FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: A 2-D GDQ solution for free vibrations

International Journal of Mechanical Sciences 53 (2011) 446–470

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FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: A 2-D GDQ solution for free vibrations Francesco Tornabene n, Alfredo Liverani, Gianni Caligiana DIEM—Department, Faculty of Engineering, viale Risorgimento 2, University of Bologna, 40100 Bologna, Italy

a r t i c l e i n f o

a b s t r a c t

Article history: Received 15 February 2011 Received in revised form 24 March 2011 Accepted 29 March 2011 Available online 9 April 2011

In this paper, the generalized differential quadrature (GDQ) method is applied to study the dynamic behavior of functionally graded materials (FGMs) and laminated doubly curved shells and panels of revolution with a free-form meridian. The First-order Shear Deformation Theory (FSDT) is used to analyze the above mentioned moderately thick structural elements. In order to include the effect of the initial curvature a generalization of the Reissner–Mindlin theory, proposed by Toorani and Lakis, is adopted. The governing equations of motion, written in terms of stress resultants, are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. Simple Rational Be´zier curves are used to define the meridian curve of the revolution structures. Firstly, the differential quadrature (DQ) rule is introduced to determine the geometric parameters of the structures with a free-form meridian. Secondly, the discretization of the system by means of the GDQ technique leads to a standard linear eigenvalue problem, where two independent variables are involved. Results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Comparisons between the Reissner–Mindlin and the Toorani–Lakis theory are presented. Furthermore, GDQ results are compared with those obtained by using commercial programs such as Abaqus, Ansys, Nastran, Straus and Pro/Mechanica. Very good agreement is observed. Finally, different lamination schemes are considered to expand the combination of the two functionally graded four-parameter power-law distributions adopted. The treatment is developed within the theory of linear elasticity, when materials are assumed to be isotropic and inhomogeneous through the lamina thickness direction. A twoconstituent functionally graded lamina consists of ceramic and metal those are graded through the lamina thickness. A parametric study is performed to illustrate the influence of the parameters on the mechanical behavior of shell and panel structures considered. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Free vibrations Doubly curved shells of revolution Rational Be´zier curves Laminated composite shells Functionally graded materials Generalized differential quadrature method

1. Introduction Shells have been widespread in many fields of engineering as they give rise to optimum conditions for dynamic behavior, strength and stability. The vibration effects on these structures caused by different phenomena can have serious consequences for their strength and safety. Therefore, an accurate frequency and mode shape determination is of considerable importance for the technical design of these structural elements. The aim of this paper is to study the dynamic behavior of doubly curved shell structures derived from shells of revolution, which are very common structural elements.

n

Corresponding author. E-mail address: [email protected] (F. Tornabene).

0020-7403/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2011.03.007

This research work is based on four aspects. The first is the improvement of the Reissner–Mindlin theory using the Toorani– Lakis theory. In this way the effect of the curvature of the shell structure is considered. The second is the generalization of the shape of the shell meridian. The free-form (Rational Be´zier or NURBS) meridian curve assumption requires the differential quadrature rule to evaluate the geometric parameters needed to describe the geometry of the structure. The third is the combination of the composite lamination scheme with functionally graded materials in order to expand the design profiles through the whole thickness of the shell. The four is the use of the generalized differential quadrature method to solve the governing equations of motion. During the last 60 years, two-dimensional linear theories of thin shells have been developed including important contribu¨ tions by Timoshenko and Woinowsky-Krieger [1], Flugge [2], Gol’Denveizer [3], Novozhilov [4], Vlasov [5], Ambartusumyan

F. Tornabene et al. / International Journal of Mechanical Sciences 53 (2011) 446–470

[6], Kraus [7], Leissa [8,9], Markuˇs [10], Ventsel and Krauthammer [11] and Soedel [12]. All these contributions are based on the Kirchhoff–Love assumptions. This theory, named Classical Shell Theory (CST), assumes that normals to the shell middle-surface remain straight and normal to it during deformations and unstretched in length. Many researchers analyzed various characteristics of thin shell structures [13–18]. When the theories of thin shells are applied to thick shells, the errors could be quite large. With the increasing use of thick shells in various engineering applications, simple and accurate theories for thick shells have been developed. With respect to thin shells, thick shell theories take the transverse shear deformation and rotary inertia into account. The transverse shear deformation has been incorporated into shell theories by following the theory of Reissner–Mindlin [19], also named First-order Shear Deformation Theory (FSDT). Abandoning the assumption on the preservation of the normals to the shell middle surface after the deformation, a comprehensive analysis for elastic isotropic shells was made by Kraus [7] and Gould [20,21]. The present work is just based on the FSDT. In order to include the effect of the initial curvature a generalization of the Reissner–Mindlin theory (RMT) has been proposed by Toorani and Lakis [22]. In this way the Reissner– Mindlin theory becomes a particular case of the Toorani–Lakis theory (TLT). As a consequence of the use of this general theory the stress resultants directly depend on the geometry of the structure in terms of the curvature coefficients and the hypothesis of the symmetry of the in-plane shearing force resultants and the torsional couples declines. In this paper the Toorani–Lakis theory is considered and improved. Comparisons between the Reissner– Mindlin and the Toorani–Lakis theory are presented. No results are available in the literature about the Toorani–Lakis theory for doubly curved shells. As for the vibration analysis of such revolution shells, several studies have been presented earlier. The most popular numerical tool in carrying out the above analyses is currently the finite element method [20,21,23]. 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-ordinates [24–30]. In other word, when dealing with a completely closed shell, the 2D problem can be reduced using standard Fourier decomposition. For a panel, however, it is not possible to perform such a reduction operation, and the two-dimensional field must be dealt with directly, as will just be done in this paper. The governing equations of motion are a set of five partial differential equations with variable coefficients, depending on two independent variables. By doing so, 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 literature available for these latter structures, compared to the literature regarding the free vibration analysis of complete shells of revolution. Complete revolution shells are obtained as special cases of shell panels by satisfying the kinematical and physical compatibility at the common meridian with W ¼0,2p. The excellent mathematical and algorithmic properties, combined with successful industrial applications, have contributed to the enormous popularity of the Rational Be´zier and Non-Uniform Rational B-Splines (NURBS) curves. These curves have become the de facto industry standard for the representation, design and data exchange of geometric information processed by computers [31–34]. Many national and international standards recognize these curves as a powerful tool for geometric design. Furthermore, these curves allow to generalize the shape of the meridian and can be used for the optimization of the structure itself. In fact,

447

by changing the control polygon it is possible to easy modify the shape and then to improve the mechanical behavior of the shell structure. By introducing the differential quadrature rule [35] and the simple mathematical formulation of the Rational Be´zier and NURBS curves [31,32] it is possible to numerically evaluate the geometric parameters of a free-form shell of revolution. For a sake of simplicity and without losing generality, only Rational Be´zier curves are used in this work. Functionally graded materials (FGMs) 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 composites. Typically, these materials consist of a mixture of ceramic and metal, or a combination of different materials. One of the advantages of using functionally graded materials is that they can survive environments with high temperature gradients, while maintaining structural 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. In this study, ceramic–metal graded shells of revolution with two different power-law variations of the volume fraction of the constituents in the thickness direction are considered. The effect of the power-law exponent and of the power-law distribution choice on the mechanical behavior of functionally graded shells and panels is investigated. In the last years, some researchers have analyzed various characteristics of functionally graded structures [36–51]. However, this paper is motivated by the lack of studies in the technical literature concerning to the free vibration analysis of functionally graded shells and panels and the effect of the power-law distribution choice on their mechanical behavior. The aim is to analyze the influence of constituent volume fractions and the effects of constituent material profiles on the natural frequencies. Concerning this purpose, two different four-parameter power-law distributions, proposed by Tornabene [51] are considered for the ceramic volume fraction. Various material profiles through the functionally graded lamina thickness are used by varying the four parameters of power-law distributions. Classical volume fracture profiles can be obtained as special cases of the general distribution laws. Furthermore, the homogeneous isotropic material can be inferred as a special case of functionally graded materials. Differently from previous work [51] the lamination scheme of laminated composite shell allows to expand the combination of the two functionally graded fourparameter power-law distributions. New profiles are presented and investigated. A parametric study is undertaken, giving insight into the effects of the material composition on the natural frequencies of doubly curved shell structures. Vibration characteristics are illustrated by varying one parameter at a time as a function of the power-law exponent. In the GDQ method the governing differential equations of motion are directly transformed in one step to obtain the final algebraic form. The interest of researches in this procedure is increasing due to its great simplicity and versatility. As shown in the literature [52], GDQ technique is a global method, which can obtain very accurate numerical results by using a considerably small number of grid points. Therefore, this simple direct procedure has been applied in a large number of cases [53–76] to circumvent the difficulties of programming complex algorithms for the computer, as well as the excessive use of storage and computer time. In conclusion, the aim of the present paper is to demonstrate an efficient and accurate application of the differential quadrature approach, by solving the equations of motion governing the free vibration of functionally graded and laminated

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F. Tornabene et al. / International Journal of Mechanical Sciences 53 (2011) 446–470

composite doubly curved moderately thick shells and panels of revolution with a free-form meridian, taking two independent coordinates into account.

Furthermore, the classical nth degree Bernstein polynomials are given by n! ui ð1uÞni i!ðniÞ!

Bi,n ðuÞ ¼ 2. Geometry description and shell fundamental systems The basic configuration of the problem considered here is a laminated composite doubly curved shell as shown in Fig. 1. The co-ordinates along the meridional and circumferential directions of the reference surface are j and s, respectively. The distance of each point from the shell mid-surface along the normal is z. Consider a laminated composite shell made of l laminae or plies, where the total thickness of the shell h is defined as h¼

l X

hk

ð1Þ

k¼1

in which hk ¼ zk þ 1 zk is the thickness of the kth lamina or ply. In this work, doubly curved shells of revolution with a freeform meridian curve are considered. The angle formed by the extended normal n to the reference surface and the axis of rotation x3, or the geometric axis x03 of the meridian curve, is defined as the meridional angle j and the angle between the radius of the parallel circle R0(j) and the x1-axis is designated as the circumferential angle W as shown in Fig. 2. For these structures the parametric co-ordinates (j,s) define, respectively, the meridional curves and the parallel circles upon the middle surface of the shell. The curvilinear abscissa s(j) of a generic parallel is related to the circumferential angle W by the relation s ¼ WR0. The horizontal radius R0(j) of a generic parallel of the shell represents the distance of each point from the axis of revolution x3. Rb is the shift of the geometric axis of the curved meridian x03 with reference to the axis of revolution x3. The position of an arbitrary point within the shell material is defined by co-ordinates j ðj0 r j r j1 Þ, sð0 rs r s0 Þ upon the middle surface, and z directed along the outward normal and measured from the reference surface ðh=2r z rh=2Þ. The geometry of shells considered is a surface of revolution with a free-form meridian (Fig. 2). A simple way to define a general meridian curve is to use the well-known Rational Be´zier representation of a plane curve [31,32]. In particular, it is possible to describe a Rational Be´zier curve in the following manner: Pn Pn 0 0 _ _ i ¼ 0 Bi,n ðuÞwi x1i i ¼ 0 Bi,n ðuÞwi x3i x 1 ðuÞ ¼ P , x ðuÞ ¼ P ð2Þ n n 3 i ¼ 0 Bi,n ðuÞwi i ¼ 0 Bi,n ðuÞwi

ð3Þ

0 x 3i Þ, In this way, only the co-ordinates of the curve ð_ x 1i ,_ i ¼ 1,2,. . .,m, are known in the co-ordinate system O0 x1 x03 . In a more general case, it is possible to suppose that the curve is described by a sufficient number of points. In order to solve the shell problem, it is important to express the horizontal radius R0(j) of a generic parallel and the radii of curvature Rj(j), RW(j) in the meridional and circumferential directions as functions of j. On the basis of differential geometry [59], the radius of curvature of the meridian curve can be described using the following expression as a function of x03 :

Rj ðx03 Þ ¼

ð1 þðdx1 =dx03 Þ2 Þ3=2 9ðd2 x1 =dx02 3 Þ9

ð4Þ

The derivatives of the curve are not known a priori, so there is need a numeric method to evaluate the first and second derivatives of the curve. The differential quadrature rule allows to approximate these derivatives using the following definition [35]:   N X dn f ðxÞ  ¼ BðnÞ f ðxj Þ, i ¼ 1,2,. . .,N ð5Þ ij dxn  j ¼ 1 x¼x i

R 0 (ϕ )

Rb

t2 = ts

O′

O

s

n

R0 (ϕ )

ϑ O

t1 = t ϕ

ϕ dϕ

x2

Rϑ

x′3

x3

where u A ½0,1 is the curve parameter, wi are the weights and ðx1i ,x03i Þ are the co-ordinates of the curve control points.

Fig. 2. Shell geometry: (a) meridional section and (b) circumferential section.

 ζ l k

 l+1 l s

x1

Rϕ

C1

C2

n

x1



k+1

1

2

k 1

hk = k − k+1

Fig. 1. Co-ordinate system of a laminated composite doubly curved shell.

2

h 2 h 2

F. Tornabene et al. / International Journal of Mechanical Sciences 53 (2011) 446–470

449

where BðijnÞ are the weighting coefficients of the nth order derivative. 0 0 By discretizing the domain I A ½_ x 31 ,_ x 3m  using the Chebyshev– Gauss–Lobatto (C–G–L) grid distribution:    _0 _0 i1 ðx 3m x 31 Þ _0 0 _ p x 3i ¼ 1cos þ x 31 , N1 2

i ¼ 1,2,. . .,N

 0 _0  0 for x^ 3 A _ x 31 , x 3m

ð6Þ and interpolating the _ x 1 co-ordinates of the curve points derived by Eq. (2) using the previous calculated points (6), the general curve can be represented by the following new co-ordinates 0 points ðx^ 1i , x^ 3i Þ, for i ¼ 1,2,. . .,N. Applying the differential quadrature definition (5), expression (4) assumes the following discrete aspect:  P 2 3=2 0 x^ 3 ð1Þ N 1þ x^ 1i j ¼ 1 Bij 0 P  Rj ðx^ 3i Þ ¼ , i ¼ 1,2,. . .,N ð7Þ 0  N  x^ ð2Þ  j ¼ 1 Bij3 x^ 1i  0 x^ ðnÞ

are the weighting coefficients evaluated in the where Bij3 0 0 domain I A ½_ x 31 ,_ x 3m . As a result of the differential geometry [59], it is possible to introduce the following expression:   p dx1 j ¼ arctan ð8Þ 0 2 dx3 By using the differential quadrature definition (5), relation (8) can be expressed in discrete form: 0 1 N X p ^ 03 ð1Þ x 0 ð9Þ j^ i ¼ j^ ðx^ 3i Þ ¼ arctan@ Bij x^ 1i A, i ¼ 1,2,. . .,N 2 j¼1 ^ 1,j ^ N  using the Chebyshev– By discretizing the domain Ij A ½j Gauss–Lobatto (C–G–L) grid distribution:

   j^ N j^ 1 i1 ^ 1 , i ¼ 1,2,. . .,N þj ji ¼ 1cos p N1 2 ^ 1,j ^ N for j A ½j

ð10Þ 0 x^ 3

co-ordinates of the curve points and interpolating the x^ 1 and using the calculated points (10), the general curve can be represented by the following new co-ordinates points ðx~ 1i , x~ 03i Þ, for i ¼ 1,2,. . .,N. Thus, all the discrete points of the curve are determined in terms of the co-ordinates ðx~ 1i , x~ 03i Þ and the angle ji. For all the numerical interpolations considered above the interp1 function of MATLAB program has been used. In Fig. 3 a Rational Be´zier curve, its control points and the curve co-ordinates ðx~ 1i , x~ 03i Þ, evaluated as above exposed, are represented. The vectors of the control points and the weights used in Fig. 3 are the following:   x1 ¼ 0:2 0:7 1:2 1:4 1:4 1:2   x03 ¼ 0 0:2 0:6 1 1:5 2 w ¼ ½1 1 1 1 1 1

ð11Þ

Based on the previous considerations, the horizontal radius R0(j) of a shells of revolution assumes the following discrete form: R0i ¼ R0 ðji Þ ¼ x~ 1i þRb ,

i ¼ 1,2,. . .,N

ð12Þ

For doubly curved shells of revolution the Gauss–Codazzi relation can be expressed as follows: dR0 ¼ Rj cos j dj

ð13Þ

By using the differential quadrature definition (5), it is possible to determine the radius of curvature Rj(j) in meridional

0 0 Fig. 3. A rational Be´zier curve ð_ x 1i ,_ x 3i Þ, its control points ðx1i ,x03i Þ and curve discrete points evaluated ðx~ 1i , x~ 03i Þ.

direction and its first derivative in discrete form: N 1 X Bjð1Þ R0i , cos ji j ¼ 1 ij

Rji ¼ Rj ðji Þ ¼

    N X dRj  dRj  ¼ ¼ Bijjð1Þ Rji ,   dj  dj  j¼1 i

i ¼ 1,2,. . .,N

i ¼ 1,2,. . .,N

ð14Þ

ð15Þ

ji

Finally, as a results of the differential geometry [59], the radius of curvature RW(j) in circumferential direction for a shell of revolution can be expressed as follows in discrete form: RWi ¼ RW ðji Þ ¼

R0i sin ji

ð16Þ

Following the previous considerations, all the useful geometric parameters describing the surface of revolution under consideration are known in discrete form. As shown, the differential quadrature rule (5) has been used to approximate the derivatives needed for the definition of the geometry of a free-form meridian shell of revolution. As concerns the shell theory, the present work is based on the following assumptions: (1) the transverse normal is inextensible so that the normal strain is equal to zero: en ¼ en ðj,s, z,tÞ ¼ 0; (2) the transverse shear deformation is considered 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, therefore it is possible to assume that the thickness-direction normal stress is negligible so that the plane assumption can be invoked: sn ¼ sn ðj,s, z,tÞ ¼ 0; (5) the linear elastic behavior of anisotropic materials is assumed and (6) the rotary inertia and the initial curvature are also taken into account. Consistent with the assumptions of a moderately thick shell theory reported above, the displacement field considered in this study is that of the First-order Shear Deformation Theory and can be put in the following form: Uj ðj,s, z,tÞ ¼ uj ðj,s,tÞ þ zbj ðj,s,tÞ Us ðj,s, z,tÞ ¼ us ðj,s,tÞ þ zbs ðj,s,tÞ Wðj,s, z,tÞ ¼ wðj,s,tÞ

ð17Þ

450

F. Tornabene et al. / International Journal of Mechanical Sciences 53 (2011) 446–470

where uj , us , w are the displacement components of points lying on the middle surface (z ¼0) of the shell, along meridional, circumferential and normal directions, respectively, while t is the time variable. bj and bs are normal-to-mid-surface rotations, respectively. The kinematic hypothesis expressed by Eq. (17) should be supplemented by the statement that the shell deflections are small and strains are infinitesimal, that is wðj,s,tÞ 5h. In-plane displacements Uj and Us vary linearly through the thickness, while W remains independent of z. Relationships between strains and displacements along the shell reference surface (z ¼0) are as follows:   1 @uj @us uj cos j w sin j 1 @us þ e0j ¼ þ w , e0s ¼ þ , g0j ¼ , Rj @j Rj @j @s R0 R0

@bj



In

3

the

02

s

ϕˆ

above

Fig. 4. A lamina with material and laminate co-ordinate systems.

@w us sin j  gsn ¼ þ bs @s R0

relations

(18),

the

first

ð18Þ four

strains

transverse shearing strains. The shell material assumed in the following is a laminated composite linear elastic one. Accordingly, the constitutive equations relate internal stress resultants and internal couples with generalized strain components (18) on the middle surface and can be written in compact form: 3

2

3

2

A

B

0

B0

D0

0

D0

E0

6 7 B6 4 M 5 ¼ @4 B T 0

D 0

6 7 0 5 þ a1 4 D0 0 C

E0 0

6 7 0 5 þ a2 4 E0 C0 0

F0 0

N



θ

1 @bs oj ¼ , R j @j

e0 ¼ ½e0j e0s g0j g0s T are the in-plane meridional, circumferential and shearing components, and v0 ¼ ½wj ws oj os T are the analogous curvature changes. The last two components c0 ¼ ½gjn gsn T are the

2

sˆ

bs cos j

R0   1 @w gjn ¼ uj þ bj , Rj @j @s

O′

ϕ

@uj us cos j  @s R0 1 @bj @bs bj cos j þ wj ¼ ,w ¼ , Rj @j s @s R0

g0s ¼

os ¼

 ≡ ˆ

2

B00 6 þb1 4 D00

D00 E00

0

0

2 00 3 0 D 7 0 5 þ b2 6 4 E00 C00 0

E00 F00 0

2 00 3 E 0 7 00 0 5 þ b3 6 4F 00 0 G

3

2

E0 7 6 F0 0 5 þ a3 4 0 G’ 0

F00 H00 0

0

3 0 07 5

F0 H0

J0

0

312 e0 3

6 07 07 5A4 v 5 J00 c0 ð19Þ

The extended notation of relations (19) can be found in the article by Tornabene [76], in which all the matrices above introduced are explicitly defined. Furthermore, the shear correction factor k is usually taken as k ¼5/6, such as in the present work. In particular, the determination of shear correction factors for composite laminated structures is still an unresolved issue, because these factors depend on various parameters [77–79]. In Eq. (19), the four components N ¼ ½Nj Ns Njs Nsj T are the in-plane meridional, circumferential and shearing force resultants, and M ¼ ½Mj Ms Mjs Msj T are the analogous couples, while T ¼ ½Tj Ts T are the transverse shear force resultants. In the above definitions (19) the symmetry of shearing force resultants Njs ,Nsj and torsional couples Mjs ,Msj is not assumed as a further hypothesis, as done in the Reissner–Mindlin theory. This hypothesis is satisfied only in the case of spherical shells and flat plates. This assumption is derived from the consideration that ratios z=Rj , z=Rs cannot be neglected with respect to unity. Thus, the curvature coefficients are introduced and determined as follows: sin j 1 1 1  ,a2 ¼  a1 ,a3 ¼ 2 a1 a1 ¼ Rj Rj R0 Rj sin j sin j a1 ,b3 ¼  a1 R0 R20

constants QijðkÞ ¼ QijðkÞ ðzÞ [51,75,76] in the material co-ordinate ^ s^ z^ (Fig. 4) are functions of thickness coordinate system O0 j zðz A ½zk , zk þ 1 Þ and are defined as ðkÞ Q11 ðzÞ ¼

1n

ð20Þ

EðkÞ 1 ðzÞ , ðkÞ ðkÞ 12 ðzÞ 21 ðzÞ

ðkÞ ðzÞ ¼ GðkÞ Q66 12 ðzÞ,

n

ðkÞ Q22 ðzÞ ¼

ðkÞ Q44 ðzÞ ¼ GðkÞ 13 ðzÞ,

1n

E2ðkÞ ðzÞ , ðkÞ ðkÞ 12 ðzÞ 21 ðzÞ

n

ðkÞ Q12 ðzÞ ¼

ðkÞ n12 ðzÞEðkÞ 2 ðzÞ ðkÞ 1n12 ðzÞnðkÞ 21 ðzÞ

ðkÞ Q55 ðzÞ ¼ GðkÞ 23 ðzÞ

ð21Þ

where the following relations have to be introduced: ðkÞ ðkÞ ðkÞ EðkÞ 1 ðzÞ ¼ E2 ðzÞ ¼ E3 ðzÞ ¼ E ðzÞ ðkÞ ðkÞ ðkÞ ðkÞ nðkÞ 12 ðzÞ ¼ n21 ðzÞ ¼ n13 ðzÞ ¼ n23 ðzÞ ¼ n ðzÞ ðkÞ ðkÞ ðkÞ G12 ðzÞ ¼ GðkÞ 13 ðzÞ ¼ G23 ðzÞ ¼ G ðzÞ

ð22Þ

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 ceramic and metal constituents. The material properties of the functionally graded lamina vary continuously and smoothly in the thickness direction z and are functions of volume fractions of constituent materials. Young’s modulus EðkÞ ðzÞ, Poisson’s ratio nðkÞ ðzÞ and mass density rðkÞ ðzÞ of the functionally graded lamina can be expressed as a linear combination of the volume fraction: ðkÞ ðkÞ ðkÞ rðkÞ ðzÞ ¼ ðrðkÞ C rM ÞVC ðzÞ þ rM ðkÞ ðkÞ ðkÞ EðkÞ ðzÞ ¼ ðEðkÞ C EM ÞVC ðzÞ þ EM ðkÞ ðkÞ ðkÞ nðkÞ ðzÞ ¼ ðnðkÞ C nM ÞVC ðzÞ þ nM

VCðkÞ

2

b1 ¼ a1 ,b2 ¼

The curvature coefficients a3 and b3 are different from those proposed by Toorani and Lakis [22]. This is due to the fact that in the work [22] a term has been forgotten in the expansion and the subsequent approximations of the curvature coefficients a3, b3. In this way the symmetry of shearing force resultants Njs ,Nsj and torsional couples Mjs ,Msj is satisfied and guaranteed for spherical shells, as previously highlighted. Thus, the Toorani–Lakis and the Reissner–Mindlin theory coincides in the case of spherical shells as well as in the case of circular and rectangular plates, due to the fact that all the curvature coefficients are equal to zero. For the functionally graded material kth lamina the elastic

ð23Þ

where is the volume fraction of the ceramic constituent ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ material, while rðkÞ C ,EC , nC and rM ,EM , nM represent mass density, Young’s modulus and Poisson’s ratio of the ceramic and metal constituent materials of the kth lamina, respectively.

F. Tornabene et al. / International Journal of Mechanical Sciences 53 (2011) 446–470

In this paper, the ceramic volume fraction VCðkÞ ðzÞ follows two simple four-parameter power-law distributions [51]: FGM1ðaðkÞ =bðkÞ =cðkÞ =pðkÞ Þ : VCðkÞ ðzÞ ¼

1aðkÞ

FGM2ðaðkÞ =bðkÞ =cðkÞ =pðkÞ Þ : VCðkÞ ðzÞ ¼

1aðkÞ



z

hk



zk hk



zk þ 1 hk





þ bðkÞ

z hk





z



hk

þ bðkÞ

zk

cðkÞ !pðkÞ

hk



zk þ 1 hk



z

cðkÞ !pðkÞ

hk

ð24Þ where the volume fraction index p ð0 r p r 1Þ and the parameters aðkÞ ,bðkÞ ,cðkÞ dictate the material variation profile through the functionally graded lamina thickness. By using the lamination scheme in combination with the two four-parameter power-law distributions it is possible to consider a simple composite shell constituted by two or three laminae: for the two laminae shell each lamina is a FGM lamina with a different power-law distribution (Fig. 5), while for the three laminae the middle is a homogeneous isotropic elastic one and the bottom and the top laminae are FGM laminae with different power-law distributions (Fig. 6). Thus, using the laminated ðkÞ

ðkÞ

451

composite material scheme a further generalization of functionally graded material profiles is introduced as represented in Figs. 5 and 6. Fig. 5 represents a possible material profile through the functionally graded shell thickness obtained with two FGM laminae, while Fig. 6 illustrates a possible material profile obtained considering a three laminae shell. In particular, Fig. 6(a), (b) and (d) present a middle lamina constituted by one of the two constituents of functionally graded material and this middle lamina is indicated with the symbol FGMC (ceramic isotropic material) or FGMM (metal isotropic material). Otherwise, Fig. 6(c) shows a middle lamina constituted by a mixture of the two constituents and then the symbol used to indicate the middle lamina is FGMCM (isotropic material obtained by a mixture of two constituents). For the sake of simplicity, the symbol FGMCM is indifferently used to indicate the three cases mentioned above (FGMC, FGMM and FGMCM), because it represents the most general case. The possible combinations are wide and the unique attention is to design a continuum profile through the thickness in order to avoid stress concentrations and geometric discontinuities. Furthermore, symmetric and asymmetric profiles can be considered.

Fig. 5. Variations of the ceramic volume fraction Vc through a two laminae thickness for different values of the power-law index p ¼ pð1Þ ¼ pð2Þ : (a) FGM1ðað1Þ ¼ 1=bð1Þ ¼ 0=cð1Þ =pð1Þ Þ = FGM2ðað2Þ ¼ 1=bð2Þ ¼ 0=cð2Þ =pð2Þ Þ and (b) FGM2ðað1Þ ¼ 1=bð1Þ ¼ 0=cð1Þ =pð1Þ Þ =FGM1ðað2Þ ¼ 1=bð2Þ ¼ 0=cð2Þ =pð2Þ Þ .

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Fig. 6. Variations of the ceramic volume fraction Vc through a three laminae thickness for different values of the power-law index p ¼ pð1Þ ¼ pð3Þ : (a) FGM1ðað1Þ ¼ 1=bð1Þ ¼ 0=cð1Þ =pð1Þ Þ=FGMM =FGM2ðað3Þ ¼ 1=bð3Þ ¼ 0=cð3Þ =pð3Þ Þ , (b) FGM2ðað1Þ ¼ 1=bð1Þ ¼ 0=cð1Þ =pð1Þ Þ =FGMC =FGM1ðað3Þ ¼ 1=bð3Þ ¼ 0=cð3Þ =pð3Þ Þ , (c) FGM1ðað1Þ ¼ 1=bð1Þ ¼ 0:5=cð1Þ ¼ 2=pð1Þ Þ =FGMCM =FGM2ðað3Þ ¼ 1=bð3Þ ¼ 0:5=cð3Þ ¼ 2=pð3Þ Þ and (d) FGM2ðað1Þ ¼ 1=bð1Þ ¼ 0:5=cð1Þ ¼ 2=pð1Þ Þ =FGMC =FGM1ðað3Þ ¼ 1=bð3Þ ¼ 0:5=cð3Þ ¼ 2=pð3Þ Þ .

Following the direct approach or the virtual work principle in dynamic version and remembering the Gauss–Codazzi relations for the shells of revolution (13), five equations of motion in terms of internal actions can be written for the revolution shell element: 1 @Nj @Nsj cos j Tj þ ðNj Ns Þ þ þ ¼ I0 u€ j þ I1 b€ j Rj @j @s R0 Rj 1 @Njs @Ns cos j sin j þ ðNjs þNsj Þ þ þ Ts ¼ I0 u€ s þ I1 b€ s Rj @j @s R0 R0 1 @Tj @Ts cos j Nj sin j € þ Tj þ  Ns ¼ I0 w Rj @j @s R0 Rj R0 1 @Mj @Msj cos j þðMj Ms Þ þ Tj ¼ I1 u€ j þ I2 b€ j Rj @j @s R0 1 @Mjs @Ms cos j þ ðMjs þMsj Þ þ Ts ¼ I1 u€ s þI2 b€ s Rj @j @s R0 where Ii are the mass inertias which are defined as    l Z zk þ 1 X z z rðkÞ zi 1 þ 1þ dz, i ¼ 0,1,2 Ii ¼ Rj RW k ¼ 1 zk

The three basic sets of equations, namely kinematic (18), constitutive (19) and motion Eq. (25) may be combined to give the fundamental system of equations, also known as the governing system of equations. By replacing the kinematic Eq. (18) into the constitutive Eq. (19) and the result of this substitution into the motion Eq. (25), the complete equations of motion in terms of displacement and rotational components can be written as 2 32 3 2 32 u€ 3 uj j L11 L12 L13 L14 L15 I0 0 0 I1 0 6L 76 7 6 76 u€ s 7 7 6 21 L22 L23 L24 L25 76 us 7 6 0 I0 0 0 I1 76 6 76 7 6 76 € 7 7 w 6 L31 L32 L33 L34 L35 76 w 7 ¼ 6 0 0 I0 0 0 76 7 6 76 7 6 76 6€ 7 6L 7 6 7 6 7 b 6 b 4 41 L42 L43 L44 L45 54 j 5 4 I1 0 0 I2 0 54 j 7 5 L51 L52 L53 L54 L55 0 I1 0 0 I2 bs b€ s

ð25Þ

ð26Þ

and rðkÞ is the mass density of the material per unit volume of the kth ply. The first three Eq. (25) represent translational equilibriums along meridional j, circumferential s and normal z directions, while the last two are rotational equilibrium equations about the s and j directions, respectively.

ð27Þ where the explicit forms of the equilibrium operators Lij ,i,j ¼ 1,. . .,5 are shown in the appendix. Two kinds of boundary conditions are considered, namely the fully clamped edge boundary condition (C) and the free edge boundary condition (F). The equations describing the boundary conditions can be written as follows: Clamped edge boundary conditions (C): uj ¼ us ¼ w ¼ bj ¼ bs ¼ 0 at j ¼ j0 and j ¼ j1 , 0 r s rs0 or at s ¼ 0 and s ¼ s0 , j0 r j r j1

ð28Þ

F. Tornabene et al. / International Journal of Mechanical Sciences 53 (2011) 446–470

Physical compatibility conditions along the closing meridian (s ¼0.2pR0)

Free edge boundary conditions (F): Nj ¼ Njs ¼ Tj ¼ Mj ¼ Mjs ¼ 0

at j ¼ j0 or j ¼ j1 , 0 r sr s0

Ns ðj,0,tÞ ¼ Ns ðj,s0 ,tÞ,Nsj ðj,0,tÞ ¼ Nsj ðj,s0 ,tÞ,Ts ðj,0,tÞ ¼ Ts ðj,s0 ,tÞ,

ð29Þ Ns ¼ Nsj ¼ Ts ¼ Ms ¼ Msj ¼ 0

Ms ðj,0,tÞ ¼ Ms ðj,s0 ,tÞ,Msj ðj,0,tÞ ¼ Msj ðj,s0 ,tÞ, j0 r j r j1

s ¼ 0 or s ¼ s0 , j0 r j r j1

at

453

ð32Þ

ð30Þ ^ 1 and j1 ¼ j ^ N. where j0 ¼ j In addition to the external boundary conditions, the kinematic and physical compatibility conditions should be satisfied at the common closing meridians with s ¼ 0,2pR 0 , if a complete shell of revolution (Fig. 7) want to be considered. The kinematic compatibility conditions include the continuity of displacement and rotation components. The physical compatibility conditions can only be the five continuous conditions for the generalized stress resultants. To consider complete revolute shells characterized by s0 ¼ 2pR 0 , it is necessary to implement the kinematic and physical compatibility conditions between the two computational meridians with s ¼0 and with s0 ¼2pR0. Kinematic compatibility conditions along the closing meridian (s¼0,2pR0) uj ðj,0,tÞ ¼ uj ðj,s0 ,tÞ,us ðj,0,tÞ ¼ us ðj,s0 ,tÞ,wðj,0,tÞ ¼ wðj,s0 ,tÞ,

bj ðj,0,tÞ ¼ bj ðj,s0 ,tÞ, bs ðj,0,tÞ ¼ bs ðj,s0 ,tÞ, j0 r j r j1 ð31Þ

 = 0

Closing meridian s = 0, 2R0

^ 1 and j1 ¼ j ^ N. where j0 ¼ j

3. Discretized equations and numerical implementation Since a brief review of the GDQ method is presented in Tornabene [51], the same approach is used in the present work about the GDQ technique. Throughout the paper, the Chebyshev–Gauss–Lobatto (C–G–L) grid distribution is assumed. Since the co-ordinates of the grid points of the reference surface in the j direction are introduced in Eq. (10), then the co-ordinates of the grid points in the s direction are the following:    j1 s0 , j ¼ 1,2,. . .,M for s A ½0,s0  ðwith s r WR0 Þ sj ¼ 1cos p M1 2 ð33Þ where M is the total number of sampling points used to discretize the domain in s direction of the doubly curved shell (Fig. 8). It has been proven that for the Lagrange interpolating polynomials, the Chebyshev–Gauss–Lobatto sampling points rule guarantees convergence and efficiency to the GDQ technique [59–61,63]. In the following, the free vibration of laminated composite doubly curved shells and panels of revolution will be studied. Using the method of variable separation, it is possible to seek solutions that are harmonic in time and whose frequency is o. The displacement field can be written as follows: uj ðj,s,tÞ ¼ U j ðj,sÞeiot us ðj,s,tÞ ¼ U s ðj,sÞeiot wðj,s,tÞ ¼ Wðj,sÞeiot

bj ðj,s,tÞ ¼ Bj ðj,sÞeiot bs ðj,s,tÞ ¼ Bs ðj,sÞeiot

ð34Þ U j ,U s ,W,Bj ,Bs

fulfill where the vibration spatial amplitude values the fundamental differential system. The GDQ procedure enables one to write the equations of motion in discrete form, transforming each space derivative into a weighted sum of node values of dependent variables. Each approximate equation is valid in a single sampling point.

 = 1 Fig. 7. Common meridians of a complete revolution shell.

1

s=0

 = 0

i

s = s0 (i,sj)

N 

s M

1

j  = 1 Fig. 8. C–G–L grid distribution on a revolution shell panel.

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F. Tornabene et al. / International Journal of Mechanical Sciences 53 (2011) 446–470

With the present approach, differing from the finite element method, no integration occurs prior to the global assembly of the linear system, and these results in a further computational cost saving in favor of the differential quadrature technique.

Thus, the whole system of differential equations has been discretized and the global assembling leads to the following set of linear algebraic equations: "

Kbb

Kbd

Kdb

Kdd

#"

#

"

0 0 db ¼ o2 0 Mdd dd

#"

db dd

# ð35Þ

4. Numerical applications and results

In the above mentioned 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, b-equations represent the discrete boundary and compatibility conditions, which are valid only for the points lying on constrained edges of the shell; while d-equations are the motion equations, assigned on interior nodes. In order to make the computation more efficient, kinematic condensation of nondomain degrees of freedom is performed: ðKdd Kdb ðKbb Þ1 Kbd Þdd ¼ o2 Mdd dd

In the present section, some results and considerations about the free vibration problem of FGM and laminated composite doubly curved shells and panels of revolution with a free-form meridian are presented. The analysis has been carried out by means of numerical procedures illustrated above. In order to verify the accuracy of the present method, some comparisons have been performed. The geometrical boundary conditions for a shell panel (Fig. 8) are identified by the following convention. For example, symbolism C–F–C–F shows that the edges j ¼ j0, s ¼0, j ¼ j1, s ¼s0 are clamped, free, clamped and free, respectively. On the contrary, for a complete shell of revolution (Fig. 7), symbolism C–F shows that the edges j ¼ j0 and j ¼ j1 are clamped and free, respectively. The missing boundary conditions are the kinematical and physical compatibility conditions that are applied at the same closing meridians for s ¼0 and s0 ¼2pR0. Tables 1–4 present new results regarding different shells and panels of revolution with a Be´zier curve meridian. Three different

ð36Þ

The natural frequencies of the structure considered can be determined by solving the standard eigenvalue problem (36). In particular, the solution procedure by means of the 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. Table 1 First ten frequencies for an F–C isotropic free-form meridian shell. Control points and weights of the Be´zier curve:      x1 ¼ 2 1:2 0:85 0:75 0:7 , x03 ¼ 0 0:3 1 1:5 2 , w ¼ 1 Geometric characteristics: W0 ¼ 3601, h ¼ 0:1 m, Rb ¼ 0 m Isotropic material properties: E ¼ 70 GPa, n ¼ 0:3, r ¼ 2707 kg=m3

1

1

1

1



Mode (Hz)

GDQ-RM 31  31

GDQ-TL 31  31

Nastran 40  80 ð4 nodesÞ

Abaqus 40  80 ð8 nodesÞ

Ansys 40  80 ð8 nodesÞ

Straus 40  80 ð8 nodesÞ

Pro/Mechanica 31  82 ðGEMÞ

f1 f2 f3 f4 f5 f6 f7 f8 f9 f10

99.139 99.139 105.351 105.351 132.777 132.777 138.515 168.497 168.497 179.511

99.225 99.225 105.478 105.478 133.087 133.087 138.009 168.230 168.230 179.999

98.935 98.935 104.854 104.854 131.839 131.839 138.549 165.901 165.901 178.904

98.945 98.946 104.800 104.800 131.760 131.760 138.950 165.810 165.810 178.950

99.189 99.190 105.160 105.160 132.500 132.500 138.710 166.730 166.730 179.940

99.058 99.058 104.890 104.890 129.049 129.049 138.673 162.648 162.648 171.142

98.897 98.897 104.768 104.768 131.737 131.740 138.586 165.772 165.776 178.919

Table 2 First ten frequencies for a C–C–F–F isotropic free-form meridian panel. Control points and weights of the Be´zier curve:      x1 ¼ 0:8 1:3 1:5 1:4 1:2 , x03 ¼ 0 0:5 1 1:5 2 , w ¼ 1 Geometric characteristics: W0 ¼ 1201, h ¼ 0:1 m, Rb ¼ 0 m Isotropic material properties: E ¼ 70 GPa, n ¼ 0:3, r ¼ 2707 kg=m3

1

1

1

1



Mode (Hz)

GDQ-RM 31  31

GDQ-TL 31  31

Nastran 40  40 ð4 nodesÞ

Abaqus 40  40 ð8 nodesÞ

Ansys 40  40 ð8 nodesÞ

Straus 80  80 ð4 nodesÞ

Pro/Mechanica 31  31 ðGEMÞ

f1 f2 f3 f4 f5 f6 f7 f8 f9 f10

73.153 97.433 191.855 247.826 368.023 431.951 513.792 586.381 614.981 639.533

73.126 96.379 192.944 248.813 367.960 433.114 513.631 586.474 615.184 639.744

72.788 97.791 192.277 247.864 370.021 433.068 514.925 587.524 616.854 642.839

72.758 97.697 192.050 247.420 368.530 431.310 514.060 586.210 615.030 639.410

72.835 97.882 192.250 248.400 370.400 434.350 515.080 588.750 617.970 644.420

74.141 99.834 195.644 252.363 379.756 445.071 517.257 590.594 622.775 653.603

72.723 97.696 191.837 247.309 368.348 431.290 513.753 585.884 614.805 639.367

F. Tornabene et al. / International Journal of Mechanical Sciences 53 (2011) 446–470

455

Table 3 First ten frequencies for a C–C isotropic free-form meridian shell. Control points and weights of the Be´zier curve:      x1 ¼ 0:8 1:3 1:5 1:4 1:2 , x03 ¼ 0 0:5 1 1:5 2 , w ¼ 1 Geometric characteristics: W0 ¼ 3601, h ¼ 0:1m, Rb ¼ 0 m Isotropic material properties: E ¼ 70GPa, n ¼ 0:3, r ¼ 2707 kg=m3

1

1

1

1



Mode (Hz)

GDQ-RM 31  31

GDQ-TL 31  31

Nastran 40  80 ð4 nodesÞ

Abaqus 40  80 ð8 nodesÞ

Ansys 40  80 ð8 nodesÞ

Straus 40  80 ð8 nodesÞ

Pro/Mechanica 31  82 ðGEMÞ

f1 f2 f3 f4 f5 f6 f7 f8 f9 f10

519.359 519.359 529.774 529.774 541.254 541.254 542.013 566.323 566.323 573.588

519.086 519.086 529.619 529.619 541.055 541.055 542.136 566.326 566.326 573.581

519.153 519.153 529.020 529.020 541.457 541.457 542.463 564.919 564.919 574.310

519.940 519.940 529.830 529.830 541.720 542.860 542.860 566.100 566.100 573.330

519.270 519.270 530.260 530.260 541.290 541.290 542.870 567.810 567.820 574.500

518.585 518.585 528.797 528.797 540.946 540.946 541.568 565.385 565.385 571.722

519.023 519.023 529.317 529.317 541.250 541.250 541.723 565.813 565.813 573.240

Table 4 First ten frequencies for a C–F–C–F isotropic free-form meridian panel. Control points and weights of the Be´zier curve:      x1 ¼ 0:8 1:3 1:5 1:3 0:8 , x03 ¼ 0 0:5 1 1:5 2 , w ¼ 1 Geometric characteristics: W0 ¼ 1201, h ¼ 0:1m, Rb ¼ 0 m Isotropic material properties: E ¼ 70 GPa, n ¼ 0:3, r ¼ 2707 kg=m3

1

1

1

1



Mode (Hz)

GDQ-RM 31  31

GDQ-TL 31  31

Nastran 40  40 ð4 nodesÞ

Abaqus 40  40 ð8 nodesÞ

Ansys 40  40 ð8 nodesÞ

Straus 80  80 ð4 nodesÞ

Pro/Mechanica 31  31 ðGEMÞ

f1 f2 f3 f4 f5 f6 f7 f8 f9 f10

283.760 305.347 381.392 433.450 545.180 584.842 590.816 599.965 628.708 691.218

283.663 304.997 382.013 433.467 544.274 584.182 590.491 599.921 628.704 691.245

284.463 306.221 382.430 434.280 546.814 587.950 592.238 602.194 629.123 691.714

283.700 305.380 382.810 433.540 546.800 585.610 590.500 600.320 628.810 690.820

284.260 305.740 382.380 433.990 546.260 587.570 592.600 602.210 629.940 694.580

292.725 314.441 386.420 440.007 552.977 596.844 602.185 617.486 635.533 696.560

283.530 305.077 381.915 433.521 545.214 584.929 590.381 599.881 628.202 690.489

1°-2° Mode Shapes

3°-4° Mode Shapes

5°-6° Mode Shapes

7° Mode Shape

8°-9° Mode Shapes

10°-11° Mode Shapes

Fig. 9. Mode shapes for the F–C free-form meridian shell of Table 1.

curves are considered and the vectors of the control points and the weights of the Be´zier curves are shown in Tables 1–4, as well as the details regarding the material properties, the geometries of the structures and the boundary conditions assumed. Results in terms of first ten frequencies obtained by the GDQ method for the

Reissner–Mindlin (RM) theory and the Toorani–Lakis (TL) theory are compared with the FEM results. Various FEM commercial codes such as Abaqus, Ansys, Straus, Nastran and Pro/Mechanica, have been considered and the finite element shell types selected in each of the commercial programs are reported in the work [75].

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For the present GDQ results, the grid distributions (10) and (33) with N ¼M¼31 have been considered. Well-converged and accurate results were obtained using different FEM meshes for shells and panels under investigation, as shown in Tables 1–4. It is

worth noting that the results achieved with the present methodology are very close to those obtained by the commercial programs for all the geometries considered. As can be seen, the numerical results show an excellent agreement. Furthermore, as

1° Mode Shape

2° Mode Shape

3° Mode Shape

4° Mode Shape

5° Mode Shape

6° Mode Shape

Fig. 10. Mode shapes for the C–C–F–F free-form meridian panel of Table 2.

1°-2° Mode Shapes

3°-4° Mode Shapes

5°-6° Mode Shapes

7° Mode Shape

8°-9° Mode Shapes

10°-11° Mode Shape

Fig. 11. Mode shapes for the C–C free-form meridian shell of Table 3.

1° Mode Shape

2° Mode Shape

3° Mode Shape

4° Mode Shape

5° Mode Shape

6° Mode Shape

Fig. 12. Mode shapes for the C–F–C–F free-form meridian panel of Table 4.

F. Tornabene et al. / International Journal of Mechanical Sciences 53 (2011) 446–470

457

Finally, in order to illustrate the GDQ convergence characteristic for free-form meridian shells, the first ten frequencies of the C–C–F–F isotropic free-form meridian panel of Table 2 are investigated by varying the number of grid points. Results are collected in Table 5 when the number of points of the Chebyshev– Gauss–Lobatto grid distributions (10) and (33) is increased from N ¼M¼11 up to N¼ M¼31. It can be seen that the proposed GDQ formulation well captures the dynamic behavior of the panel by using only 21 points in two co-ordinate directions. It can also be seen that for the considered structure, 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, as shown in the Ph.D. Thesis

regarding the influence of the initial curvature, the difference between the RMT and TLT results is low for all the laminated composite doubly curved structures considered. In Figs. 9–12, there are reported the first six mode shapes for the structures with a free-form meridian considered above. In particular, for the complete shells of revolution there are some symmetrical mode shapes due to the symmetry of the problem considered in 3D space. In these cases, the symmetrical mode shapes are summarized in one figure. The mode shapes of all the structures under discussion have been evaluated by authors. By using the authors’ MATLAB code, these mode shapes have been reconstructed in three-dimensional view by means of considering the displacement field (17) after solving the eigenvalue problem (36).

Table 5 First ten frequencies for the C–C–F–F isotropic free-form meridian panel of Table 2 for an increasing the number of grid points N ¼ M of the Chebyshev–Gauss–Lobatto distribution. Mode (Hz)

N ¼M ¼11

N ¼ M ¼15

N ¼ M¼ 17

N ¼M ¼ 21

N ¼M ¼ 25

N¼ M ¼29

N ¼ M ¼31

GDQ-RM f1 f2 f3 f4 f5 f6 f7 f8 f9 f10

72.115 95.815 190.226 242.855 363.562 424.634 513.585 584.260 613.416 640.392

73.153 96.800 191.894 247.103 367.505 431.237 513.879 586.074 614.395 638.604

73.268 97.173 192.067 247.722 367.959 432.047 513.863 586.372 614.865 639.249

73.257 97.414 192.044 247.979 368.162 432.278 513.831 586.479 615.082 639.608

73.195 97.439 191.936 247.909 368.101 432.107 513.808 586.430 615.040 639.593

73.161 97.435 191.872 247.845 368.041 431.984 513.795 586.392 614.994 639.548

73.153 97.433 191.855 247.826 368.023 431.951 513.792 586.381 614.981 639.533

GDQ-TL f1 f2 f3 f4 f5 f6 f7 f8 f9 f10

72.159 94.487 191.230 243.644 363.399 425.686 513.462 584.391 613.633 640.595

73.024 95.761 192.935 248.007 367.406 432.336 513.725 586.173 614.584 638.810

73.189 96.126 193.123 248.652 367.879 433.160 513.705 586.463 615.054 639.454

73.219 96.360 193.118 248.941 368.096 433.417 513.670 586.567 615.277 639.815

73.166 96.383 193.019 248.888 368.038 433.261 513.647 586.521 615.239 639.803

73.134 96.380 192.959 248.830 367.979 433.146 513.634 586.484 615.196 639.759

73.126 96.379 192.944 248.813 367.960 433.114 513.631 586.474 615.184 639.744

Table 6 The first ten frequencies for the functionally graded free-form panel C–C–F–F of Table 2 (W0 ¼ 1201,h ¼ 0:1 m,Rb ¼ 0 m) as a function of the power-law exponent p ¼ pð1Þ ¼ pð2Þ . Mode (Hz)

p ¼0

p ¼0.6

p ¼1

p ¼5

p ¼20

p ¼ 50

p ¼ 100

p¼N

FGM1ðað1Þ ¼ 1=bð1Þ ¼ 0=cð1Þ ¼ 2=pð1Þ Þ =FGM2ðað2Þ ¼ 1=bð2Þ ¼ 0=cð2Þ ¼ 2=pð2Þ Þ power-law distribution 78.170 80.625 81.175 f1 f2 102.874 106.716 107.612 f3 206.175 210.527 211.298 f4 265.772 274.032 275.943 f5 392.983 406.716 409.887 f6 462.720 478.622 482.109 f7 548.399 547.996 546.791 f8 626.224 628.740 628.129 f9 656.875 659.473 658.396 f10 683.073 694.777 695.201

79.788 105.867 207.298 271.270 403.116 473.969 534.664 614.624 643.927 680.729

76.097 100.564 199.298 258.639 383.520 451.479 521.855 597.995 627.483 659.323

74.553 98.313 195.990 253.414 375.249 441.841 517.341 591.776 621.015 649.478

73.922 97.390 194.634 251.294 371.867 437.870 515.583 589.299 618.332 645.050

73.219 96.360 193.118 248.941 368.096 433.417 513.670 586.567 615.277 639.815

FGM2ðað1Þ ¼ 1=bð1Þ ¼ 0=cð1Þ ¼ 2=pð1Þ Þ =FGM1ðað2Þ ¼ 1=bð2Þ ¼ 0=cð2Þ ¼ 2=pð2Þ Þ power-law distribution f1 78.170 72.837 71.322 f2 102.874 95.087 92.905 f3 206.175 194.203 190.616 f4 265.772 248.215 243.279 f5 392.983 364.631 356.697 f6 462.720 428.617 418.929 f7 548.399 532.703 527.277 f8 626.224 603.357 595.289 f9 656.875 627.086 615.850 f10 683.073 640.379 629.167

70.552 92.097 188.126 240.431 353.166 415.129 516.181 584.574 607.459 620.314

72.263 94.843 191.367 245.825 362.729 426.977 514.287 585.848 613.555 632.713

72.814 95.717 192.383 247.611 365.816 430.697 513.928 586.277 614.659 636.920

73.013 96.033 192.746 248.263 366.935 432.035 513.802 586.423 614.982 638.373

73.219 96.360 193.118 248.941 368.096 433.417 513.670 586.567 615.277 639.815

458

F. Tornabene et al. / International Journal of Mechanical Sciences 53 (2011) 446–470

are EC ¼ 168 GPa, nC ¼ 0:3, rC ¼ 5700 kg=m3 , and for the aluminum are EM ¼ 70 GPa, nM ¼ 0:3 and rM ¼ 2707 kg=m3 , respectively. Tables 6–10 illustrate the first ten frequencies of different structures with a free-form meridian. 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 shells and panels. For the GDQ results shown in Tables 6–10, the grid distributions (10) and (33) with N ¼M ¼21 are considered. Furthermore, two and three layered shells and panels of revolution with a free-form meridian have been considered in order to illustrate the influence of the volume fraction profiles shown in Figs. 5 and 6. The influence of the index p on the vibration frequencies is shown in Figs. 13–17. As can be seen from figures, natural frequencies of FGM shells and panels often present an

by Tornabene [59]. In addition, Hosseini-Hashemi et al. [80] (see Table 3 of the work [80]) have compared their results obtained with a semi analytical method with those results presented in the article by Tornabene [51]. Since the code used to obtained the previous results [51] is exactly the same of the code used to obtained all the results presented in this paper, the results of the work [80] represent another proof of the validity and the accuracy of the present procedure. As shown, the exact results by HosseiniHashemi et al. [80] are in good agreement with those reported by Tornabene [51]. The discrepancy between results obtained from two methods is closely zero. 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

Table 7 The first ten frequencies for the functionally graded free-form panel C–F–C–F of Table 4 (W0 ¼ 1201,h ¼ 0:1 m,Rb ¼ 0 m) as a function of the power-law exponent p ¼ pð1Þ ¼ pð3Þ . Mode (Hz)

p¼0

p ¼0.6

p¼1

p ¼5

p¼ 20

p ¼50

p¼ 100

p¼N

FGM1ðað1Þ ¼ 1=bð1Þ ¼ 0:5=cð1Þ ¼ 2=pð1Þ Þ =FGMCM =FGM2ðað3Þ ¼ 1=bð3Þ ¼ 0:5=cð3Þ ¼ 2=pð3Þ Þ power-law distribution 302.868 307.559 309.470 305.694 f1 f2 325.703 330.387 332.275 327.992 f3 407.901 405.629 403.909 392.648 f4 462.809 461.597 460.236 448.207 f5 581.145 580.178 578.430 562.962 f6 623.734 628.692 628.242 613.132 f7 630.423 629.451 629.878 617.256 f8 640.571 644.045 645.538 634.560 f9 671.233 667.691 665.740 649.782 f10 738.008 734.648 732.131 712.890

291.768 313.459 385.643 438.728 551.418 596.918 598.165 611.530 634.687 698.359

287.199 308.716 383.621 435.796 547.538 589.809 593.781 604.822 631.145 694.292

285.495 306.949 382.868 434.687 546.013 587.111 592.179 602.445 629.941 692.815

283.688 305.077 382.069 433.500 544.342 584.234 590.500 600.005 628.725 691.271

FGM2ðað1Þ ¼ 1=bð1Þ ¼ 0:5=cð1Þ ¼ 2=pð1Þ Þ =FGMCM =FGM1ðað3Þ ¼ 1=bð3Þ ¼ 0:5=cð3Þ ¼ 2=pð3Þ Þ power-law distribution 302.868 299.911 297.810 285.463 f1 f2 325.703 322.504 320.248 307.110 f3 407.901 403.481 400.733 387.234 f4 462.809 457.870 454.738 438.856 f5 581.145 574.996 571.054 550.668 f6 623.734 617.419 613.134 589.295 f7 630.423 623.709 619.440 597.779 f8 640.571 633.938 629.565 606.311 f9 671.233 663.927 659.411 637.531 f10 738.008 730.034 725.059 700.509

283.171 304.601 383.158 434.415 545.248 584.077 591.722 600.524 630.695 693.163

283.403 304.807 382.496 433.841 544.665 584.060 590.947 600.121 629.498 692.003

283.533 304.929 382.281 433.666 544.498 584.129 590.716 600.047 629.109 691.633

283.688 305.077 382.069 433.500 544.342 584.234 590.500 600.005 628.725 691.271

Table 8 The first ten frequencies for the functionally graded free-form panel F–F–C–C of Table 1 (W0 ¼ 1201,h ¼ 0:1 m,Rb ¼ 0 m) as a function of the power-law exponent p ¼ pð1Þ ¼ pð3Þ . Mode (Hz)

p¼ 0

p ¼ 0.6

p¼1

p ¼5

p ¼20

p¼ 50

p ¼ 100

p¼N

FGM1ðað1Þ ¼ 0=bð1Þ ¼ 0:6=cð1Þ ¼ 2=pð1Þ Þ =FGMCM =FGM2ðað3Þ ¼ 0=bð3Þ ¼ 0:6=cð3Þ ¼ 2=pð3Þ Þ power-law 30.556 31.221 31.528 f1 f2 83.906 85.692 86.526 f3 177.390 180.959 182.592 f4 204.823 206.327 206.993 f5 236.942 239.330 240.373 f6 290.465 296.946 300.060 f7 339.938 346.555 349.629 f8 363.542 371.615 375.425 f9 414.743 426.401 431.999 f10 465.840 472.783 475.884

distribution 32.081 88.056 185.345 207.267 241.234 306.200 354.979 382.635 443.683 479.955

31.298 85.897 180.972 203.188 236.349 298.412 346.553 373.089 431.567 469.614

30.645 84.100 177.363 200.211 232.679 291.859 339.617 365.072 421.025 461.411

30.195 82.867 174.874 198.257 230.226 287.377 334.862 359.563 413.737 455.802

28.621 78.593 166.157 191.853 221.938 272.072 318.413 340.522 388.482 436.342

FGM2ðað1Þ ¼ 0=bð1Þ ¼ 0:6=cð1Þ ¼ 2=pð1Þ Þ =FGMCM =FGM1ðað3Þ ¼ 0=bð3Þ ¼ 0:6=cð3Þ ¼ 2=pð3Þ Þ power-law f1 30.556 30.591 30.547 f2 83.906 83.985 83.861 f3 177.390 177.501 177.218 f4 204.823 204.112 203.499 f5 236.942 236.336 235.699 f6 290.465 290.811 290.412 f7 339.938 340.064 339.495 f8 363.542 363.981 363.479 f9 414.743 416.000 415.692 f10 465.840 465.366 464.359

distribution 29.592 81.256 171.776 198.161 229.283 281.305 329.158 352.077 401.826 450.934

28.753 78.981 167.051 194.082 224.193 273.357 320.262 342.086 389.229 439.750

28.594 78.543 166.121 192.963 222.911 271.839 318.475 340.190 387.103 437.268

28.561 78.447 165.907 192.526 222.457 271.514 318.042 339.792 386.812 436.540

28.621 78.592 166.156 191.852 221.937 272.070 318.410 340.520 388.478 436.340

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459

Table 9 The first ten frequencies for the functionally graded free-form panel F–C–F–C of Table 1 (W0 ¼ 1201, h ¼ 0:1 m, Rb ¼ 0 m) as a function of the power-law exponent p ¼ pð1Þ ¼ pð3Þ . Mode (Hz)

p¼ 0

p ¼ 0.6

p ¼1

FGM1ðað1Þ ¼ 0:6=bð1Þ ¼ 0:2=cð1Þ ¼ 3=pð1Þ Þ =FGMCM =FGM2ðað3Þ ¼ 0:6=bð3Þ ¼ 0:2=cð3Þ ¼ 3=pð3Þ Þ power-law 187.034 187.543 187.765 f1 f2 214.487 214.466 214.343 f3 292.151 293.868 294.753 f4 350.950 353.713 355.147 f5 401.696 405.932 408.205 f6 411.864 412.758 413.060 f7 541.686 546.512 549.048 f8 551.139 556.695 559.563 f9 564.376 569.841 572.827 f10 635.461 637.301 638.006

p¼ 5

p ¼ 20

p ¼50

p¼ 100

p¼N

distribution 186.610 211.251 295.212 356.987 409.350 413.015 553.217 564.397 579.202 633.023

180.220 205.258 283.554 342.061 393.923 396.200 529.209 539.605 552.615 612.223

177.472 202.901 278.161 334.858 384.379 390.576 517.430 527.113 539.559 603.089

176.384 201.955 276.014 331.952 380.528 388.307 512.669 521.987 534.357 599.374

175.190 200.904 273.650 328.725 376.257 385.782 507.382 516.237 528.635 595.218

175.471 201.513 273.619 328.298 375.227 386.466 506.430 514.861 527.501 596.008

175.234 201.101 273.482 328.330 375.530 385.916 506.624 515.267 527.766 595.294

175.200 200.995 273.539 328.489 375.840 385.826 506.939 515.681 528.129 595.217

175.190 200.904 273.650 328.725 376.257 385.782 507.382 516.237 528.635 595.218

FGM2ðað1Þ ¼ 0:6=bð1Þ ¼ 0:2=cð1Þ ¼ 3=pð1Þ Þ =FGMCM =FGM1ðað3Þ ¼ 0:6=bð3Þ ¼ 0:2=cð3Þ ¼ 3=pð3Þ Þ power-law distribution f1 f2 f3 f4 f5 f6 f7 f8 f9 f10

187.034 214.487 292.151 350.950 401.696 411.864 541.686 551.139 564.376 635.461

185.790 213.063 290.205 348.609 399.013 409.126 538.067 547.458 560.608 631.233

184.967 212.130 288.903 347.030 397.187 407.316 535.619 544.955 558.053 628.432

178.873 205.370 279.008 334.833 382.794 393.950 516.557 525.242 538.079 607.598

Table 10 The first ten frequencies for the functionally graded free-form panel C–C of Table 4 (W0 ¼ 3601, h ¼ 0:1 m, Rb ¼ 0m) as a function of the power-law exponent p ¼ pð1Þ ¼ pð3Þ . Mode (Hz)

p ¼0

p ¼0.6

p ¼1

p ¼5

p ¼20

p ¼ 50

p ¼ 100

p¼N

488.224 513.492 546.405 546.405 566.738 566.738 620.451 620.451 639.171 639.171

481.544 510.620 539.265 539.265 563.297 563.297 616.022 616.022 630.030 630.030

479.638 509.947 537.243 537.243 562.469 562.469 614.893 614.893 627.416 627.416

478.934 509.712 536.499 536.499 562.176 562.176 614.488 614.488 626.451 626.451

478.192 509.470 535.714 535.714 561.875 561.875 614.066 614.066 625.433 625.433

FGM2ðað1Þ ¼ 1=bð1Þ ¼ 1=cð1Þ ¼ 5=pð1Þ Þ =FGMCM =FGM1ðað3Þ ¼ 1=bð3Þ ¼ 1=cð3Þ ¼ 5=pð3Þ Þ power-law distribution 503.738 499.180 496.620 484.618 f1 f2 522.847 520.696 519.479 513.531 f3 563.237 558.333 555.580 542.679 f4 563.237 558.333 555.580 542.679 f5 577.530 574.976 573.534 566.530 f6 577.530 574.976 573.534 566.530 f7 633.303 630.121 628.323 619.614 f8 633.303 630.121 628.323 619.614 f9 660.211 653.958 650.452 634.085 f10 660.211 653.958 650.452 634.085

479.784 510.639 537.453 537.453 563.197 563.197 615.599 615.599 627.561 627.561

478.829 509.956 536.411 536.411 562.422 562.422 614.697 614.697 626.283 626.283

478.510 509.716 536.063 536.063 562.152 562.152 614.384 614.384 625.858 625.858

478.192 509.470 535.714 535.714 561.875 561.875 614.066 614.066 625.433 625.433

FGM1ðað1Þ ¼ 1=bð1Þ ¼ 1=cð1Þ ¼ 5=pð1Þ Þ =FGMCM =FGM2ðað3Þ ¼ 1=bð3Þ ¼ 1=cð3Þ ¼ 5=pð3Þ Þ power 503.738 500.481 498.581 f1 f2 522.847 520.680 519.456 f3 563.237 559.685 557.616 f4 563.237 559.685 557.616 f5 577.530 575.052 573.649 f6 577.530 575.052 573.649 f7 633.303 630.408 628.758 f8 633.303 630.408 628.758 f9 660.211 655.809 653.238 f10 660.211 655.809 653.238

intermediate value between the natural frequencies of the limit cases of homogeneous shells of zirconia (p¼0) and of aluminum (p¼ N), as expected. However, natural frequencies sometimes exceed limit cases, as can be seen from Figs. 13–17. This fact can depend on various parameters, such as the geometry of the shell, the boundary conditions, the power-law distribution profile, the lamination scheme, 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 figures under consideration, even if the contents of ceramic is not much. Increasing the values of the parameter index p up to infinity reduces the contents of ceramic and at the same time increases the percentage of metal. Thus, it is possible to obtain dynamic characteristics similar or better than the isotropic ceramic or metal limit case by choosing suitable values of the four parameters a, b, c and p.

Figs. 13–17 are divided into two parts. On the left, part (a) shows the first four frequencies versus the power-law index p obtained using FGM1ðað1Þ =bð1Þ =cð1Þ =pð1Þ Þ =FGM2ðað2Þ =bð2Þ =cð2Þ =pð2Þ Þ or FGM1ðað1Þ =bð1Þ =cð1Þ =pð1Þ Þ =FGMCM =FGM2ðað3Þ =bð3Þ =cð3Þ =pð3Þ Þ distributions, while on the right, part (b) illustrates the first four frequencies versus the power-law index p obtained using FGM2ðað1Þ =bð1Þ =cð1Þ =pð1Þ Þ = FGM1ðað2Þ =bð2Þ =cð2Þ =pð2Þ Þ or FGM2ðað1Þ =bð1Þ =cð1Þ =pð1Þ Þ =FGMCM =FGM1ðað3Þ =bð3Þ =cð3Þ =pð3Þ Þ distributions. The symbol FGMCM indicates that the middle ply can be constituted by a mixture of the two constituents. If one of the two constituents has a zero volume fraction, the isotropic material lamina is inferred as a special case. Fig. 13 shows the first four natural frequencies of the C–C–F–F free-form meridian panel of Table 2 versus the power-law index p for various values of the parameter bð1Þ ¼ bð2Þ . Fig. 13(a) illustrates the variation of the first four frequencies

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Fig. 13. First four frequencies of the functionally graded C–C–F–F free-form meridian panel of Table 2 (W0 ¼ 1201, h ¼ 0:1 m, Rb ¼ 0 m) versus the power-law exponent p ¼ pð1Þ ¼ pð2Þ for various values of the parameter b ¼ bð1Þ ¼ bð2Þ : (a) FGM1ðað1Þ ¼ 1=0 r bð1Þ r 1=cð1Þ ¼ 2=pð1Þ Þ =FGM2ðað2Þ ¼ 1=0 r bð2Þ r 1=cð2Þ ¼ 2=pð2Þ Þ and (b) FGM2ðað1Þ ¼ 1=0 r bð1Þ r 1=cð1Þ ¼ 2=pð1Þ Þ = FGM1ðað2Þ ¼ 1=0 r bð2Þ r 1=cð2Þ ¼ 2=pð2Þ Þ .

obtained using the FGM1ðað1Þ =bð1Þ =cð1Þ =pð1Þ Þ =FGM2ðað2Þ =bð2Þ =cð2Þ =pð2Þ Þ distribution, while in Fig. 13(b) the first four frequencies for the FGM2ðað1Þ =bð1Þ =cð1Þ =pð1Þ Þ = FGM1ðað2Þ =bð2Þ =cð2Þ =pð2Þ Þ distribution are reported. For (a, b) the same parameters að1Þ ¼ að2Þ ¼ 1,cð1Þ ¼ cð2Þ ¼ 2 are kept. It is interesting to note that frequencies attain the value for a shell made only of metal, due to the fact that aluminum has a much

smaller Young’s modulus than zirconia. In particular, it can be noted that in Fig. 13(a) for low values of the parameter b the most of frequencies exceeds the ceramic limit case (p ¼0) varying the power-law index from p ¼0 to pE1, while for values of p greater than unity frequencies decrease until a minimum value. After the maximum, frequencies slowly decrease by increasing the

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461

Fig. 14. First four frequencies of the functionally graded C–F–C–F free-form meridian panel of Table 4 (W0 ¼ 1201,h ¼ 0:1 m, Rb ¼ 0m) versus the power-law exponent p ¼ pð1Þ ¼ pð3Þ for various values of the parameter b ¼ bð1Þ ¼ bð3Þ : (a) FGM1ðað1Þ ¼ 1=0 r bð1Þ r 1=cð1Þ ¼ 2=pð1Þ Þ =FGMCM =FGM2ðað3Þ ¼ 1=0 r bð3Þ r 1=cð3Þ ¼ 2=pð3Þ Þ and (b) FGM2ðað1Þ ¼ 1=0 r bð1Þ r 1=cð1Þ ¼ 2=pð1Þ Þ = FGMCM =FGM1ðað3Þ ¼ 1=0 r bð3Þ r 1=cð3Þ ¼ 2=pð3Þ Þ .

power-law exponent p and tend to the metal limit case (p ¼N). 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. On the contrary, in Fig. 13(b) for values of the parameter b approaching the unity the behavior described above is not present. In particular for b¼0 all the frequencies present a knee as previously described, but show a fast

descending behavior up to a minimum value by increasing the power-law index p and exceed the metal limit case. After this minimum, frequencies gradually tend to the metal limit case. This behavior depends on the type of vibration mode. Some frequencies do not present a knee or a maximum value as described above, but decrease gradually from the ceramic limit case (p ¼0) to the metal limit case (p ¼N) by increasing the power-law

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Fig. 15. First four frequencies of the functionally graded F–F–C–C free-form meridian shell of Table 1 (W0 ¼ 1201, h ¼ 0:1 m, Rb ¼ 0 m) versus the power-law exponent p ¼ pð1Þ ¼ pð3Þ for various values of the parameter b ¼ bð1Þ ¼ bð3Þ : (a) FGM1ðað1Þ ¼ 0=1 r bð1Þ r 0=cð1Þ ¼ 2=pð1Þ Þ =FGMCM =FGM2ðað3Þ ¼ 0=1 r bð3Þ r 0=cð3Þ ¼ 2=pð3Þ Þ and (b) FGM2ðað1Þ ¼ 0=1 r bð1Þ r 0=cð1Þ ¼ 2=pð1Þ Þ = FGMCM =FGM1ðað3Þ ¼ 0=1 r bð3Þ r 0=cð3Þ ¼ 2=pð3Þ Þ .

exponent p. In particular, the types of vibration mode that can present this monotone gradually decrease of frequency are torsional, bending and axisymmetric mode shapes, while the circumferential and radial mode shapes are characterized by a knee or a maximum value, as can be seen by comparing the mode shapes represented with variations of frequencies as functions of

the power-law exponent p. However, this behavior depends on the geometry of the shell and boundary conditions. In the same way, Fig. 14 shows the first four natural frequencies of the C–F–C–F free-form meridian panel of Table 3 versus the power-law index p for various values of the parameter bð1Þ ¼ bð3Þ . Fig. 14(a) illustrates the variation of the first

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463

Fig. 16. First four frequencies of the functionally graded F–C–F–C free-form meridian panel of Table 1 (W0 ¼ 1201, h ¼ 0:1 m, Rb ¼ 0 m) versus the power-law exponent p ¼ pð1Þ ¼ pð3Þ for various values of the parameter a ¼ að1Þ ¼ að3Þ : (a) FGM1ð0:2 r að1Þ r 1:2=bð1Þ ¼ 0:2=cð1Þ ¼ 3=pð1Þ Þ =FGMCM =FGM2ð0:2 r að3Þ r 1:2=bð3Þ ¼ 0:2=cð3Þ ¼ 3=pð3Þ Þ and (b) FGM2ð0:2 r að1Þ r 1:2=bð1Þ ¼ 0:2=cð1Þ ¼ 3=pð1Þ Þ =FGMCM =FGM1ð0:2 r að3Þ r 1:2=bð3Þ ¼ 0:2=cð3Þ ¼ 3=pð3Þ Þ .

four frequencies obtained using the FGM1ðað1Þ =bð1Þ =cð1Þ =pð1Þ Þ =FGMCM = FGM2ðað3Þ =bð3Þ =cð3Þ =pð3Þ Þ distribution, while in Fig. 14(b) the first four frequencies for the FGM2ðað1Þ =bð1Þ =cð1Þ =pð1Þ Þ =FGMCM =FGM1ðað3Þ =bð3Þ =cð3Þ =pð3Þ Þ distribution are reported. For (a, b) the same parameters að1Þ ¼ að3Þ ¼ 1,cð1Þ ¼ cð3Þ ¼ 2 are kept. Differently from the previous case a new limit has been introduced. This new limit represents

the laminated shell made of three laminae: the first and the third laminae are fully metal plies, while the middle lamina is a fully ceramic ply. As can be seen, by varying parameters b and p frequencies tend to different limit behaviors: ceramic, metal and metal/ceramic/metal limit cases. b ¼ bð1Þ ¼ bð3Þ is contained in the interval [0,1] for this case.

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Fig. 17. First four frequencies of the functionally graded C–C free-form meridian shell of Table 4 (W0 ¼ 3601, h ¼ 0:1 m, R b ¼ 0 m) versus the power-law exponent p ¼ pð1Þ ¼ pð3Þ for various values of the parameter c ¼ cð1Þ ¼ cð3Þ : (a) FGM1ðað1Þ ¼ 1=bð1Þ ¼ 1=1 r cð1Þ r 13=pð1Þ Þ =FGMCM =FGM2ðað3Þ ¼ 1=bð3Þ ¼ 1=1 r cð3Þ r 13=pð3Þ Þ and (b) FGM2ðað1Þ ¼ 1=bð1Þ ¼ 1=1 r cð1Þ r 13=pð1Þ Þ =FGMCM =FGM1ðað3Þ ¼ 1=bð3Þ ¼ 1=1 r cð3Þ r 13=pð3Þ Þ .

Fig. 15 presents the first four natural frequencies of the F–F–C– C free-form meridian shell of Table 1 versus the power-law index p for various values of the parameter bð1Þ ¼ bð3Þ . As for a previous case a three laminae shell is considered. For (a, b) the same parameters að1Þ ¼ að3Þ ¼ 0,cð1Þ ¼ cð3Þ ¼ 2 are kept, while ð1Þ ð3Þ b ¼ b ¼ b is contained in the interval [ 1,0]. As can be seen,

in Fig. 15(a) frequencies exceed the ceramic limit case for low values of the power-law index p up to a maximum value and after this maximum decrease by increasing the power-law index p. For high value of the power-law index p frequencies slowly tend to the metal limit case. On the contrary, different behavior can be seen in Fig. 15(b) due to the change of the lamination scheme profile.

F. Tornabene et al. / International Journal of Mechanical Sciences 53 (2011) 446–470

Moreover, the influence of the parameter a on the dynamic vibration of the F–C–F–C free-form meridian panel of Table 1 is investigated in Fig. 16 by considering bð1Þ ¼ bð3Þ ¼ 0:2,cð1Þ ¼ cð3Þ ¼ 3. In this case the parameter a ¼ að1Þ ¼ að3Þ varies from 0.2 to 1.2. As can be seen, by varying parameters a and p frequencies tend to different limit behaviors: ceramic, metal and metal/ceramic/ metal limit cases. Finally, Fig. 17 the influence of the parameter c on the dynamic vibration of the C–C free-form meridian shell of Table 3 is presented by considering að1Þ ¼ að3Þ ¼ 1,bð1Þ ¼ bð3Þ ¼ 1. The parameter c ¼ cð1Þ ¼ cð3Þ varies from 1 to 13. It can be noted that the influence of the parameter c is poor and frequencies gradually decrease by increasing the power-law exponent p and tend to the metal/ceramic/metal limit case.

5. Conclusion remarks and summary A generalized differential quadrature method application to the free vibration analysis of laminated composite and functionally graded doubly curved shells and panels of revolution with a free-form meridian has been presented to illustrate the versatility and the accuracy of this methodology. The differential quadrature rule is introduced to determine the geometric parameters of the structures with a free-form meridian. Simple Rational Be´zier curves are used. Various lamination schemes with different layers have been considered. New functionally graded material profiles have been introduced. Ceramic–metal graded shells of revolution with two four parameter power-law distributions of the volume fraction of the constituents in the thickness direction have been considered. Various material profiles through the functionally graded shell thickness have been illustrated by varying the four parameters of power-law distributions. 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. Extensive numerical results have been presented, showing the effect of the choice of the four parameters on shell

frequencies. 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, lamination scheme, power-law distribution, power-law exponent and the curvature of the structure. The adopted shell theory is the First-order Shear Deformation Theory. In particular, the Toorani–Lakis theory has been used. By doing so, the Reissner–Mindlin theory becomes a special case of the Toorani–Lakis theory when the curvature coefficients are set to zero. The motion equations have been discretized with the GDQ method giving a standard linear eigenvalue problem. The vibration results have been obtained without the modal expansion methodology. In this way, the complete 2D differential system, governing the structural problem has been solved. By doing so, complete revolution shells have been obtained as special cases of shell panels by satisfying the kinematic and physical compatibility conditions. The examples presented show that the generalized differential quadrature method can produce accurate results by using a small number of sampling points. Numerical solutions has been compared with those presented in literature and the ones obtained using commercial programs such as Abaqus, Ansys, Nastran, Straus and Pro/Mechanica. The comparisons conducted with FEM codes confirm how the GDQ simple numerical method provides accurate and computationally low cost results for all the structures considered. The GDQ method provides converging results for all the cases as the number of grid points increases. Convergence and stability have been shown. Furthermore, discretizing and programming procedures are quite easy. The GDQ results show to be precise and reliable. The numerical tests demonstrate and confirm the favorable precision of the generalized differential quadrature method.

Acknowledgments This research was supported by the Italian Ministry for University and Scientific, Technological Research MIUR (40% and 60%).

Appendix The equilibrium operators in Eq. (27) assume the following aspect: L11 ¼

1 @2 ðA11 þ a1 B11 þ a2 D11 þ a3 E11 Þ 2 2 Rj @j þ 

!   1 @Rj 1 @a1 @a2 @a3 cos j @ ðA þa B þa D þ a E Þ þ B þD þ E ðA þa B þ a D þa E Þ þ 11 1 11 2 11 3 11 11 11 11 11 1 11 2 11 3 11 @j @j @j @j Rj R0 R3j @j R2j

þ ðA66 þ b1 B66 þ b2 D66 þb3 E66 Þ

L12 ¼

@2 2A16 @2 sin j  þ A12 2 Rj @j@s Rj R0 @s

cos2 j k ðA22 þ b1 B22 þb2 D22 þ b3 E22 Þ 2 ðA44 þ a1 B44 þ a2 D44 þ a3 E44 Þ Rj R20



1 @2 ðA16 þ a1 B16 þ a2 D16 þ a3 E16 Þ 2 2 Rj @j

!   1 @Rj 1 @a1 @a2 @a3 cos j @ þ  3 ðA16 þa1 B16 þa2 D16 þ a3 E16 Þ þ 2 B16 þD16 þ E16 ðA26 þa1 B16 þa2 D16 þa3 E16 Þ þ @j @j @j @j Rj R0 Rj @j Rj   @2 cos j @ þ ðA26 þ b1 B26 þ b2 D26 þb3 E26 Þ 2 þ  ðA66 þ b1 B66 þ b2 D66 þb3 E66 þ A22 þ b1 B22 þ b2 D22 þ b3 E22 Þ @s R0 @s þ

L13 ¼

465

A12 þ A66 @2 sin j cos2 j sin j þ A16 þ ðA26 þ b1 B26 þb2 D26 þ b3 E26 Þk A45 Rj @j@s Rj R0 Rj R0 R20

! 1 sin j k @ ðA þa B þa D þ a E Þ þ A þ ðA þa B þa D þa E Þ 11 1 11 2 11 3 11 12 44 1 44 2 44 3 44 @j Rj R0 R2j R2j

466

F. Tornabene et al. / International Journal of Mechanical Sciences 53 (2011) 446–470

  sin j A16 A45 @ 1 @Rj  ðA26 þ b1 B26 þb2 D26 þ b3 E26 Þ þ þk ðA11 þ a1 B11 þ a2 D11 þa3 E11 Þ R0 Rj Rj @s R3j @j !   1 @a1 @a2 @a3 cos j sin j cos j þ  þD11 þ E11 þ A12 þ 2 B11 @j @j @j Rj R0 Rj R20 þ

þ

L14 ¼

cos j cos j sin j ðA11 A12 þ a1 B11 þ a2 D11 þ a3 E11 Þ þ ðA12 A22 b1 B22 b2 D22 b3 E22 Þ Rj R0 R20

1 @2 ðB11 þ a1 D11 þa2 E11 þ a3 F11 Þ @ j2 R2j þ 

!   1 @Rj 1 @a1 @a2 @a3 cos j @ ðB þ a D þ a E þa F Þ þ D þ E þF ðB þ a D þ a E þa F Þ þ 11 1 11 2 11 3 11 11 11 11 11 1 11 2 11 3 11 @j @j @j @j Rj R0 R3j @j R2j

þ ðB66 þb1 D66 þ b2 E66 þ b3 F66 Þ 

L15 ¼

@2 2B16 @2 sin j  þ B12 Rj @j@s Rj R0 @s2

cos2 j k ðB22 þb1 D22 þ b2 E22 þ b3 F22 Þ þ ðA44 þ a1 B44 þ a2 D44 þ a3 E44 Þ Rj R20

1 @2 ðB16 þ a1 D16 þa2 E16 þ a3 F16 Þ 2 Rj @ j2

!   1 @Rj 1 @a1 @a2 @a3 cos j @ þ  3 ðB16 þ a1 D16 þ a2 E16 þa3 F16 Þ þ 2 D16 þ E16 þF16 ðB26 þ a1 D16 þ a2 E16 þa3 F16 Þ þ @j @j @j @j Rj R0 Rj @j Rj   @2 cos j @ þ ðB26 þb1 D26 þ b2 E26 þ b3 F26 Þ 2 þ  ðB66 þ b1 D66 þb2 E66 þb3 F66 þ B22 þ b1 D22 þb2 E22 þ b3 F22 Þ @s R0 @s þ

L21 ¼

B12 þ B66 @2 sin j cos2 j A45 þ B16 þ ðB26 þ b1 D26 þb2 E26 þ b3 F26 Þ þ k Rj @j@s Rj R0 Rj R20

1 @2 ðA16 þ a1 B16 þ a2 D16 þ a3 E16 Þ 2 2 Rj @j

!   1 @Rj 1 @a1 @a2 @a3 cos j @ ðA þa B þa D þa E Þ þ B þ D þ E ð2A þA þ a B þ a D þ a E Þ þ 16 1 16 2 16 3 16 16 16 16 16 26 1 16 2 16 3 16 @j @j @j @j Rj R0 R3j @j R2j   @2 cos j @ þ ðA26 þ b1 B26 þb2 D26 þ b3 E26 Þ 2 þ ðA66 þ b1 B66 þ b2 D66 þ b3 E66 þ A22 þ b1 B22 þ b2 D22 þb3 E22 Þ @s R0 @s þ 

þ

L22 ¼

A12 þ A66 @2 sin j cos2 j sin j  A26 þ ðA26 þ b1 B26 þ b2 D26 þ b3 E26 Þk A45 Rj @j@s Rj R0 Rj R0 R20

1 @2 ðA66 þ a1 B66 þ a2 D66 þ a3 E66 Þ 2 @j R2j þ 

!   1 @Rj 1 @a1 @a2 @a3 cos j @ ðA þa B þa D þa E Þ þ B þ D þ E ðA þ a B þ a D þa E Þ þ 66 1 66 2 66 3 66 66 66 66 66 1 66 2 66 3 66 @j @j @j @j Rj R0 R3j @j R2j

þ ðA22 þ b1 B22 þb2 D22 þ b3 E22 Þ 

@2 2A26 @2 sin j þ þ A66 Rj @j@s Rj R0 @s2

cos2 j sin2 j ðA66 þ b1 B66 þb2 D66 þ b3 E66 Þk ðA55 þ b1 B55 þb2 D55 þ b3 E55 Þ 2 R0 R20

! 1 sin j sin j @ ðA16 þa1 B16 þa2 D16 þ a3 E16 Þ þ A26 þ k A45 L23 ¼ @j Rj R0 Rj R0 R2j   sin j A12 sin j @ þ ðA22 þ b1 B22 þb2 D22 þ b3 E22 Þ þ þk ðA55 þ b1 B55 þ b2 D55 þb3 E55 Þ @s R0 Rj R0 !   1 @Rj 1 @a1 @a2 @a3 cos jsin j cos j  3 ðA16 þ a1 B16 þ a2 D16 þ a3 E16 Þ þ 2 B16 þ D16 þE16 þ þ  A26 @j @j @j Rj R0 Rj @j Rj R20 ! cos j cos j sin j þ ð2A16 þ a1 B16 þ a2 D16 þ a3 E16 Þ þ ð2A þ b B þb D þ b E Þ 26 1 26 2 26 3 26 Rj R0 R20 L24 ¼

1 @2 ðB16 þ a1 D16 þa2 E16 þ a3 F16 Þ R2j @ j2 þ 

!   1 @Rj 1 @a1 @a2 @a3 cos j cos j ðB þ a D þ a E þa F Þ þ D þ E þ F B þ ð2B þa D þ a E þ a F Þ þ 16 1 16 2 16 3 16 16 16 16 26 16 1 16 2 16 3 16 @j @j Rj R0 Rj R0 R3j @j R2j @j

F. Tornabene et al. / International Journal of Mechanical Sciences 53 (2011) 446–470

L25 ¼



  @ @2 cos j @ þ ðB26 þ b1 D26 þb2 E26 þb3 F26 Þ 2 þ ðB22 þ b1 D22 þb2 E22 þ b3 F22 þB66 þb1 D66 þ b2 E66 þ b3 F66 Þ @j @s R0 @s

þ

B12 þB66 @2 sin j cos2 j sin j  B26 þ ðB26 þb1 D26 þ b2 E26 þ b3 F26 Þ þ k A45 Rj @j@s Rj R0 R0 R20

1 @2 ðB66 þa1 D66 þa2 E66 þ a3 F66 Þ 2 Rj @ j2 þ 

L31 ¼

467

!   1 @Rj 1 @a1 @a2 @a3 cos j ðB þ a D þ a E þa F Þ þ D þ E þ F ðB þ a D þ a E þa F Þ þ 66 1 66 2 66 3 66 66 66 66 66 1 66 2 66 3 66 @j @j Rj R0 R3j @j R2j @j



@ @2 2B26 @2 sin j þ þ ðB22 þ b1 D22 þb2 E22 þb3 F22 Þ 2 þ B66 @j Rj @j@s Rj R0 @s



cos2 j sin j ðB66 þb1 D66 þ b2 E66 þ b3 F66 Þ þ k ðA55 þ b1 B55 þ b2 D55 þb3 E55 Þ R0 R20

1 sin j ðA11 þ a1 B11 þ a2 D11 þa3 E11 Þ A12 Rj R0 R2j R2j   A45 A16 sin j @ þ k   ðA26 þ b1 B26 þ b2 D26 þ b3 E26 Þ @s Rj Rj R0   k @Rj k @a1 @a2 @a3 ðA44 þa1 B44 þa2 D44 þa3 E44 Þ 2 B44 þ D44 þ E44 þ 3 @j @j Rj @j Rj @j 

k

k

!

@ @j

cos j cos j cos j sin j ðA44 þ a1 B44 þ a2 D44 þ a3 E44 Þ A12  ðA22 þb1 B22 þ b2 D22 þb3 E22 Þ Rj R0 Rj R0 R20

k

L32 ¼

ðA44 þ a1 B44 þ a2 D44 þ a3 E44 Þ

sin j 1 sin j A45  2 ðA16 þa1 B16 þ a2 D16 þa3 E16 Þ A26 Rj R0 Rj R0 Rj

!

@ @j

  sin j A12 sin j @ þ k ðA55 þ b1 B55 þ b2 D55 þ b3 E55 Þ  ðA22 þb1 B22 þ b2 D22 þb3 E22 Þ @s R0 Rj R0 k12

L33 ¼

k R2

cos j cos j cos j sin j A45 þ A16 þ ðA26 þ b1 B26 þb2 D26 þ b3 E26 Þ Rj R0 Rj R0 R20

ðA44 þ a1 B44 þ a2 D44 þ a3 E44 Þ

j



þ 



!

k @Rj k @a1 @a2 @a3 cos j @ ðA44 þa1 B44 þa2 D44 þ a3 E44 Þ þ 2 B44 þ D44 þ E44 þ k ðA44 þa1 B44 þa2 D44 þa3 E44 Þ @j @j @j Rj R0 R3j @j Rj @j

þ kðA55 þ b1 B55 þ b2 D55 þb3 E55 Þ 

@2 @j2

@2 2A45 @2 2 sin j  þk A12 2 Rj @j@s Rj R0 @s

1 sin2 j ðA11 þ a1 B11 þ a2 D11 þ a3 E11 Þ ðA22 þb1 B22 þ b2 D22 þb3 E22 Þ R2j R20

! 1 sin j @ L34 ¼ ðA44 þa1 B44 þa2 D44 þ a3 E44 Þ 2 ðB11 þ a1 D11 þa2 E11 þ a3 F11 Þ B12 @j Rj Rj R0 Rj     B16 sin j @ k @a1 @a2 @a3 cos j þ þ kA45   ðB26 þ b1 D26 þb2 E26 þb3 F26 Þ B44 þ D44 þ E44 þ k ðA44 þ a1 B44 þ a2 D44 þ a3 E44 Þ @s Rj @j Rj R0 @j @j R0

k



L35 ¼

cos j cos j sin j B12  ðB22 þ b1 D22 þ b2 E22 þ b3 F22 Þ Rj R0 R20

! A45 1 sin j @  2 ðB16 þa1 D16 þ a2 E16 þ a3 F16 Þ B26 @j Rj Rj Rj R0   B12 sin j @ þ kðA55 þ b1 B55 þ b2 D55 þ b3 E55 Þ  ðB22 þ b1 D22 þ b2 E22 þ b3 F22 Þ @s Rj R0

k

þk

L41 ¼

cos j cos j cos j sin j A45 þ B16 þ ðB26 þ b1 D26 þ b2 E26 þ b3 F26 Þ R0 Rj R0 R20

1 @2 ðB11 þa1 D11 þa2 E11 þ a3 F11 Þ R2j @ j2 þ 

!   1 @Rj 1 @a1 @a2 @a3 cos j @ ðB þ a D þ a E þa F Þ þ D þ E þ F ðB þ a D þ a E þa F Þ þ 11 1 11 2 11 3 11 11 11 11 11 1 11 2 11 3 11 @j @j @j Rj R0 R3j @j R2j @j

468

F. Tornabene et al. / International Journal of Mechanical Sciences 53 (2011) 446–470

þ ðB66 þb1 D66 þ b2 E66 þ b3 F66 Þ 

L42 ¼

@2 2B16 @2 sin j  þ B12 Rj @j@s Rj R0 @s2

cos2 j k ðB22 þb1 D22 þ b2 E22 þ b3 F22 Þ þ ðA44 þ a1 B44 þ a2 D44 þ a3 E44 Þ Rj R20

1 @2 ðB16 þ a1 D16 þa2 E16 þ a3 F16 Þ R2j @ j2

!   1 @Rj 1 @a1 @a2 @a3 cos j @ ðB þ a D þ a E þa F Þ þ D þ E þ F ðB þ a D þ a E þa F Þ þ 16 1 16 2 16 3 16 16 16 16 26 1 16 2 16 3 16 @j @j @j Rj R0 R3j @j R2j @j   @2 cos j @ þ ðB26 þb1 D26 þ b2 E26 þ b3 F26 Þ 2 þ  ðB66 þ b1 D66 þb2 E66 þb3 F66 þ B22 þ b1 D22 þb2 E22 þ b3 F22 Þ @s R0 @s þ 

þ

L43 ¼

L44 ¼

B12 þ B66 @2 sin j cos2 j sin j þ B16 þ ðB26 þ b1 D26 þb2 E26 þ b3 F26 Þ þ k A45 Rj @j@s Rj R0 R0 R20

! 1 sinj k @ ðB þ a D þ a E þa F Þ þ B  ðA þa B þa D þa E Þ 11 1 11 2 11 3 11 12 44 1 44 2 44 3 44 @j Rj R0 Rj R2j   sin j B16 @ þ ðB26 þb1 D26 þ b2 E26 þ b3 F26 Þ þ kA45 @s R0 Rj   1 @Rj 1 @a1 @a2 @a3 ðB11 þa1 D11 þ a2 E11 þ a3 F11 Þ þ 2 D11 þ E11 þ F11  3 @j @j Rj @j Rj @j ! cos j sin j cos j cos j cos j sin j þ ðB11 B12 þ a1 D11 þa2 E11 þ a3 F11 Þ þ ðB12 B22 b1 D22 b2 E22 b3 F22 Þ B12 þ þ  Rj R0 Rj R0 R20 R20 1 @2 ðD11 þa1 E11 þ a2 F11 þ a3 H11 Þ R2j @ j2 þ 

!   1 @Rj 1 @a1 @a2 @a3 cos j @ ðD þ a E þa F þ a H Þ þ E þ F þ H ðD þa E þ a F þa H Þ þ 11 1 11 2 11 3 11 11 11 11 11 1 11 2 11 3 11 @j @j @j Rj R0 R3j @j R2j @j

þ ðD66 þ b1 E66 þ b2 F66 þ b3 H66 Þ 

L45 ¼

@2 2D16 @2 sin j  þ D12 2 Rj @j@s Rj R0 @s

cos2 j ðD22 þb1 E22 þb2 F22 þ b3 H22 ÞkðA44 þ a1 B44 þ a2 D44 þa3 E44 Þ R20

1 @2 ðD16 þa1 E16 þ a2 F16 þ a3 H16 Þ 2 Rj @ j2 þ 

!   1 @Rj 1 @a1 @a2 @a3 cos j @ ðD þ a E þa F þ a H Þ þ E þ F þ H ðD þa E þ a F þa H Þ þ 16 1 16 2 16 3 16 16 16 16 26 1 16 2 16 3 16 @j @j @j Rj R0 R3j @j R2j @j

@2 þ ðD26 þ b1 E26 þ b2 F26 þ b3 H26 Þ 2 @s   cos j @ þ  ðD66 þ b1 E66 þ b2 F66 þ b3 H66 þ D22 þ b1 E22 þ b2 F22 þ b3 H22 Þ @s R0 þ

L51 ¼

D12 þD66 @2 sin j cos2 j þ D16 þ ðD26 þ b1 E26 þb2 F26 þ b3 H26 ÞkA45 Rj @j@s Rj R0 R20

1 @2 ðB16 þ a1 D16 þa2 E16 þ a3 F16 Þ 2 Rj @ j2 þ 

!   1 @Rj 1 @a1 @a2 @a3 cos j ðB þ a D þ a E þa F Þ þ D þ E þ F ð2B þ B þ a D þ a E þa F Þ þ 16 1 16 2 16 3 16 16 16 16 16 26 1 16 2 16 3 16 @j @j Rj R0 R3j @j R2j @j

@ @2 þ ðB26 þ b1 D26 þb2 E26 þ b3 F26 Þ 2 @j @s   cos j @ þ ðB66 þ b1 D66 þ b2 E66 þ b3 F66 þ B22 þ b1 D22 þ b2 E22 þ b3 F22 Þ @s R0



þ

L52 ¼

B12 þ B66 @2 sin j cos2 j A45  B26 þ ðB26 þb1 D26 þ b2 E26 þ b3 F26 Þ þ k Rj @j@s Rj R0 Rj R20

1 @2 ðB66 þ a1 D66 þa2 E66 þ a3 F66 Þ R2j @ j2 þ 

!   1 @Rj 1 @a1 @a2 @a3 cos j ðB þ a D þ a E þa F Þ þ D þ E þ F ðB þ a D þ a E þa F Þ þ 66 1 66 2 66 3 66 66 66 66 66 1 66 2 66 3 66 @j @j Rj R0 R3j @j R2j @j

F. Tornabene et al. / International Journal of Mechanical Sciences 53 (2011) 446–470

L53 ¼



@ @2 2B26 @2 sin j þ þ ðB22 þ b1 D22 þb2 E22 þb3 F22 Þ 2 þ B66 @j Rj @j@s Rj R0 @s



cos2 j sin j ðB66 þ b1 D66 þb2 E66 þb3 F66 Þ þ k ðA55 þb1 B55 þ b2 D55 þ b3 E55 Þ R0 R20

! 1 sin j k @ ðB þ a D þ a E þa F Þ þ B  A 16 1 16 2 16 3 16 26 45 @j Rj R0 Rj R2j   sin j B12 @ þ ðB22 þ b1 D22 þb2 E22 þ b3 F22 Þ þ kðA55 þ b1 B55 þb2 D55 þ b3 E55 Þ @s R0 Rj   1 @Rj 1 @a1 @a2 @a3  3 ðB16 þ a1 D16 þ a2 E16 þ a3 F16 Þ þ 2 D16 þ E16 þ F16 @j @j Rj @j Rj @j ! cos j sin j cos j cos j þ ð2B16 þ a1 D16 þ a2 E16 þ a3 F16 Þ þ  B26 þ Rj R0 Rj R0 R20 þ

L54 ¼

cos jsin j ð2B26 þ b1 D26 þb2 E26 þ b3 F26 Þ R20

1 @2 ðD16 þa1 E16 þ a2 F16 þ a3 H16 Þ 2 R2j @j þ 

L55 ¼

469

!   1 @Rj 1 @a1 @a2 @a3 cos j þ ðD þ a E þ a F þa H Þ þ E þ F þ H ð2D þ D þ a E þ a F þ a H Þ 16 1 16 2 16 3 16 16 16 16 16 26 1 16 2 16 3 16 @j @j Rj R0 R3j @j R2j @j



@ @2 cos j @ þ ðD26 þ b1 E26 þ b2 F26 þ b3 H26 Þ 2 þ ðD22 þ b1 E22 þ b2 F22 þ b3 H22 þ D66 þ b1 E66 þ b2 F66 þb3 H66 Þ @j @s R0 @s

þ

D12 þD66 @2 sin j cos2 j  D26 þ ðD26 þb1 E26 þb2 F26 þ b3 H26 ÞkA45 Rj @j@s Rj R0 R20

1 @2 ðD66 þa1 E66 þ a2 F66 þ a3 H66 Þ 2 R2j @j þ 

!   1 @Rj 1 @a1 @a2 @a3 cos j @ ðD þ a E þ a F þa H Þ þ E þ F þ H ðD þa E þ a F þa H Þ þ 66 1 66 2 66 3 66 66 66 66 66 1 66 2 66 3 66 @j @j @j Rj R0 R3j @j R2j @j

þ ðD22 þ b1 E22 þ b2 F22 þ b3 H22 Þ 

@2 2D26 @2 sin j þ þ D66 Rj @j@s Rj R0 @s2

cos2 j ðD66 þ b1 E66 þb2 F66 þ b3 H66 ÞkðA55 þb1 B55 þ b2 D55 þb3 E55 Þ R20

References [1] Timoshenko S, Woinowsky-Krieger S. Theory of plates and shells. New York: McGraw-Hill; 1959. ¨ [2] Flugge W. Stresses in shells. Berlin: Springer-Verlag; 1960. [3] Gol’Denveizer AL. Theory of elastic thin shells. Oxford: Pergamon Press; 1961. [4] Novozhilov VV. Thin shell theory, Groningen: P. Noordhoff; 1964. [5] Vlasov VZ. General theory of shells and its application in engineering, NASATT-F-99, 1964. [6] Ambartusumyan SA. Theory of anisotropic shells, NASA-TT-F-118, 1964. [7] Kraus H. Thin elastic shells. New York: John Wiley & Sons; 1967. [8] Leissa AW. Vibration of plates, NASA-SP-160, 1969. [9] Leissa AW. Vibration of shells, NASA-SP-288, 1973. ˇ The mechanics of vibrations of cylindrical shells. New York: [10] Markuˇs S. Elsevier; 1988. [11] Ventsel E, Krauthammer T. Thin plates and shells. New York: Marcel Dekker; 2001. [12] Soedel W. Vibrations of shells and plates. New York: Marcel Dekker; 2004. [13] Shu C. An efficient approach for free vibration analysis of conical shells. Int J Mech Sci 1996;38:535–49. [14] Hua L, Lam KY. The generalized quadrature method for frequency analysis of a rotating conical shell with initial pressure. Int J Num Methods Eng 2000;48: 1703–22. [15] Ng TY, Li H, Lam KY. Generalized differential quadrature method for the free vibration of rotating composite laminated conical shell with various boundary conditions. Int J Mech Sci 2003;45:567–87. [16] Redekop D. Buckling analysis of an orthotropic thin shell of revolution using differential quadrature. Int J Pressure Ves Pip 2005;82:618–24. [17] Zhang L, Xiang Y, Wei GW. Local adaptive differential quadrature for free vibration analysis of cylindrical shells with various boundary conditions. Int J Mech Sci 2006;48:1126–38.

[18] Wang X, Wang X, Shi X. Accurate buckling loads of thin rectangular plates under parabolic edge compressions by the differential quadrature method. Int J Mech Sci 2007;49:447–53. [19] Reissner E. The effect of transverse shear deformation on the bending of elastic plates. J Appl Mech—Trans ASME 1945;12:66–77. [20] Gould PL. Finite element analysis of shells of revolution. Boston: Pitman Publishing; 1984. [21] Gould PL. Analysis of plates and shells. Upper Saddle River: Prentice-Hall; 1999. [22] Toorani MH, Lakis AA. General equations of anisotropic plates and shells including transverse shear deformations, rotary inertia and initial curvature effects. J. Sound Vib 2000;237:561–615. [23] Reddy JN. Mechanics of laminated composites plates and shells. New York: CRC Press; 2003. [24] Wu CP, Tarn JQ, Tang SC. A refined asymptotic theory of dynamic analysis of doubly curved laminated shells. Int J Solid Struct 1998;35:1953–79. [25] Wu CP, Lee CY. Differential quadrature solution for the free vibration analysis of laminated conical shells with variable stiffness. Int J Mech Sci 2001;43: 1853–69. [26] Messina A. Free vibrations of multilayered doubly curved shells based on a mixed variational approach and global piecewise-smooth functions. Int J Solid Struct 2003;40:3069–88. [27] Viola E, Artioli E. The G.D.Q. method for the harmonic dynamic analysis of rotational shell structural elements. Struct Eng Mech 2004;17:789–817. [28] Artioli E, Gould P, Viola E. A differential quadrature method solution for shear-deformable shells of revolution. Eng Struct 2005;27:1879–92. [29] Artioli E, Viola E. Static analysis of shear-deformable shells of revolution via G.D.Q. method. Struct Eng Mech 2005;19:459–75. [30] Artioli E, Viola E. Free vibration analysis of spherical caps using a G.D.Q. numerical solution. J Pressure Ves-Technol—Trans ASME 2006;128: 370–8. [31] Farin G. Curves and surfaces for computer aided geometric design. Boston: Academic Press; 1990. [32] Piegl L, Tiller W. The NURBS book. Berlin: Springer; 1997.

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F. Tornabene et al. / International Journal of Mechanical Sciences 53 (2011) 446–470

[33] Beccari CV, Farella E, Liverani A, Morigi S, Rucci M. A fast interactive reverseengineering system. CAD 2010;42:860–73. [34] Amati G, Liverani A, Caligiana G. From spline to Class-A curves through multiscale analysis filtering. Comput. Graphics 2006;30:345–52. [35] Shu C. Differential quadrature and its application in engineering. Berlin: Springer; 2000. [36] Yang J, Kitipornchai S, Liew KM. Large amplitude vibration of thermo-electromechanically stressed FGM laminated plates. Comput Methods Appl Mech Eng 2003;192:3861–85. [37] Wu CP, Tsai YH. Asymptotic DQ solutions of functionally graded annular spherical shells. Eur J Mech A—Solids 2004;23:283–99. [38] Elishakoff I, Gentilini C, Viola E. Forced vibrations of functionally graded plates in the three-dimensional setting. AIAA J 2005;43:2000–7. [39] Elishakoff I, Gentilini C, Viola E. Three-dimensional analysis of an all-around clamped plate made of functionally graded materials. Acta Mech 2005;180: 21–36. [40] Patel BP, Gupta SS, Loknath MS, Kadu CP. Free vibration analysis of functionally graded elliptical cylindrical shells using higher-order theory. Compos Struct 2005;69:259–70. [41] Abrate S. Free vibration, buckling, and static deflection of functionally graded plates. Compos Sci Technol 2006;66:2383–94. [42] Pelletier JL, Vel SS. An exact solution for the steady-state thermoelastic response of functionally graded orthotropic cylindrical shells. Int J Solids Struct 2006;43:1131–58. [43] Arciniega RA, Reddy JN. Large deformation analysis of functionally graded shells. Int J Solids Struct 2007;44:2036–52. [44] Roque CMC, Ferreira AJM, Jorge RMN. A radial basis function for the free vibration analysis of functionally graded plates using refined theory. J Sound Vib 2007;300:1048–70. [45] Allahverdizadeh A, Naei MH, Nikkhah Bahrami M. Vibration amplitude and thermal effects on the nonlinear behavior of thin circular functionally graded plates. Int J Mech Sci 2008;50:445–54. [46] Zhao X, Liew KM. Geometrically nonlinear analysis of functionally graded shells. Int J Mech Sci 2009;51:131–44. [47] Shen HS. Postbuckling of shear deformable FGM cylindrical shells surrounded by an elastic medium. Int J Mech Sci 2009;51:372–83. [48] Zenkour AM. The refined sinusoidal theory for FGM plates on elastic foundations. Int J Mech Sci 2009;51:869–80. [49] Sobhani Aragh B, Yas MH. Three-dimensional analysis of thermal stresses in four-parameter continuous grading fiber reinforced cylindrical panels. Int J Mech Sci 2010;52:1047–63. [50] Hosseimi-Hashemi S, Fadaee M, Atashipour SR. A new exact analytical approach for free vibration of Reissner–Mindlin functionally graded rectangular plates. Int J Mech Sci 2011;53:11–22. [51] Tornabene F. Vibration analysis of functionally graded conical, cylindrical and annular shell structures with a four-parameter power-law distribution. Comput Methods Appl Mech Eng 2009;198:2911–35. [52] Bert C, Malik M. Differential quadrature method in computational mechanics. Appl Mech Rev 1996;49:1–27. [53] Lam KY, Hua L. Influence of initial pressure on frequency characteristics of a rotating truncated circular conical shell. Int J Mech Sci 2000;42:213–36. [54] Liew KM, Teo TM, Han JB. Three-dimensional static solutions of rectangular plates by variant differential quadrature method. Int J Mech Sci 2001;43: 1611–28. [55] Wang X, Wang Y. Free vibration analyses of thin sector plates by the new version of differential quadrature method. Comput Methods Appl Mech Eng 2004;193:3957–71. [56] Malekzadeh P, Karami G, Farid M. A semi-analytical DQEM for free vibration analysis of thick plates with two opposite edges simply supported. Comput Methods Appl Mech Eng 2004;193:4781–96. [57] Viola E, Tornabene F. Vibration analysis of damaged circular arches with varying cross-section. Struct Integr Durab (SID-SDHM) 2005;1:155–69.

[58] Viola E, Tornabene F. Vibration analysis of conical shell structures using GDQ method. Far East J Appl Math 2006;25:23–39. [59] Tornabene F. Modellazione e Soluzione di Strutture a Guscio in Materiale Anisotropo. PhD thesis, University of Bologna—DISTART Department, 2007. [60] Tornabene F, Viola E. Vibration analysis of spherical structural elements using the GDQ method. Comput Math Appl 2007;53:1538–60. [61] Viola E, Dilena M, Tornabene F. Analytical and numerical results for vibration analysis of multi-stepped and multi-damaged circular arches. J Sound Vib 2007;299:143–63. [62] Marzani A, Tornabene F, Viola E. Nonconservative stability problems via generalized differential quadrature method. J Sound Vib 2008;315:176–96. [63] Tornabene F, Viola E. 2-D solution for free vibrations of parabolic shells using generalized differential quadrature method. Eur J Mech A/Solids 2008;27: 1001–25. [64] Alibeigloo A, Modoliat R. Static analysis of cross-ply laminated plates with integrated surface piezoelectric layers using differential quadrature. Comput Struct 2009;88:342–53. [65] Tornabene F, Viola E. Free vibrations of four-parameter functionally graded parabolic panels and shell of revolution. Eur J Mech A/Solids 2009;28: 991–1013. [66] Tornabene F, Viola E. Free vibration analysis of functionally graded panels and shells of revolution. Meccanica 2009;44:255–81. [67] Tornabene F, Viola E, Inman DJ. 2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical and annular shell structures. J Sound Vib 2009;328:259–90. [68] Viola E, Tornabene F. Free vibrations of three parameter functionally graded parabolic panels of revolution. Mech Res Commun 2009;36:587–94. [69] Tornabene F, Marzani A, Viola E, Elishakoff I. Critical flow speeds of pipes conveying fluid by the generalized differential quadrature method. Adv Theor Appl Mech 2010;3:121–38. [70] Alibeigloo A, Nouri V. Static analysis of functionally graded cylindrical shell with piezoelectric layers using differential quadrature method. Compos Struct 2010;92:1775–85. [71] Andakhshideh A, Maleki S, Aghdam MM. Non-linear bending analysis of laminated sector plates using generalized differential quadrature. Compos Struct 2010;92:2258–64. [72] Malekzadeh P, Alibeygi Beni A. Free vibration of functionally graded arbitrary straight-sided quadrilateral plates in thermal environment. Compos Struct 2010;92:2758–67. [73] Sepahi O, Forouzan MR, Malekzadeh P. Large deflection analysis of thermomechanical loaded annular FGM plates on nonlinear elastic foundation via DQM. Compos Struct 2010;92:2369–78. [74] Yas MH, Sobhani Aragh B. Three-dimensional analysis for thermoelastic response of functionally graded fiber reinforced cylindrical panel. Compos Struct 2010;92:2391–9. [75] Tornabene F. Free Vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ method. Comput Methods Appl Mech Eng 2011;200:931–52. [76] Tornabene F. 2-D GDQ solution for free vibrations of anisotropic doublycurved shells and panels of revolution. Compos Struct 2011;in press, doi:10. 1016/j.compstruct.2011.02.006. [77] Auricchio F, Sacco E. A mixed-enhanced finite-element for the analysis of laminated composite plates. Int J Num Methods Eng 1999;44:1481–504. [78] Alfano G, Auricchio F, Rosati L, Sacco E. MITC finite elements for laminated composite plates. Int J Num Methods Eng 2001;50:707–38. [79] Auricchio F, Sacco E. Refined first-order shear deformation theory models for composite laminates. J Appl Mech 2003;70:381–90. [80] Hosseini-Hashemi Sh, Fadaee M, Es’haghi M. A novel approach for in-plane/ out-of-plane frequency analysis of functionally graded circular/annular plates. Int J Mech Sci 2010;52:1025–35.