analogous and similar
convergence results can be obtained for all the shell structures considered in this work. A wide convergence study of
Tornabene and Viola (2007, 2008).
6.
Conclusion Remarks and Summary
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GDQ solutions for shell structures has been shown in the PhD thesis by Tornabene (2007) and in the articles by
The Generalized Differential Quadrature Method has been used to study the free vibration analysis of functionally
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graded parabolic thick panels and shells of revolution. The First-order Shear Deformation Theory has been adopted.
The dynamic equilibrium equations discretized with the present method lead to a standard linear eigenvalue
problem. The complete 2D differential system, governing the structural problem, has been solved. The vibration
results are obtained without the modal expansion methodology. Thus, complete revolution shells are obtained as
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special cases of shell panels by satisfying the kinematic and physical compatibility. The GDQ method provides
converging results for all the cases as the number of grid points increases. Convergence and stability have been
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shown for one of the four structures considered.
In this study, ceramic-metal graded shells of revolution with four parameter power-law distributions of the volume
fraction of the constituents in the thickness direction have been worked out. Various material profiles through the
functionally graded shell thickness have been illustrated by varying the four parameters of power-law distributions.
Symmetric and asymmetric volume fraction profiles have been presented. The numerical results have shown the
influence of the power-law exponent, of the power-law distribution choice and of the choice of the four parameters
on the free vibrations of functionally graded shells considered. The analysis provides information about the dynamic
response of parabolic shell structures for different proportions of the ceramic and metal. For curved shells and
panels, it has been observed that the influence of the distribution choice is marked and can be considered from the
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structural design point of view. In general, it can be pointed out that the frequency vibration of functionally graded
shells and panels of revolution depends on the type of vibration mode, thickness, power-law distribution, power-law
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exponent and the curvature of the structure.
Acknowledgments
This research was supported by the Italian Ministry for University and Scientific, Technological Research MIUR
(40% and 60%). The research topic is one of the subjects of the Centre of Study and Research for the Identification
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of Materials and Structures (CIMEST)-“M. Capurso” of the University of Bologna (Italy).
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Pelletier, J.L., Vel, S.S., 2006. An exact solution for the steady-state thermoelastic response of functionally graded
orthotropic cylindrical shells, Int. J. Solids Struct. 43, 1131-1158.
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TE
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Roque, C.M.C., Ferreira, A.J.M., Jorge, R.M.N., 2007. A radial basis function for the free vibration analysis of
functionally graded plates using refined theory, J. Sound Vib. 300, 1048-1070.
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Toorani, M.H., Lakis, A.A., 2000. General equations of anisotropic plates and shells including transverse shear
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deformations, rotary inertia and initial curvature effects, J. Sound Vib. 237, 561-615.
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University of Bologna – DISTART Department.
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Math. Appl. 53, 1538-1560.
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quadrature method, Eur. J. Mech. A/Solids, Available online 4 march 2008.
Viola, E., Artioli, E., 2004. The G.D.Q. method for the harmonic dynamic analysis of rotational shell structural
elements, Struct. Eng. Mech. 17, 789-817.
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cylindrical panels, J. Sound Vib. 261, 871-893.
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Zenkour, A.M., 2006. Generalized shear deformation theory for bending analysis of functionally graded plates,
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Appendix
The following are the equilibrium operators in equations (10):
⎛ cos ϕ
A11 ∂ 2
A12 sin ϕ A11 cos 2 ϕ
A 66
1 dR ⎞ ∂
∂2
+ A11 ⎜
− 3 ϕ⎟
+ A 66 2 −
−
−κ 2
2
2
2
⎜
⎟
∂
∂
ϕ
ϕ
Rϕ ∂ϕ
R
R
R
d
s
R
R
R
Rϕ
ϕ
ϕ
ϕ
0
0
0
⎝
⎠
TE
D
L 11 =
⎛ A11 cos ϕ A 66 cos ϕ ⎞ ∂ ⎛ A12 A 66
L 12 = − ⎜
+
+
⎟ +⎜
⎜
⎜ R0
⎟
R0
Rϕ
⎝
⎠ ∂s ⎝ Rϕ
⎞ ∂2
⎟
⎟
⎠ ∂ϕ ∂s
2
⎛ cos ϕ
B 11 ∂ 2
A 66
1 dR ⎞ ∂
∂ 2 B 12 sin ϕ B 11 cos ϕ
+ B 11 ⎜
− 3 ϕ⎟
+ B 66 2 −
−
+κ
2
2
2
⎜ Rϕ R 0 Rϕ dϕ ⎟ ∂ϕ
∂
Rϕ ∂ϕ
s
R
R
R
Rϕ
ϕ 0
0
⎝
⎠
AC
C
L 14 = L 41 =
EP
⎛ A11 A12 sin ϕ
⎛ cos ϕ
A 66 ⎞ ∂
1 dR ⎞ A11 sin ϕ cos ϕ
L 13 = ⎜ 2 +
+κ 2 ⎟
+ A11 ⎜
− 3 ϕ ⎟−
⎜ Rϕ
⎟
⎜
⎟
∂
R
R
R
R
R
R
R 20
ϕ
ϕ 0
ϕ ⎠
ϕ dϕ ⎠
⎝
⎝ ϕ 0
⎛ B 11 cos ϕ B 66 cos ϕ ⎞ ∂ ⎛ B 12 B 66 ⎞ ∂ 2
+
+
L 15 = L 42 = − ⎜
⎟ +⎜
⎟
⎜
⎜ R0
⎟
R0
Rϕ ⎠⎟ ∂ϕ ∂s
⎝
⎠ ∂s ⎝ Rϕ
⎛ A11 cos ϕ A 66 cos ϕ ⎞ ∂ ⎛ A12 A 66
L 21 = ⎜
+
+
⎟ +⎜
⎜ R0
⎟ ∂s ⎜ Rϕ
R0
Rϕ
⎝
⎝
⎠
L 22 =
⎞ ∂2
⎟
⎟ ∂ϕ ∂s
⎠
⎛ cos ϕ
⎛ cos 2 ϕ sin ϕ
A 66 ∂ 2
1 dR ⎞ ∂
∂2
+ A 66 ⎜
− 3 ϕ⎟
+ A11 2 − A 66 ⎜
−
2
2
⎜ Rϕ R 0 Rϕ dϕ ⎟ ∂ϕ
⎜ R 20
Rϕ ∂ϕ
Rϕ R 0
∂s
⎝
⎠
⎝
⎞
A 66 sin 2 ϕ
⎟ −κ
⎟
R 20
⎠
⎛ A12 A11 sin ϕ
A 66 sin ϕ ⎞ ∂
+
+κ
L 23 = − L 32 = ⎜
⎟
⎜ Rϕ
R
R 0 ⎟⎠ ∂s
0
⎝
22
ARTICLE IN PRESS
⎛ B 11 cos ϕ B 66 cos ϕ ⎞ ∂ ⎛ B 12 B 66 ⎞ ∂ 2
L 24 = L 51 = ⎜
+
+
⎟ +⎜
⎟
⎜ R0
⎟ ∂s ⎜ Rϕ
R0
Rϕ ⎟⎠ ∂ϕ ∂s
⎝
⎝
⎠
⎛ cos ϕ
⎛ cos 2 ϕ sin ϕ
B 66 ∂ 2
1 dR ⎞ ∂
∂2
+ B 66 ⎜
− 3 ϕ⎟
+ B 11 2 − B 66 ⎜
−
2
2
⎜ Rϕ R 0 Rϕ dϕ ⎟ ∂ϕ
⎜ R 20
Rϕ ∂ϕ
Rϕ R 0
∂s
⎝
⎠
⎝
⎞
A 66 sin ϕ
⎟+κ
⎟
R0
⎠
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L 25 = L 52 =
⎛ A11 A12 sin ϕ
⎛ cos ϕ
A 66 ⎞ ∂
A12 cos ϕ A11 sin ϕ cos ϕ
1 dR ⎞
L 31 = − ⎜ 2 +
+κ 2 ⎟
−
−
− κ A 66 ⎜
− 3 ϕ⎟
2
⎜ Rϕ
⎟
⎜
⎟
R
R
R
R
R
R
R
R
R
∂
ϕ
ϕ 0
ϕ ⎠
ϕ 0
ϕ dϕ ⎠
0
⎝
⎝ ϕ 0
L 33 = κ
⎛ cos ϕ
A 66 ∂ 2
A11 2 A12 sin ϕ A11 sin 2 ϕ
1 dR ⎞ ∂
∂2
+ κ A 66 ⎜
− 3 ϕ⎟
+ κ A 66 2 − 2 −
−
2
2
⎜
⎟
Rϕ ∂ϕ
Rϕ
Rϕ R 0
R 20
∂s
⎝ Rϕ R 0 Rϕ dϕ ⎠ ∂ϕ
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⎛ B 12 B 11 sin ϕ
⎞∂
−
+ κ A 66 ⎟
L 35 = − L 53 = ⎜ −
⎜ Rϕ
⎟
R
0
⎝
⎠ ∂s
SC
⎛ B 11 B 12 sin ϕ
A 66 ⎞ ∂
B 12 cos ϕ B 11 sin ϕ cos ϕ
A 66 cos ϕ
+κ
−
−
+κ
L 34 = ⎜ − 2 −
⎟
2
⎜ Rϕ
⎟
R 0 Rϕ
Rϕ ⎠ ∂ϕ
Rϕ R 0
R0
R0
⎝
⎛ B 11 B 12 sin ϕ
⎛ cos ϕ
A 66 ⎞ ∂
1 dR ⎞ B 11 sin ϕ cos ϕ
L 43 = ⎜ 2 +
−κ
+ B 11 ⎜
− 3 ϕ ⎟−
⎟
⎜ Rϕ
⎟
⎜
⎟
R
R
R
R
R
R
R 20
∂
ϕ
ϕ 0
ϕ ⎠
ϕ dϕ ⎠
⎝
⎝ ϕ 0
L 44 =
2
⎛ cos ϕ
D 11 ∂ 2
1 dR ⎞ ∂
∂ 2 D12 sin ϕ D 11 cos ϕ
+ D 11 ⎜
− 3 ϕ⎟
+ D 66 2 −
−
− κ A 66
2
2
2
⎜
⎟
∂s
Rϕ ∂ϕ
Rϕ R 0
R0
⎝ Rϕ R 0 Rϕ dϕ ⎠ ∂ϕ
⎞ ∂2
⎟
⎟ ∂ϕ ∂s
⎠
TE
D
⎛ D 11 cos ϕ D 66 cos ϕ ⎞ ∂ ⎛ D 12 D 66
L 45 = − ⎜
+
+
⎟ +⎜
⎜ R0
⎟ ∂s ⎜ Rϕ
R0
Rϕ
⎝
⎝
⎠
⎛ cos ϕ
⎛ cos 2 ϕ sin ϕ
D 66 ∂ 2
∂2
1 dR ⎞ ∂
+ D 66 ⎜
− 3 ϕ⎟
+ D11 2 − D 66 ⎜
−
2
2
⎜ Rϕ R 0 Rϕ dϕ ⎟ ∂ϕ
⎜ R 20
∂s
Rϕ ∂ϕ
Rϕ R 0
⎝
⎠
⎝
⎞
⎟ − κ A 66
⎟
⎠
AC
C
L 55 =
EP
⎛ D11 cos ϕ D 66 cos ϕ ⎞ ∂ ⎛ D12 D 66 ⎞ ∂ 2
L 54 = ⎜
+
+
⎟ +⎜
⎟
⎜ R0
⎟ ∂s ⎜ Rϕ
R0
Rϕ ⎟⎠ ∂ϕ ∂s
⎝
⎝
⎠
23
ARTICLE IN PRESS
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ζ
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ϕ
s
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Figure 1. Co-ordinate system of doubly curved shell.
R 0 (ϕ )
Rb
O′
O
ts
d
n
ϕ
ϑ
O
TE
D
s0
R0 (ϕ )
n
x1
s
x1
tϕ
Rϕ
S
C1
C2
AC
C
x3
dϕ
EP
Rs
x2
s1
ϕ
x′3
Figure 2. Shell geometry.
24
(a)
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ARTICLE IN PRESS
(b)
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index p : (a) FGM 1 ( a =1/ b = 0 / c / p ) , (b) FGM 2 ( a =1/ b = 0 / c / p ) .
SC
Figure 3. Variations of the ceramic volume fraction VC through the thickness for different values of the power-law
(b)
EP
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D
(a)
(c)
(d)
Figure 4. Variations of the ceramic volume fraction VC through the thickness for different values of the power-law
AC
C
index p : (a) FGM 1 ( a =1/ b =1/ c = 2 / p ) , (b) FGM 2 ( a =1/ b =1/ c = 2 / p ) , (c) FGM 1 ( a =1/ b =1/ c = 4 / p ) , (d) FGM 2 ( a =1/ b =1/ c = 4 / p ) .
25
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ARTICLE IN PRESS
(b)
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SC
(a)
(d)
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D
(c)
(e)
(f)
Figure 5. Variations of the ceramic volume fraction VC through the thickness for different values of the three
parameters
a , b, c
and the power-law index
p : (a)
FGM 1 ( a =1/ b = 0.5 / c = 2 / p ) , (b)
FGM 2 ( a =1/ b = 0.5 / c = 2 / p ) , (c)
AC
C
EP
FGM 1 ( a = 0.8 / b = 0.2 / c = 3 / p ) , (d) FGM 2 ( a = 0.8 / b = 0.2 / c =3 / p ) , (e) FGM 1 ( a = 0 / b = 0.5 / c = 2 / p ) , (f) FGM 2 ( a = 0 / b = 0.5 / c = 2 / p ) .
26
ARTICLE IN PRESS
1
i
s = s0
(ϕ , s )
i
j
N
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s=0
ϕ =ϕ0
s
ϕ
j
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ϕ = ϕ1
SC
M
1
Figure 6. C-G-L grid distribution on a parabolic panel.
Mode shape 3
Mode shape 5
Mode shape 6
Mode shape 8
Mode shape 9
AC
C
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Mode shape 4
Mode shape 2
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D
Mode shape 1
Mode shape 7
Figure 7. Mode shapes for the FGM 1 ( a =1/ b = 0.5 / c = 2 / p =1) toro-parabolic panel C-F-C-F.
27
Mode shape 4
Mode shape 5
Mode shape 3
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Mode shape 2
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Mode shape 1
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ARTICLE IN PRESS
Mode shape 7
Mode shape 8
Mode shape 6
Mode shape 9
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Figure 8. Mode shapes for the FGM 1 ( a =1/ b =1/ c = 3 / p =1) parabolic panel C-F-F-F.
Mode shapes 1-2
Mode shapes 5-6
Mode shape 7
Mode shapes 8-9
Mode shapes 10-11
Mode shapes 12-13
Mode shapes 14-15
Mode shapes 16-17
AC
C
EP
Mode shapes 3-4
Figure 9. Mode shapes for the FGM 1 ( a = 0 / b =−0.5 / c = 2 / p =1) parabolic toroid F-C.
28
Mode shape 7
Mode shapes 8-9
Mode shapes 5-6
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Mode shapes 3-4
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Mode shapes 1-2
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ARTICLE IN PRESS
Mode shape 12
Mode shapes 13-14
Mode shapes 10-11
Mode shapes 15-16
AC
C
EP
TE
D
Figure 10. Mode shapes for the FGM 1 ( a = 0.8 / b = 0.2 / c = 3 / p =1) parabolic dome C-F.
29
ARTICLE IN PRESS
b =1
b = 0.8
b = 0.8
b=0
b=0
(a)
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b =1
SC
(b)
b = 0.8
b =1
b=0
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b = 0.8
b =1
b=0
TE
D
(a)
b = 0.8
b = 0.8
b =1
AC
C
EP
b=0
b=0
(b)
b =1
b=0
(a)
b =1
b =1
b = 0.8
b=0
(a)
b = 0.8
(b)
Figure 11. The first four frequencies of functionally graded toro-parabolic panel (C-F-C-F) versus the power-law
exponent p for various values of the parameter b : (a) FGM 1 ( a =1/ 0≤b ≤1/ c = 2 / p ) , (b) FGM 2 ( a =1/ 0≤b ≤1/ c = 2 / p ) .
30
ARTICLE IN PRESS
b =1
b =1
b = 0.8
b = 0.6
b = 0.4
b = 0.4
b = 0.6
b = 0.8
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b = 0.95
b = 0.95
b=0
b = 0.2
b = 0.2
(a)
b =1
b =1
b = 0.95
b = 0.6
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b = 0.8
b=0
(b)
SC
b=0
b = 0.95
b = 0.6
b = 0.2
b = 0.4
b = 0.2
b = 0.4
(a)
(b)
b =1
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D
b =1
b = 0.95
b = 0.8
b = 0.95
b = 0.6
b = 0.2
b = 0.6
b = 0.8
b=0
b = 0.4
AC
C
EP
b=0
b = 0.8
b=0
b = 0.2
b = 0.4
(a)
(b)
b =1
b =1
b = 0.95
b = 0.95
b = 0.8
b = 0.8
b=0
b = 0.2
b = 0.6
b = 0.6
b = 0.4
(a)
b=0
b = 0.2
b = 0.4
(b)
Figure 12. The first four frequencies of functionally graded toro-parabolic panel (S-F-S-F) versus the power-law
exponent p for various values of the parameter b : (a) FGM 1 ( a =1/ 0≤b ≤1/ c =1/ p ) , (b) FGM 2 ( a =1/ 0≤b ≤1/ c =1/ p ) .
31
ARTICLE IN PRESS
b =1
b =1
b = 0.95
b = 0.95
b = 0.8
b = 0.6
b = 0.6
b = 0.8
b = 0.4
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b = 0.4
b=0
b=0
b = 0.2
b = 0.2
(a)
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b =1
(b)
b =1
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b = 0.95
b = 0.95
b = 0.8
b = 0.6
b=0
b = 0.8
b = 0.6
b=0
b = 0.4
b = 0.2
b = 0.4
(a)
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b =1
(b)
b =1
b = 0.95
b = 0.95
b = 0.6
b=0
b = 0.8
b = 0.8
b = 0.4
b=0
b = 0.2
b = 0.2
AC
C
EP
b = 0.4
b =1
(a)
(b)
b =1
b = 0.95
b = 0.6
b = 0.6
b = 0.8
b = 0.95
b = 0.8
b = 0.4
b=0
b=0
b = 0.2
b = 0.6
b = 0.2
b = 0.4
(a)
(b)
Figure 13. The first four frequencies of functionally graded parabolic panel (C-F-F-F) versus the power-law exponent
p for various values of the parameter b : (a) FGM 1 ( a =1/ 0≤ b ≤1/ c =1/ p ) , (b) FGM 2 ( a =1/ 0≤b ≤1/ c =1/ p ) .
32
ARTICLE IN PRESS
b=0
b=0
b = −0.2
b = −0.2
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b = −1
b = −0.6
b = −1
(a)
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(b)
b=0
b=0
b = −0.6
b = −0.2
b = −0.6
b = −0.6
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b = −0.2
b = −1
b = −1
(a)
b = −0.2
b = −1
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D
b=0
(b)
b=0
b = −0.2
AC
C
EP
b = −0.6
b = −0.6
(a)
b = −1
(b)
b=0
b=0
b = −0.2
b = −1
b = −0.2
b = −0.6
(a)
b = −0.6
b = −1
(b)
Figure 14. The first four frequencies of functionally graded parabolic toroid (F-C) versus the power-law exponent p
for various values of the parameter b : (a) FGM 1 ( a = 0 / −1≤b ≤ 0 / c = 2 / p ) , (b) FGM 2 ( a = 0 / −1≤ b ≤ 0 / c = 2 / p ) .
33
ARTICLE IN PRESS
a = 1.2
a = 1.2
a = 0.8
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a = 0.8
a =1
a = 0.2
a =1
a = 0.2
a = 0.4
a = 0.6
(a)
a = 0.6
a = 0.4
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(b)
a = 1.2
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a = 1.2
a = 0.8
a = 0.8
a =1
a = 0.2
a = 0.4
a = 0.6
a = 0.2
a =1
a = 0.6
a = 0.4
(a)
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a = 1.2
(b)
a = 1.2
a = 0.8
a = 0.8
a =1
a = 0.2
a = 0.4
a = 0.6
AC
C
EP
a = 0.4
a =1
a = 0.2
a = 0.6
(a)
(b)
a = 1.2
a = 1.2
a = 0.8
a = 0.8
a =1
a =1
a = 0.2
a = 0.2
a = 0.4
a = 0.6
a = 0.4
(a)
a = 0.6
(b)
Figure 15. The first four frequencies of functionally graded parabolic dome (C-F) versus the power-law exponent p
for various values of the parameter a : (a) FGM 1 (0.2≤ a ≤1.2 / b = 0.2 / c = 3 / p ) , (b) FGM 2 (0.2≤ a ≤1.2 / b = 0.2 / c =3 / p ) .
34
ARTICLE IN PRESS
c=3
c=3
c=5
c=5
c =1
c = 11
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c =1
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c = 11
c =1
c =1
c=3
c=3
c=5
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c=5
c = 11
c = 11
c=3
c=3
TE
D
c=5
c=5
c =1
c = 11
AC
C
EP
c = 11
c =1
c=3
c=3
c=5
c = 11
c=5
c =1
c =1
c = 11
Figure 16. The first four frequencies of functionally graded parabolic panel (C-F-F-F) versus the power-law exponent
p for various values of the parameter c : (a) FGM 1 ( a =1/ b =1/1≤ c ≤11/ p ) , (b) FGM 2 ( a =1/ b =1/1≤ c ≤11/ p ) .
35
ARTICLE IN PRESS
Table 1. The first ten frequencies for the functionally graded toro-parabolic panel (C-F-C-F) as a function of the
power-law exponent p , for a = 1, b = 0.5, c = 2 .
FGM 1 ( a =1/ b = 0.5 / c = 2 / p ) power-law distribution
p = 0.6
p=5
p = 20
p = 50
p = 100
p=∞
98.95
102.57
110.06
128.77
155.49
181.53
189.01
189.94
199.68
205.00
97.51
101.03
108.42
126.84
153.14
178.67
185.90
186.85
196.50
201.62
95.59
99.15
106.46
124.69
150.72
175.96
182.89
184.15
193.07
198.76
94.66
98.24
105.52
123.67
149.57
174.66
181.43
182.86
191.40
197.38
93.56
97.17
104.41
122.45
148.21
173.09
179.65
181.29
189.38
195.70
p =1
99.88
99.73
99.57
103.74
103.55
103.36
f3
111.46
111.23
111.01
f4
130.73
130.40
130.11
f5
158.23
157.76
157.38
f6
184.79
184.26
183.82
f7
191.80
191.39
191.02
f8
193.54
192.97
192.51
f9
202.18
201.81
201.45
f10
208.93
208.32
207.83
FGM 2 ( a =1/ b = 0.5 / c = 2 / p ) power-law distribution
f1
f2
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p=0
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Frequencies
[Hz]
p=0
p = 0.6
p =1
p=5
p = 20
p = 50
p = 100
p=∞
f1
99.88
103.74
111.46
130.73
158.23
184.79
191.80
193.54
202.18
208.93
99.24
103.05
110.73
129.88
157.19
183.53
190.43
192.18
200.79
207.45
98.81
102.59
110.24
129.30
156.50
182.70
189.54
191.29
199.86
206.48
97.36
100.96
108.46
127.10
153.69
179.27
185.98
187.51
196.38
202.32
96.78
100.29
107.69
126.07
152.32
177.65
184.51
185.74
194.98
200.40
95.27
98.82
106.14
124.36
150.36
175.51
182.28
183.66
192.39
198.22
94.50
98.08
105.36
123.50
149.39
174.42
181.11
182.60
191.05
197.10
93.56
97.17
104.41
122.45
148.21
173.09
179.65
181.29
189.38
195.70
f3
f4
f5
f6
f7
f8
f9
AC
C
EP
f10
TE
D
f2
M
AN
U
Frequencies
[Hz]
36
ARTICLE IN PRESS
Table 2. The first ten frequencies for the functionally graded toro-parabolic panel (S-F-S-F) as a function of the
power-law exponent p , for a = 1, b = 0.5, c = 1 .
FGM 1 ( a =1/ b = 0.5 / c =1/ p ) power-law distribution
p = 0.6
p=5
p = 20
p = 50
p = 100
p=∞
70.69
75.74
84.57
105.03
133.14
150.65
153.20
161.66
164.70
172.14
71.10
75.93
85.04
105.92
134.08
152.21
154.73
163.45
165.55
174.42
69.96
74.80
84.12
105.11
132.92
149.02
151.56
160.26
163.98
171.28
69.14
74.00
83.38
104.36
131.89
146.65
149.25
157.86
162.69
168.86
67.94
72.85
82.30
103.19
130.28
143.21
145.94
154.36
160.73
165.33
p =1
72.54
72.08
71.79
77.77
77.31
77.01
f3
87.86
87.06
86.57
f4
110.17
108.93
108.18
f5
139.09
137.67
136.81
f6
152.89
152.32
151.89
f7
155.81
155.13
154.65
f8
164.79
163.98
163.42
f9
171.59
169.95
168.95
f10
176.50
175.32
174.56
FGM 2 ( a =1/ b = 0.5 / c =1/ p ) power-law distribution
f1
f2
RI
PT
p=0
SC
Frequencies
[Hz]
p=0
p = 0.6
p =1
p=5
p = 20
p = 50
p = 100
p=∞
f1
72.54
77.77
87.86
110.17
139.09
152.89
155.81
164.79
171.59
176.50
71.97
77.17
87.47
109.92
138.58
151.09
154.15
163.09
170.91
175.04
71.62
76.80
87.22
109.72
138.21
149.99
153.15
162.04
170.42
174.14
70.25
75.19
86.12
108.75
136.51
145.97
149.47
158.24
168.16
171.12
70.71
75.45
85.99
108.26
136.39
148.89
151.78
160.94
167.92
173.11
69.79
74.58
84.60
106.29
134.08
147.38
150.13
159.03
165.18
170.68
69.05
73.89
83.65
105.00
132.51
145.78
148.52
157.21
163.34
168.58
67.94
72.85
82.30
103.19
130.28
143.21
145.94
154.36
160.73
165.33
f3
f4
f5
f6
f7
f8
f9
AC
C
EP
f10
TE
D
f2
M
AN
U
Frequencies
[Hz]
37
ARTICLE IN PRESS
Table 3. The first ten frequencies for the functionally graded parabolic panel (C-F-F-F) as a function of the power-law
exponent p , for a = 1, b = 0.5, c = 1 .
FGM 1 ( a =1/ b = 0.5 / c =1/ p ) power-law distribution
p = 0.6
p=5
p = 20
p = 50
p = 100
p=∞
59.19
74.75
112.72
150.55
166.19
210.11
238.87
276.89
283.04
295.03
59.56
73.98
113.60
152.57
164.95
210.62
241.38
279.48
285.08
298.26
58.59
73.14
111.69
149.76
162.88
207.39
237.12
274.67
280.34
293.07
57.86
72.63
110.25
147.57
161.57
205.07
233.87
271.00
276.80
289.04
56.80
71.90
108.17
144.32
159.75
201.74
229.08
265.61
271.62
282.97
p =1
60.64
60.27
60.04
76.77
76.36
76.11
f3
115.49
114.77
114.31
f4
154.07
153.07
152.43
f5
170.55
169.65
169.08
f6
215.38
214.11
213.31
f7
244.57
243.02
242.03
f8
283.57
281.78
280.64
f9
289.98
288.19
287.04
f10
302.10
300.14
298.89
FGM 2 ( a =1/ b = 0.5 / c =1/ p ) power-law distribution
f1
f2
RI
PT
p=0
SC
Frequencies
[Hz]
p=0
p = 0.6
p =1
p=5
p = 20
p = 50
p = 100
p=∞
f1
60.64
76.77
115.49
154.07
170.55
215.38
244.57
283.57
289.98
302.10
60.15
76.23
114.54
152.72
169.33
213.69
242.48
281.19
287.60
299.50
59.84
75.91
113.95
151.89
168.58
212.64
241.19
279.72
286.13
297.89
58.71
74.27
111.84
149.23
164.94
208.45
236.82
274.59
280.78
292.63
59.25
73.68
113.01
151.69
164.10
209.49
239.98
277.93
283.54
296.72
58.43
72.99
111.40
149.33
162.46
206.83
236.44
273.90
279.59
292.31
57.78
72.55
110.10
147.34
161.35
204.78
233.50
270.59
276.40
288.62
56.80
71.90
108.17
144.32
159.75
201.74
229.08
265.61
271.62
282.97
f3
f4
f5
f6
f7
f8
f9
AC
C
EP
f10
TE
D
f2
M
AN
U
Frequencies
[Hz]
38
ARTICLE IN PRESS
Table 4. The first ten frequencies for the functionally graded parabolic panel (C-F-F-F) as a function of the power-law
exponent p , for a = 1, b = 0.5, c = 3 .
FGM 1 ( a =1/ b =1/ c =3 / p ) power-law distribution
p = 0.6
p=5
p = 20
p = 50
p = 100
p=∞
62.43
76.22
119.29
160.90
170.51
219.93
253.90
293.66
298.94
313.13
60.07
74.07
114.67
154.31
165.34
212.07
243.83
282.21
287.60
301.21
58.40
72.96
111.33
149.24
162.44
206.75
236.32
273.76
279.44
292.11
57.66
72.48
109.86
146.97
161.19
204.43
232.97
270.00
275.83
287.93
56.80
71.90
108.17
144.32
159.75
201.74
229.08
265.61
271.62
282.97
p =1
60.64
61.18
61.48
76.77
76.86
76.87
f3
115.49
116.58
117.20
f4
154.07
156.00
157.06
f5
170.55
170.94
171.11
f6
215.38
216.88
217.69
f7
244.57
247.25
248.73
f8
283.57
286.52
288.15
f9
289.98
292.68
294.17
f10
302.10
305.58
307.46
FGM 2 ( a =1/ b =1/ c =3 / p ) power-law distribution
f1
f2
RI
PT
p=0
SC
Frequencies
[Hz]
p=0
p = 0.6
p =1
p=5
p = 20
p = 50
p = 100
p=∞
f1
60.64
76.77
115.49
154.07
170.55
215.38
244.57
283.57
289.98
302.10
61.16
76.83
116.54
155.93
170.88
216.79
247.14
286.40
292.57
305.45
61.44
76.83
117.12
156.95
171.01
217.56
248.56
287.96
293.98
307.26
62.31
76.11
119.07
160.57
170.20
219.51
253.39
293.09
298.38
312.59
59.99
74.00
114.52
154.08
165.12
211.77
243.47
281.81
287.21
300.82
58.36
72.93
111.26
149.13
162.33
206.61
236.14
273.56
279.25
291.92
57.64
72.46
109.83
146.91
161.13
204.35
232.88
269.89
275.72
287.83
56.80
71.90
108.17
144.32
159.75
201.74
229.08
265.61
271.62
282.97
f3
f4
f5
f6
f7
f8
f9
AC
C
EP
f10
TE
D
f2
M
AN
U
Frequencies
[Hz]
39
ARTICLE IN PRESS
Table 5. The first ten frequencies for the functionally graded parabolic toroid (F-C) as a function of the power-law
exponent p , for a = 0, b = −0.5, c = 2 .
FGM 1 ( a = 0 / b =−0.5 / c = 2 / p ) power-law distribution
Frequencies
[Hz]
p=5
p = 20
p = 50
p = 100
p=∞
52.59
52.37
52.24
52.59
52.37
52.24
f3
59.27
58.97
58.79
f4
59.27
58.97
58.79
f5
59.69
59.46
59.32
f6
59.69
59.46
59.32
f7
74.38
74.04
73.83
f8
74.76
74.29
73.99
f9
74.77
74.29
74.00
f10
96.51
95.78
95.33
FGM 2 ( a = 0 / b =−0.5 / c = 2 / p ) power-law distribution
51.47
51.47
57.79
57.79
58.47
58.47
72.53
72.53
72.68
93.20
51.16
51.16
57.75
57.75
58.08
58.08
72.61
73.01
73.01
94.42
50.99
50.99
57.75
57.75
57.85
57.85
72.53
73.36
73.36
95.26
50.77
50.77
57.56
57.56
57.58
57.58
72.24
73.21
73.22
95.17
49.26
49.26
55.52
55.52
55.91
55.91
69.66
70.03
70.03
90.39
Frequencies
[Hz]
p=0
p = 0.6
p =1
p=5
p = 20
p = 50
p = 100
p=∞
f1
52.59
52.59
59.27
59.27
59.69
59.69
74.38
74.76
74.77
96.51
52.28
52.28
58.86
58.86
59.34
59.34
73.84
74.15
74.15
95.60
52.09
52.09
58.62
58.62
59.12
59.12
73.52
73.78
73.78
95.06
51.05
51.05
57.34
57.34
57.94
57.94
71.83
71.95
71.95
92.47
50.70
50.70
57.24
57.24
57.50
57.50
71.65
72.38
72.38
93.62
50.61
50.61
57.34
57.34
57.37
57.37
71.74
72.85
72.85
94.60
50.47
50.47
57.20
57.20
57.24
57.24
71.61
72.80
72.81
94.65
49.26
49.26
55.52
55.52
55.91
55.91
69.67
70.03
70.04
90.40
f1
f3
f4
f5
f6
f7
f8
f9
AC
C
EP
f10
TE
D
f2
M
AN
U
f2
SC
p = 0.6
RI
PT
p =1
p=0
40
ARTICLE IN PRESS
Table 6. The first ten frequencies for the functionally graded parabolic dome (C-F) as a function of the power-law
exponent p , for a = 0.8, b = 0.2, c = 3 .
FGM 1 ( a = 0.8 / b = 0.2 / c = 3 / p ) power-law distribution
p = 0.6
p=5
p = 20
p = 50
p = 100
p=∞
122.15
122.15
165.63
165.63
175.08
175.08
201.30
216.48
216.49
219.28
120.81
120.81
162.83
162.83
173.28
173.28
198.19
214.70
214.71
216.40
119.14
119.14
161.72
161.72
170.80
170.80
196.43
211.17
211.18
213.97
118.29
118.29
161.25
161.25
169.52
169.52
195.62
209.32
209.33
212.77
117.23
117.23
160.72
160.72
167.93
167.93
194.66
206.99
206.99
211.32
p =1
125.16
124.31
123.78
125.16
124.31
123.78
f3
171.58
170.40
169.66
f4
171.58
170.40
169.66
f5
179.28
178.06
177.29
f6
179.28
178.06
177.29
f7
207.82
206.41
205.53
f8
220.98
219.46
218.50
f9
220.99
219.47
218.51
f10
225.61
224.07
223.10
FGM 2 ( a = 0.8 / b = 0.2 / c =3 / p ) power-law distribution
f1
f2
RI
PT
p=0
SC
Frequencies
[Hz]
p=0
p = 0.6
p =1
p=5
p = 20
p = 50
p = 100
p=∞
f1
125.16
125.16
171.58
171.58
179.28
179.28
207.82
220.98
220.99
225.61
123.96
123.96
170.07
170.07
177.58
177.58
205.92
218.86
218.87
223.52
123.25
123.25
169.16
169.16
176.55
176.55
204.79
217.58
217.59
222.27
121.17
121.17
164.75
164.75
173.72
173.72
199.93
214.77
214.78
217.74
120.34
120.34
162.41
162.41
172.62
172.62
197.53
213.86
213.87
215.66
118.93
118.93
161.52
161.52
170.49
170.49
196.13
210.79
210.80
213.63
118.18
118.18
161.15
161.15
169.36
169.36
195.46
209.12
209.13
212.60
117.23
117.23
160.72
160.72
167.93
167.93
194.66
206.99
206.99
211.32
f3
f4
f5
f6
f7
f8
f9
f10
TE
D
f2
M
AN
U
Frequencies
[Hz]
EP
Table 7. The first ten frequencies for the functionally graded toro-parabolic panel (C-F-C-F) for an increasing the
number of grid points N = M of the Chebyshev-Gauss-Lobatto distribution.
FGM 1 ( a =1/ b = 0 / c = 0 / p = 5) power-law distribution
Frequencies
[Hz]
N = M =11
N = M =15
N = M =17
N = M = 21
N = M = 25
N = M = 29
N = M = 31
f1
98.12
101.71
109.11
129.04
154.10
181.86
188.10
191.16
198.51
203.82
99.18
102.70
110.16
128.96
155.43
181.65
189.01
189.96
199.81
204.87
99.17
102.71
110.18
128.92
155.50
181.60
189.05
189.92
199.85
204.91
99.14
102.71
110.20
128.87
155.55
181.53
189.07
189.88
199.86
204.91
99.13
102.71
110.20
128.84
155.56
181.50
189.07
189.86
199.86
204.92
99.12
102.72
110.20
128.83
155.56
181.49
189.07
189.85
199.86
204.92
99.12
102.72
110.20
128.83
155.56
181.49
189.07
189.85
199.86
204.92
AC
C
f2
f3
f4
f5
f6
f7
f8
f9
f10
41