RETRACTED: Free vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ method

Comput. Methods Appl. Mech. Engrg. 200 (2011) 931–952

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Free vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ method Francesco Tornabene DISTART – Department, Faculty of Engineering, University of Bologna, Italy

a r t i c l e

i n f o

Article history: Received 28 March 2010 Received in revised form 23 September 2010 Accepted 22 November 2010 Available online 30 November 2010 Keywords: Free vibrations Doubly-curved shells of revolution Laminated composite shells First-order shear deformation theory Generalized differential quadrature method

a b s t r a c t In this paper, the Generalized Differential Quadrature (GDQ) method is applied to study the dynamic behaviour of laminated composite doubly-curved shells of revolution. The First-order Shear Deformation Theory (FSDT) is used to analyze the above mentioned moderately thick structural elements. 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. The discretization of the system by means of the Differential Quadrature (DQ) 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. Examples of hyperbolic, catenary, cycloid, parabolic, elliptic and circular shell and panel structures are presented to illustrate the validity and the accuracy of the GDQ method. Furthermore, GDQ results are compared with those presented in literature and the ones obtained by using commercial programs such as Abaqus, Ansys, Nastran, Straus and Pro/Mechanica. Very good agreement is observed. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction Thin and thick shells as structural elements occupy a leadership position in many branches of engineering technologies and, in particular, in civil, mechanical, architectural, aeronautical, and marine engineering. Examples of shell structures in civil and architectural engineering are large-span roofs, cooling towers, liquid-retaining structures and water tanks, containment shells of nuclear power plants and concrete arch domes. In mechanical engineering, shell shapes are used in piping systems, turbine disks and pressure vessels technology. Aircrafts, missiles, rockets, ships and submarines are examples of the use of shells in aeronautical and marine engineering. Shells have been widespread in many fields of engineering as they give rise to optimum conditions for dynamic behaviour, strength and stability. These structures support applied external forces efficiently thanks to their geometrical shape. In other words, shells are much stronger and stiffer than other structural shapes. 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

E-mail addresses: [email protected], francesco.tornabene3@ unibo.it 0045-7825/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2010.11.017

dynamic behaviour of doubly-curved shell structures derived from shells of revolution, which are very common structural elements. It is well known that a shell may be considered as a threedimensional body and the methods of the linear theory of elasticity may be applied. However, a calculation based on these methods will generally be difficult and computationally expensive. In the theory of shells, an alternative simplified method is therefore used. Adapting some hypotheses, the 3D problem of shell equilibrium may be reduced to the analysis of its middle surface only and the given shell may be regarded as a 2D body. The approximations necessary for the development of an adequate theory of shells have been the subject of considerable discussions among investigators. Starting from Love’s theory for thin shells, which dates back to about 100 years ago, many contributions on this topic have been made since then, in order to seek better and better approximations for the exact three-dimensional elasticity solutions for shells. During the last sixty years, two-dimensional linear theories of thin shells have been developed including important contributions by Timoshenko and Woinowsky-Krieger [1], Flügge [2], Gol’Denveizer [3], Novozhilov [4], Vlasov [5], Ambartusumyan [6], Kraus [7], Leissa [8,9], Markuš [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].

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It is worth noting that, 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. Furthermore, due to the significant developments that have taken place in composite materials [22,23], the increase in the their use in a lot of types of engineering structures in the last decades calls for improved analysis and design tools for these types of structures. Thus, in this paper, the laminated composite doubly-curved shells of revolution are considered. 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. These fundamental equations are expressed in terms of kinematic parameters and can be obtained by combining three basic sets of equations, namely the equilibrium, kinematic and constitutive ones. Referring to the formulation of the dynamic equilibrium in terms of harmonic amplitudes of mid-surface displacements and rotations, in this paper the system of second-order linear partial differential equations is solved, without resorting to the onedimensional formulation of the dynamic equilibrium of the shell. Now, the discretization of the system leads to a standard linear eigenvalue problem, where two independent variables are involved. By so doing, 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 # = 0, 2p. In this paper, the analysis will be performed by following two different investigations. In the first one, the solution is obtained by using the numerical technique termed Generalized Differential Quadrature (GDQ) method, which leads to a generalized eigenvalue problem. The mathematical fundamentals and recent developments of the GDQ method as well as its major applications in engineering are discussed in detail in the book by Shu [31]. The solution is given in terms of generalized displacement components of the points lying on the middle surface of the shell. Then numerical results will also be computed by using commercial programs in order to verify the accuracy of the present method. It is worth noting that 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 [32], 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 [33–70] 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 laminated composite doubly-curved moderately thick shells and panels of revolution, taking two independent co-ordinates into account. 2. Laminated composite shells and fundamental systems The basic configuration of the problem considered here is a laminated composite doubly-curved shell as shown in Fig. 1. The coordinates along the meridional and circumferential directions of the reference surface are u and s, respectively. The distance of each point from the shell mid-surface along the normal is f. 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 = fk+1  fk is the thickness of the kth lamina or ply. In this work, doubly-curved shells of revolution with hyperbolic (Fig. 2), catenary (Fig. 3), cycloid (Fig. 4), parabolic (Fig. 5) and

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

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F. Tornabene / Comput. Methods Appl. Mech. Engrg. 200 (2011) 931–952

R 0 (ϕ )

c

Rb

π −ϕt

t2 = ts O′

O

d Rb

C

a

O′

O

x1

R0 (ϕ )

dϕ

R0 (ϕ )

n

x1

O

x1

O

t1 = tϕ Rϕ

C1

t1 = tϕ

Rϑ

s

ϑ

ϕ

b

s

ϑ

n

ϕ

D

t2 = ts

C1

Rϕ

n

c

R 0 (ϕ )

n

x1

dϕ

x2

Rϑ

C2

(b)

a

C2 x2

d

(a)

x′3

x3

(b)

Fig. 5. Parabolic shell geometry: meridional section (a); circumferential section (b).

ϕb

(a)

R 0 (ϕ )

x′3

x3

t2 = ts

Rb

Fig. 2. Hyperbolic shell geometry: meridional section (a); circumferential section (b).

O′

O

n x1

R0 (ϕ )

t2 = ts

Rb O′

O

ϑ t1 = tϕ

x1

O

n

ϕ

R0 (ϕ )

b

t1 = tϕ

x1

n

s

ϑ

b

R 0 (ϕ )

n

Rϕ

x2

s x1

O

(b)

(a)

dϕ

C1

a

ϕ (b)

x3

dϕ

C1

Fig. 6. Elliptic shell geometry: meridional section (a); circumferential section (b).

a (a)

x′3

Rϑ C2 x3

Fig. 3. Catenary shell geometry: meridional section (a); circumferential section (b).

R 0 (ϕ )

t2 = ts

Rb O′

O

n

ϑ t1 = tϕ

rc

ϕ

O

s x1

x2

Rϕ

(b)

dϕ

C1 Rϑ

n

x1

R0 (ϕ )

b = 2rc

Rϑ

C2

x2

Rϕ

x′3

x′3

a = π rc

(a)

C2

rotation x3, or the geometric axis x03 of the meridian curve, is defined as the meridional angle u and the angle between the radius of the parallel circle R0(u) and the x1 axis is designated as the circumferential angle # as shown in Figs. 2–6. For all these structures the parametric co-ordinates (u, s) define, respectively, the meridional curves and the parallel circles upon the middle surface of the shell. The curvilinear abscissa s(u) of a generic parallel is related to the circumferential angle # by the relation s = 0R0. The horizontal radius R0(u) 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 u(u0 6 u 6 u1), s(0 6 s 6 s0) upon the middle surface, and f directed along the outward normal and measured from the reference surface (h/2 6 f 6 h/2). The geometry of shells considered is a surface of revolution with a hyperbolic, catenary, cycloid, parabolic and elliptic curved meridian. The hyperbolic meridian (Fig. 2) can be described by means of the following equation: 2

x3 Fig. 4. Cycloid shell geometry: meridional section (a); circumferential section (b).

elliptic (Fig. 6) meridian curves are considered. The angle formed by the extended normal n to the reference surface and the axis of

2 ðR0  Rb Þ2  ðk  1Þx02 3 ¼ a ;

ð2Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi .pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi .pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 d  a2 and k ¼ 1 þ a2 =b are where b ¼ aC c2  a2 ¼ aD characteristic parameters of the hyperbolic meridian. The horizontal radius R0(u) of a generic parallel of the hyperbolic shell assumes the form:

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F. Tornabene / Comput. Methods Appl. Mech. Engrg. 200 (2011) 931–952

R0 ðuÞ ¼ a sin u

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u k2  1 t 2

2

k sin u  1

þ Rb :

ð3Þ

For the hyperbolic shell of revolution, the radii of curvature Ru(u), R#(u) in the meridional and circumferential directions, and the first derivative of Ru(u) with respect to u can be expressed, respectively, as follows:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 2 u k2  1 u k  1 Rb t t ; R þ Ru ðuÞ ¼ a ð u Þ ¼ a ; # 2 2 2 2 3 sin u ðk sin u  1Þ k sin u  1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 dRu 3ak sin u cos u k  1 k sin u  1 ¼ : 2 2 du ðk sin u  1Þ3 ð4Þ The catenary meridian (Fig. 3) can be described with the following parametric equation:

    R0  Rb x03 ¼ d cosh 1 ; d

ð5Þ

where d is the curvature radius at the apex of the catenary curve. The horizontal radius R0(u) of a generic parallel of the catenary shell assumes the form:

R0 ðuÞ ¼ d arcsinhðtan uÞ þ Rb :

d ; cos2 u dRu 2d sin u ¼ : du cos3 u

R# ðuÞ ¼

d arcsinhðtan uÞ þ Rb ; sin u

The cycloid meridian (Fig. 4) can be described with the following parametric equations:

ð8Þ

where rc is the radius of the generator circle of the cycloid curve. For the cycloid shell of revolution, the radii of curvature Ru(u), R#(u) in the meridional and circumferential directions, and the first derivative of Ru(u) with respect to u can be expressed, respectively, as follows:

Ru ðuÞ ¼ 4rc cos u;

R# ðuÞ ¼

dRu ¼ 4rc sin u: du

r c ð2u þ sin 2uÞ Rb þ ; sin u sin u ð9Þ

The parabolic meridian (Fig. 5) can be described with the following equation: 2

ðR0  Rb Þ 

0 kx3 2

¼ 0;

ð10Þ

2

where k = (a  d )/b is a characteristic parameter of the parabolic curve. The horizontal radius R0(u) of a generic parallel of the parabolic shell assumes the form:

R0 ðuÞ ¼

k tan u þ Rb : 2

R# ðuÞ ¼

k Rb þ ; 2 cos u sin u

2 2 ðR0  Rb Þ2 þ k b  x03 ¼ a2 ;

ð11Þ

For the parabolic shell of revolution, the radii of curvature Ru(u), R#(u) in the meridional and circumferential directions, and the first derivative of Ru(u) with respect to u can be expressed, respectively, as follows:

dRu 3k sin u ¼ : du 2 cos2 u ð12Þ

ð13Þ

where a, b and k = a/b are the semimajor and semiminor axis of the elliptic curve and their ratio, respectively. The horizontal radius R0(u) of a generic parallel of the elliptic shell assumes the form:

ak tan u R0 ðuÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ Rb : 2 1 þ k tan2 u

ð14Þ

For the elliptic shell of revolution, the radii of curvature Ru(u), R#(u) in the meridional and circumferential directions, and the first derivative of Ru(u) with respect to u can be expressed, respectively, as follows:

Ru ðuÞ ¼

ak rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  3 ; 2 2 3 1 þ k tan u cos u

ak R qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ b ; sin u 2 2 cos u 1 þ k tan u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 3ak sin u ð1 þ k tan2 uÞ3 ð1  k Þ

R# ð u Þ ¼

dRu ¼ du

2

cos4 uð1 þ k tan2 uÞ3

ð15Þ

:

Finally, for doubly-curved shells of revolution the Gauss–Codazzi relation can be expressed as follows:

dR0 ¼ Ru cos u: du ð7Þ

x03 ¼ r c ð1  cos 2uÞ; R0 ¼ rc ð2u þ sin 2uÞ þ Rb ;

k ; 2 cos3 u

The elliptic meridian (Fig. 6) can be described with the following equation:

ð6Þ

For the catenary shell of revolution, the radii of curvature Ru(u), R#(u) in the meridional and circumferential directions, and the first derivative of Ru(u) with respect to u can be expressed, respectively, as follows:

Ru ðuÞ ¼

Ru ðuÞ ¼

ð16Þ

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 = en(u, s, f, t) = 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: rn = rn(u, s, f, t) = 0; (5) the linear elastic behaviour of anisotropic materials is assumed; (6) the rotary inertia is also taken into account. Consistent with the assumptions of a moderately thick shell theory reported above, the displacement field considered in this study is that of the First-order Shear Deformation Theory and can be put in the following form:

U u ðu; s; f; tÞ ¼ uu ðu; s; tÞ þ fbu ðu; s; tÞ; U s ðu; s; f; tÞ ¼ us ðu; s; tÞ þ fbs ðu; s; tÞ;

ð17Þ

Wðu; s; f; tÞ ¼ wðu; s; tÞ; where uu, us, w are the displacement components of points lying on the middle surface (f = 0) of the shell, along meridional, circumferential and normal directions, respectively, while t is the time variable. bu 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 w(u, s, t)  h. It is worth noting that in-plane displacements Uu and Us vary linearly through the thickness, while W remains independent of f. Relationships between strains and displacements along the shell reference surface (f = 0) are as follows:

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F. Tornabene / Comput. Methods Appl. Mech. Engrg. 200 (2011) 931–952



 1 @uu @us uu cos u w sin u e0u ¼ þ w ; e0s ¼ þ ; þ Ru @ u @s R0 R0 1 @us @uu us cos u 1 @bu c0us ¼ þ ; vu ¼ ;  Ru @ u Ru @ u @s R0 b cos u @b 1 @bs @bu bs cos u vs ¼ s þ u ; vus ¼ þ ;  Ru @ u @s R0 @s R0   1 @w @w us sin u  cun ¼  uu þ bu ; csn ¼ þ bs : Ru @ u @s R0

ðkÞ

Q 26

ðkÞ

ð18Þ

3

2

A11 Nu 6 N s 7 6 A12 7 6 6 7 6 6 6 Nus 7 6 A16 7 6 6 6 M 7 6B 6 u 7 6 11 7¼6 6 6 Ms 7 6 B12 7 6 6 6M 7 6B 6 us 7 6 16 7 6 6 4 Tu 5 4 0 Ts

A12

A16

B11

B12

B16

0

A22

A26

B12

B22

B26

0

A26

A66

B16

B26

B66

0

B12

B16

D11

D12

D16

0

B22

B26

D12

D22

D26

0

B26

B66

D16

D26

D66

0

jA44 jA45

0

0

0

0

0

0

0

0

0

0

0

32 0 3 eu 0 6 0 7 0 7 e 7 76 76 0s 7 7 0 76 c 7 76 6 us 7 7 0 76 vu 7 7; 6 76 0 7 vs 7 7 76 7 6 0 7 vus 7 76 7 6 76 jA45 54 cun 7 5 jA55 csn ð19Þ

where j is the shear correction factor, which is usually taken as j = 5/6, such as in the present work. In particular, it is worth noting that the determination of shear correction factors for composite laminated structures is still an unresolved issue, because these factors depend on various parameters [71–73]. In Eq. (19), the three components Nu, Ns, Nus are in-plane meridional, circumferential and shearing force resultants, and Mu, Ms, Mus are the analogous couples, while Tu, Ts are the transverse shear force resultants. We notice that, in the above definitions (19) the symmetry of shearing force resultants Nus, Nsu and torsional couples Mus, Msu is assumed as a further hypothesis, even if it is satisfied only in the case of spherical shells and flat plates [22]. This assumption is derived from the consideration that ratios f/Ru, f/Rs can be neglected with respect to unity (f/Ru, f/Rs  1). The extensional stiffnesses Aij, the bending stiffnesses Dij and the bending–extensional coupling stiffnesses Bij are defined as:

Aij ¼

l Z X k¼1

Dij ¼

fkþ1

fk

l Z X k¼1

ðkÞ

Q ij df;

Bij ¼

l Z X k¼1

fkþ1

fk

fkþ1

fk

ðkÞ

ðkÞ

ðkÞ

2

ðkÞ

ðkÞ

ðkÞ

2

Q 44 ¼ Q 44 cos2 hðkÞ þ Q 55 sin hðkÞ ;   ðkÞ ðkÞ ðkÞ Q 45 ¼ Q 44  Q 55 cos hðkÞ sin hðkÞ ;

In the above Eq. (18), the first three strains e0u ; e0s ; c0us are in-plane meridional, circumferential and shearing components, and vu, vs, vus are the analogous curvature changes. The last two components cun, csn are transverse shearing strains. The shell material assumed in the following is a laminated composite linear elastic one. Accordingly, the following constitutive equations relate internal stress resultants and internal couples with generalized strain components (18) on the middle surface:

2

Q 66

  3 ðkÞ ðkÞ ðkÞ ¼ Q 11  Q 12  2Q 66 sin hðkÞ cos hðkÞ   ðkÞ ðkÞ ðkÞ þ Q 12  Q 22 þ 2Q 66 sin hðkÞ cos3 hðkÞ ;   2 ðkÞ ðkÞ ðkÞ ðkÞ ¼ Q 11 þ Q 22  2Q 12  2Q 66 sin hðkÞ cos2 hðkÞ   4 ðkÞ þ Q 66 sin hðkÞ þ cos4 hðkÞ ;

Q 55 ¼ Q 55 cos2 hðkÞ þ Q 44 sin hðkÞ ;

ð21Þ

(k)

where h is the orientation angle of the principal material co^ ^s^f of the kth orthotropic ply with respect to ordinate system O0 u the laminate co-ordinate system O0 usf, as shown in Fig. 7. ðkÞ Furthermore, the elastic constants Q ij in the material co0 ^ ^ ^ ordinate system O usf are expressed as follows: ðkÞ

ðkÞ Q 66

ðkÞ

E1

ðkÞ

Q 11 ¼

ðkÞ ðkÞ 12 21 ðkÞ Q 44

1m ðkÞ G12 ;

¼

m

E2

ðkÞ

Q 22 ¼

; ¼

ðkÞ ðkÞ 1  12 21 ðkÞ ðkÞ Q 55 ¼ G23 ;

ðkÞ G13 ;

m m

;

ðkÞ

Q 12 ¼

ðkÞ mðkÞ 12 E2 ; ðkÞ ðkÞ 1  m12 m21

ð22Þ

where E1, E2, G13, G23, G12, m12 are the engineering parameters of the kth lamina. It should be noted that for a complete characterization of an orthotropic material, parameters E3, m13, m23 have to be taken into account as well. Following the direct approach or the virtual work principle in dynamic version and remembering the Gauss–Codazzi relations for the shells of revolution (16), five equations of dynamic equilibrium in terms of internal actions can be written for the revolution shell element:

1 Ru 1 Ru 1 Ru 1 Ru 1 Ru

 cos u T u @Nu @Nus  €u ; € u þ I1 b þ þ ¼ I0 u þ Nu  Ns @u @s R0 Ru @Nus @Ns cos u sin u €s ; € s þ I1 b þ þ Ts ¼ I0 u þ 2Nus @u @s R0 R0 @T u @T s cos u Nu sin u € þ   Ns ¼ I0 w; þ Tu @u @s R0 Ru R0  cos u @M u @Mus  €u ; € u þ I2 b þ  T u ¼ I1 u þ Mu  Ms @u @s R0 @M us @M s cos u €s ; € s þ I2 b þ  T s ¼ I1 u þ 2M us @u @s R0

ð23Þ

where:

Ii ¼

l Z X k¼1

fkþ1



qðkÞ fi 1 þ

fk

f Ru

 1þ

 f df; R#

i ¼ 0; 1; 2;

ð24Þ

ζ ≡ ζˆ

ðkÞ

Q ij fdf;

ðkÞ Q ij f2 df:

ð20Þ ðkÞ

For the kth orthotropic lamina the elastic constants Q ij in the laminate co-ordinate system Os0u sf can be written as:

Q 11

  2 ðkÞ ðkÞ ðkÞ ¼ Q 11 cos4 hðkÞ þ 2 Q 12 þ 2Q 66 sin hðkÞ cos2 hðkÞ

ðkÞ Q 12

þ Q cos4 hðkÞ ;  22  2 ðkÞ ðkÞ ðkÞ ¼ Q 11 þ Q 22  4Q 66 sin hðkÞ cos2 hðkÞ

ðkÞ

O′

ðkÞ

ðkÞ

ðkÞ

Q 22

ðkÞ

ðkÞ

Q 16

4

þ Q 12 ðsin hðkÞ þ cos4 hðkÞ Þ;   4 2 ðkÞ ðkÞ ðkÞ ¼ Q 11 sin hðkÞ þ 2 Q 12 þ 2Q 66 sin hðkÞ cos2 hðkÞ þ Q cos4 hðkÞ ;  22  ðkÞ ðkÞ ðkÞ ¼ Q 11  Q 12  2Q 66 sin hðkÞ cos3 hðkÞ   3 ðkÞ ðkÞ ðkÞ þ Q 12  Q 22 þ 2Q 66 sin hðkÞ cos hðkÞ ;

ϕ

sˆ

θ

θ

s

ϕˆ Fig. 7. A lamina with material and laminate co-ordinate systems.

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F. Tornabene / Comput. Methods Appl. Mech. Engrg. 200 (2011) 931–952

are the mass inertias and q(k) is the mass density of the material per unit volume of the kth ply. The first three equation (23) represent translational equilibriums along meridional u, circumferential s and normal f directions, while the last two are rotational equilibrium equations about the s and u directions, respectively. The three basic sets of equations, namely the kinematic (18), constitutive (19) and equilibrium (23) equations may be combined to give the fundamental system of equations, also known as the governing system of equations. By replacing the kinematic equation (18) into the constitutive equation (19) and the result of this substitution into the equilibrium equation (23), the complete equations of motion in terms of displacements can be written as:

2

L11

6 6 L21 6 6 L31 6 6 4 L41 L51

L12

L13

L14

L22

L23

L24

L32

L33

L34

L42

L43

L52

L53

L15

32

uu

3

2

I0

0

0

I0

0

0

0

I0

0

L44

76 7 6 L25 76 us 7 6 0 76 7 6 6 7 6 L35 7 76 w 7 ¼ 6 0 76 7 6 L45 54 bu 5 4 I1

I1

0

0

I2

L54

L55

0

I1

0

0

bs

3 €u u 76 € 7 I1 76 u s 7 76 7 6w € 7; 07 76 7 76 € 7 0 54 b u5 €s I2 b 0

32

ð25Þ

complete shell of revolution (Fig. 8(a)). The kinematic compatibility conditions include the continuity of displacements. The physical compatibility conditions can only be the five continuous conditions for the generalized stress resultants. To consider complete revolute shells characterized by s0 = 2pR0, it is necessary to implement the kinematic and physical compatibility conditions between the two computational meridians with s = 0 and with s0 = 2pR0: Kinematic compatibility conditions along the closing meridian (s = 0, 2pR0)

uu ðu; 0; tÞ ¼ uu ðu; s0 ; tÞ;

us ðu; 0; tÞ ¼ us ðu; s0 ; tÞ;

wðu; 0; tÞ ¼ wðu; s0 ; tÞ;

bu ðu; 0; tÞ ¼ bu ðu; s0 ; tÞ;

bs ðu; 0; tÞ ¼ bs ðu; s0 ; tÞ;

u0 6 u 6 u1 :

Physical compatibility (s = 0, 2pR0)

conditions

Nus ðu; 0; tÞ ¼ Nus ðu; s0 ; tÞ; M s ðu; 0; tÞ ¼ Ms ðu; s0 ; tÞ;

M us ðu; 0; tÞ ¼ M us ðu; s0 ; tÞ;

meridian

ð31Þ

u0 6 u 6 u1 :

In analogous way, in order to consider a toroidal shell of revolution (Fig. 8(b)) it is necessary to implement the kinematic and physical compatibility conditions between the two computational parallels with u0 = 0 and with u1 = 2p:

Clamped edge boundary conditions (C)

uu ð0; s; tÞ ¼ uu ð2p; s0 ; tÞ;

ð26Þ

closing

T s ðu; 0; tÞ ¼ T s ðu; s0 ; tÞ;

Kinematic compatibility (u = 0, 2p)

6 s0 ;

the

Ns ðu; 0; tÞ ¼ Ns ðu; s0 ; tÞ;

where the explicit forms of the equilibrium operators Lij, i, j = 1, . . . , 5 are listed in Appendix A. 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:

uu ¼ us ¼ w ¼ bu ¼ bs ¼ 0 at u ¼ u0 or u ¼ u1 ; 0 6 s

along

ð30Þ

conditions

along

the

closing

parallel

us ð0; s; tÞ ¼ us ð2p; s0 ; tÞ;

wð0; s; tÞ ¼ wð2p; s0 ; tÞ;

bu ð0; s; tÞ ¼ bu ð2p; s0 ; tÞ;

bs ð0; s; tÞ ¼ bs ð2p; s0 ; tÞ;

0 6 s 6 s0 :

ð32Þ

Physical compatibility conditions along the closing parallel (u = 0, 2p)

uu ¼ us ¼ w ¼ bu ¼ bs ¼ 0 at s ¼ 0 or s ¼ s0 ; u0 6 u 6 u1 :

ð27Þ

Nu ð0; s; tÞ ¼ Nu ð2p; s0 ; tÞ;

Nus ð0; s; tÞ ¼ Nus ð2p; s0 ; tÞ;

T u ð0; s; tÞ ¼ T u ð2p; s0 ; tÞ;

M u ð0; s; tÞ ¼ Mu ð2p; s0 ; tÞ;

M us ð0; s; tÞ ¼ M us ð2p; s0 ; tÞ;

Free edge boundary conditions (F)

Nu ¼ Nus ¼ T u ¼ M u ¼ Mus ¼ 0 at u ¼ u0 or u ¼ u1 ; 0 6 s 6 s0 ;

ð28Þ

Ns ¼ Nus ¼ T s ¼ Ms ¼ M us ¼ 0 at s ¼ 0 or s ¼ s0 ; u0 6 u 6 u1 :

ð29Þ

In addition to the external boundary conditions, the kinematic and physical compatibility conditions should be satisfied at the common closing meridians with s = 0, 2pR0, if we want to consider a

ð33Þ

0 6 s 6 s0 :

3. Discretized equations and numerical implementation The Generalized Differential Quadrature method will be used to discretize the derivatives in the governing equations in terms of displacements as well as boundary and compatibility conditions. Since a brief review of the GDQ Method is presented in Tornabene [65], the same approach is used in the present work about the GDQ technique.

Closing parallel ϕ = 0, 2π

Closing meridian s = 0, 2π R 0

(b) (a) Fig. 8. Common meridians of a complete revolution shell (a) and common parallels of a toroidal shell (b).

F. Tornabene / Comput. Methods Appl. Mech. Engrg. 200 (2011) 931–952

þ A26

M X

937

s 1sð2Þ jm U im

m¼1

  M A22 cos ui A66 cos ui X s þ 1sð1Þ jm U im R0i R0i m¼1   N M X A12 A66 X s þ þ 1ikuð1Þ 1sð1Þ jm U km Rui Rui k¼1 m¼1 

! A16 sin ui A26 cos2 ui A45 sin ui þ þ j U sij Rui R0i Rui R0i R20i ! N A11 A12 sin ui A44 X þ þ þ j 1ikuð1Þ W kj Rui R0i R2ui R2ui k¼1   M A16 A26 sin ui A45 X þ þ þj 1sð1Þ W im Rui R0i Rui m¼1 jm ! ! cos ui 1 dRu A22 sin ui cos ui  W ij þ A11  Rui R0i R3ui du i R20i

Fig. 9. C–G–L grid distribution on a parabolic shell.

Throughout the paper, the Chebyshev–Gauss–Lobatto (C–G–L) grid distribution is assumed, for which the co-ordinates of grid points (ui, sj) along the reference surface are:



ui ¼ 1  cos



i1 p N1



ðu1  u0 Þ þ u0 ; 2

i ¼ 1; 2; . . . ; N; for u 2 ½u0 ; u1 ;    j1 s0 sj ¼ 1  cos p ; M1 2

þ ð34Þ

N B11 X

R2ui

þ B11

j ¼ 1; 2; . . . ; M; for s 2 ½0; s0  ðwith s 6 #R0 Þ; where N, M are the total number of sampling points used to discretize the domain in u and s directions, respectively, of the doublycurved shell (Fig. 9). 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 x. The displacement field can be written as follows: u

ix t

uu ðu; s; tÞ ¼ U ðu; sÞe

;

wðu; s; tÞ ¼ Wðu; sÞeixt ;

ð35Þ

bu ðu; s; tÞ ¼ Bu ðu; sÞeixt ; bs ðu; s; tÞ ¼ Bs ðu; sÞeixt ; where the vibration spatial amplitude values Uu, Us, W, Bu, Bs fulfil 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. The governing equations can be discretized and for the domain points, i = 2, 3, . . . , N  1, j = 2, 3, . . . , M  1, we have: (1) Translational equilibrium along the meridional direction u

2

Rui

1ikuð2Þ U ukj þ A11

k¼1 M X

! N cos ui 1 dRu X  3 1uð1Þ U ukj Rui R0i Rui du i k¼1 ik

N M X 2A16 X sð1Þ u þ A66 1 þ 1ikuð1Þ 1jm U km R u i m¼1 m¼1 k¼1 ! A12 sin ui A22 cos2 ui A44 A16 Uu  þ þ j ij þ 2 Rui R0i R20i R2ui Rui ! N N X uð2Þ s A cos ui A16 dRu X uð1Þ s  1ik U kj  26 þ 3 1ik U kj Rui R0i Rui du i k¼1 k¼1 sð2Þ u jm U im

! N cos ui 1 dRu X  3 1uð1Þ Bu þ B66 Rui R0i Rui du i k¼1 ik kj

N M X 2B16 X u 1ikuð1Þ 1sð1Þ jm Bkm Rui k¼1 m¼1 m¼1 ! B12 sin ui B22 cos2 ui A44 u B16 B þ  þ  j Rui R0i Rui ij R2ui R20i ! N N X B26 cos ui B16 dRu X uð2Þ s  1ik Bkj  þ 3 1ikuð1Þ Bskj Rui R0i Rui du i k¼1 k¼1



þ B26

u 1sð2Þ jm Bim þ

M X

sð2Þ s 1jm Bim

m¼1

  M B22 cos ui B66 cos ui X s þ 1sð1Þ jm Bim R0i R0i m¼1   N M X B12 B66 X uð1Þ þ þ 1 1sð1Þ Bs Rui Rui k¼1 ik m¼1 jm km ! B16 sin ui B26 cos2 ui A45 s B þ þ þj Rui R0i Rui ij R20i   u ¼ x2 I0 U u ij þ I 1 Bij : 

us ðu; s; tÞ ¼ U s ðu; sÞeixt ;

N A11 X

M X

1ikuð2Þ Bukj

k¼1

ð36Þ

(2) Translational equilibrium along the circumferential direction s N A16 X

R2ui

k¼1

þ

1ikuð2Þ U ukj ! N 2A16 cos ui A26 cos ui A16 dRu X þ  3 1uikð1Þ U ukj Rui R0i Rui R0i Rui du i k¼1

þ A26

M X m¼1

u 1sð2Þ jm U im

  M A22 cos ui A66 cos ui X u þ þ 1sð1Þ jm U im R0i R0i m¼1   N M X A12 A66 X uð1Þ þ þ 1 1sð1Þ U u Rui Rui k¼1 ik m¼1 jm km ! ! cos2 ui sin ui A45 sin ui A66 j Uu þ A26  ij þ 2 Rui R0i Rui R0i R20i Rui

938

F. Tornabene / Comput. Methods Appl. Mech. Engrg. 200 (2011) 931–952



N X

uð2Þ

1ik

U skj

þ A66

k¼1

N M X 2A45 X 1uikð1Þ 1sð1Þ jm W km Rui k¼1 m¼1 m¼1 ! 2 A11 2A12 sin ui A22 sin ui  þ þ 2 2 Rui R0i Rui R0i ! N B12 sin ui A44 X  B11 þ j 1ikuð1Þ Bukj  W ij 2 R R R ui 0i ui k¼1 Rui  X M B16 B26 sin ui u  þ  jA45 1sð1Þ jm Bim Rui R0i m¼1

sð2Þ s 1jm U im þ

m¼1

M X

N M X 2B26 X sð1Þ s 1ikuð1Þ 1jm Bkm R u i m¼1 m¼1 k¼1 ! ! cos2 ui sin ui A55 sin ui s j Bij  B66  Rui R0i R0i R20i   ¼ x2 I0 U sij þ I1 Bsij :

! B12 cos ui B22 sin ui cos ui A44 cos ui u þ  j Bij Rui R0i R0i R20i ! N B16 B26 sin ui A45 X  þ  j 1uð1Þ Bs Rui R0i Rui k¼1 ik kj R2ui  X M B12 B22 sin ui s  þ  jA55 1sð1Þ jm Bim Rui R0i m¼1 ! B16 cos ui B26 sin ui cos ui A45 cos ui s þ þ þj Bij Rui R0i R0i R20i ¼ x2 I0 W ij :

N B11 X

R2ui

sð2Þ s 1jm Bim þ

ð37Þ

! N A26 sin ui A45 sin ui X  þ þj 1ikuð1Þ U skj 2 Rui R0i Rui R0i Rui k¼1   M A12 A22 sin ui A55 sin ui X sð1Þ s  þ þj 1jm U im Rui R0i R0i m¼1

A44 R2ui

k¼1

þ jA44

1ikuð2Þ W kj

! N cos ui 1 dRu X  3 1uð1Þ W kj þ jA55 Rui R0i Rui du i k¼1 ik

u 1sð2Þ jm U im þ

m¼1

! B16 sin ui B26 cos2 ui A45 sin ui U sij þ þ þj Rui R0i R0i R20i ! N B11 B12 sin ui A44 X þ þ j 1uð1Þ W kj 2 Rui R0i Rui k¼1 ik Rui  X M B16 B26 sin ui þ þ  jA45 1sð1Þ jm W im Rui R0i m¼1 ! ! cos ui 1 dRu B22 sin ui cos ui  W ij þ B11  Rui R0i R3ui du i R20i N D11 X þ 2 1ikuð2Þ Bukj Rui k¼1 ! N cos ui 1 dRu X þ D11  3 1uð1Þ Bu þ D66 Rui R0i Rui du i k¼1 ik kj

A16

þj

M X

! N cos ui 1 dRu X  1uð1Þ U ukj Rui R0i R3ui du i k¼1 ik

 M B22 cos ui B66 cos ui X s  þ 1sð1Þ jm U im R0i R0i m¼1   N M X B12 B66 X sð1Þ s þ þ 1ikuð1Þ 1jm U km Rui Rui k¼1 m¼1

A12 cos ui A22 sin ui cos ui þ Rui R0i R20i !! cos ui 1 dRu Uu  þjA44 ij Rui R0i R3ui du i

N X

1ik U kj þ B11

k¼1





! A16 cos ui A26 sin ui cos ui A45 cos ui U sij þ  j Rui R0i Rui R0i R20i

u

N M X 2B16 X sð1Þ u 1ikuð1Þ 1jm U km R u i m¼1 m¼1 k¼1 ! B12 sin ui B22 cos2 ui A44 B16  þ j Uu ij þ 2 Rui R0i Rui R20i Rui !X N N X dR B cos u B u 16 i  1uikð2Þ U skj  26 þ 3 1uikð1Þ U skj Rui R0i Rui du i k¼1 k¼1 M X s þ B26 1sð2Þ jm U im

þ

þ

uð2Þ

þ B66

! N A12 sin ui A44 X þ j 1ikuð1Þ U ukj 2 2 Rui R0i Rui Rui k¼1   M A16 A26 sin ui A45 X  þ þj 1sð1Þ U u Rui R0i Rui m¼1 jm im

A11

ð38Þ

(4) Rotational equilibrium about the meridional direction s

(3) Translational equilibrium along the normal direction f



1sð2Þ jm W im þ j



  M B22 cos ui B66 cos ui X sð1Þ u þ þ 1jm Bim R0i R0i m¼1   N M X B12 B66 X þ þ 1uð1Þ 1sð1Þ Bu Rui Rui k¼1 ik m¼1 jm km ! ! cos2 ui sin ui A45 sin ui u B66 þj Bij þ 2 þ B26  Rui R0i R0i R20i Rui ! X N N X dR cos u 1 u i  1ikuð2Þ Bskj þ B66  1uð1Þ Bs Rui R0i R3ui du i k¼1 ik kj k¼1 þ B22

M X



N M X 2A26 X sð1Þ s 1ikuð1Þ 1jm U km R u i m¼1 m¼1 k¼1 ! ! 2 cos2 ui sin ui A55 sin ui  A66  j þ U sij Rui R0i R20i R20i ! N A16 A26 sin ui A45 sin ui X þ þ þj 1ikuð1Þ W kj 2 R R R R ui 0i ui 0i Rui k¼1   M A12 A22 sin ui A55 sin ui X sð1Þ þ þ þj 1jm W im Rui R0i R0i m¼1 ! sin ui cos ui cos ui þ A26 þ Rui R0i R20i !! N 2 cos ui 1 dRu B16 X þA16  3 1ikuð2Þ Bukj W ij þ 2 Rui R0i Rui du i Rui k¼1 !! X N B26 cos ui 2 cos ui 1 dRu þ þ B16  3 1ikuð1Þ Bukj Rui R0i Rui R0i d u Rui i k¼1 M X sð2Þ u þ B26 1jm Bim

þ A22

M X

! N cos ui 1 dRu X  1uð1Þ U skj Rui R0i R3ui du i k¼1 ik



M X

u 1sð2Þ jm Bim þ

m¼1

N M X 2D16 X u 1ikuð1Þ 1sð1Þ jm Bkm Rui k¼1 m¼1

Table 1 Finite element shell types used in commercial programs. FEM

Abaqus

Ansys

Straus

Nastran

Pro/ Mechanica

Number of nodes Element type

8 S8R6

8 SHELL99

8 Quad8

4 CQUAD4

4 GEM-Quad4

939

F. Tornabene / Comput. Methods Appl. Mech. Engrg. 200 (2011) 931–952

!

D12 sin ui D22 cos2 ui D16 þ þ jA44 Bu ij þ 2 2 Rui R0i R0i Rui ! N N X X D cos ui D16 dRu  1ikuð2Þ Bskj  26 þ 3 1ikuð1Þ Bskj Rui R0i Rui du i k¼1 k¼1 M X

sð2Þ s 1jm Bim

m¼1

  M D22 cos ui D66 cos ui X sð1Þ s þ 1jm Bim R0i R0i m¼1   N M X D12 D66 X sð1Þ s þ þ 1ikuð1Þ 1jm Bkm Rui Rui k¼1 m¼1 ! D16 sin ui D26 cos2 ui þ þ  j A Bsij 45 Rui R0i R20i   u ¼ x2 I1 U u ij þ I2 Bij : 

ð39Þ

(5) Rotational equilibrium about the circumferential direction u N B16 X

R2ui

1ikuð2Þ U ukj

þ

k¼1

þ

B26 cos ui þ B16 Rui R0i

þ B26

M X

sin ui Rui R0i

!

A45 B66 Uu ij 2 Rui Rui ! N N X cos ui 1 dRu X uð2Þ s  1ik U kj þ B66  3 1ikuð1Þ U skj R R d u u i 0i R i ui k¼1 k¼1 M N M X X 2B26 X sð2Þ s uð1Þ sð1Þ s þ B22 1jm U im þ 1 1 U Rui k¼1 ik m¼1 jm km m¼1 ! ! cos2 ui sin ui A55 sin ui  B66  j U sij Rui R0i R0i R20i ! N B16 B26 sin ui A45 X þ þ  j 1uð1Þ W kj 2 Rui R0i Rui k¼1 ik Rui  X M B12 B22 sin ui þ þ  jA55 1sð1Þ jm W im Rui R0i m¼1 ! sin ui cos ui cos ui þ B26 þ Rui R0i R20i !! 2 cos ui 1 dRu W ij þB16  3 Rui R0i Rui du i þ B26



þ D26

cos2 ui

!

!! X N 2 cos ui 1 dRu  3 1ikuð1Þ U ukj Rui R0i Rui du i k¼1

R20i

N D16 X

R2ui



þj

1ikuð2Þ Bukj

k¼1

!! X N D26 cos ui 2 cos ui 1 dRu þ þ D16  3 1ikuð1Þ Bukj Rui R0i Rui R0i Ru i d u i k¼1

u 1sð2Þ jm U im

þ D26

m¼1

M X

u 1sð2Þ jm Bim

m¼1

  M D22 cos ui D66 cos ui X sð1Þ u þ 1jm Bim R0i R0i m¼1  X M D12 D66 N uð1Þ X u þ þ 1ik 1sð1Þ jm Bkm Rui Rui k¼1 m¼1

  M B22 cos ui B66 cos ui X sð1Þ u þ 1jm U im R0i R0i m¼1   N M X B12 B66 X þ þ 1uð1Þ 1sð1Þ U u Rui Rui k¼1 ik m¼1 jm km

þ

þ

Table 2 First ten frequencies for a C–C–C–C single layer orthotropic hyperbolic panel with h = 30°. Mode [Hz]

GDQ 31  31

Nastran 100  60 (4 nodes)

Abaqus 60  40 (8 nodes)

Ansys 60  40 (8 nodes)

Straus 60  40 (8 nodes)

Pro/Mechanica 31  21 (GEM)

f1 f2 f3 f4 f5 f6 f7 f8 f9 f10

195.729 233.480 277.392 304.990 318.883 351.754 362.594 372.420 400.404 416.361

195.130 233.349 276.901 304.599 319.453 352.753 362.485 371.312 401.002 418.753

195.196 233.499 277.113 304.756 319.493 352.944 362.612 371.400 401.222 419.038

194.896 232.797 276.227 303.811 318.455 351.224 361.600 370.610 399.389 415.984

195.249 233.634 276.744 304.762 318.786 352.178 361.795 371.322 400.024 417.657

194.860 232.779 276.187 303.769 318.425 351.188 361.536 370.559 399.358 415.960

Geometric characteristics: a = 1 m, d = 2 m, D = 4 m, C = 1 m, Rb = 0 m, h = 0.1 m, #0 = 120°. Material properties: E1 = 137.9 GPa, E2 = 8.96 GPa, G12 = G13 = 7.1 GPa, G23 = 6.21 GPa, m12 = 0.3, q = 1450 Kg/m3.

Table 3 First ten frequencies for a C–F asymmetric laminated hyperbolic shell (30/60). Mode [Hz]

GDQ 31  31

Nastran 100  160 (4 nodes)

Abaqus 30  80 (8 nodes)

Ansys 30  80 (8 nodes)

Straus 60  160 (8 nodes)

Pro/Mechanica 31  82 (GEM)

f1 f2 f3 f4 f5 f6 f7 f8 f9 f10

55.374 55.374 58.372 58.372 68.060 68.060 77.457 77.457 82.409 82.409

54.728 54.728 57.951 57.957 67.537 67.573 77.428 77.428 82.195 82.195

54.741 54.741 57.970 57.970 67.550 67.550 77.383 77.383 82.230 82.230

54.709 54.814 58.029 58.038 67.418 67.677 77.335 77.507 82.251 82.324

55.445 55.445 58.752 58.752 68.418 68.418 78.056 78.056 83.405 83.405

54.719 54.719 57.998 57.998 67.519 67.519 77.417 77.417 82.272 82.272

Geometric characteristics: a = 1 m, d = 2 m, D = 4 m, C = 1 m, Rb = 2 m, #0 = 360°, h1 = h2 = 0.05 m. Material properties: E1 = 137.9 GPa, E2 = 8.96 GPa, G12 = G13 = 7.1 GPa, G23 = 6.21 GPa, m12 = 0.3, q = 1450 Kg/m3.

940

F. Tornabene / Comput. Methods Appl. Mech. Engrg. 200 (2011) 931–952

þ D26

cos2 ui R20i

sin ui  Rui R0i

!

!  jA45 Bu ij þ

D66

R2ui ! N N X cos ui 1 dRu X  1ikuð2Þ Bskj þ D66  3 1ikuð1Þ Bskj R R d u ui 0i Rui i k¼1 k¼1 M N M X X 2D26 X sð1Þ s s þ D22 1sð2Þ 1ikuð1Þ 1jm Bkm jm Bim þ R u i m¼1 m¼1 k¼1 ! ! cos2 ui sin ui  D66  þ jA55 Bsij Rui R0i R20i   ð40Þ ¼ x2 I1 U sij þ I2 Bsij :

uð1Þ

sð1Þ

uð2Þ

sð2Þ

In Eqs. (36)–(40), 1ik ; 1jm ; 1ik and 1jm are the weighting coefficients of the first and second derivatives in u and s directions, respectively. Furthermore, N and M are the total number of grid points in u and s directions. By applying the GDQ methodology, the discretized forms of the boundary (26)–(29) and compatibility (30) and (31) conditions are given as follows: Clamped edge boundary condition (C) s ¼ u s Uu aj ¼ U aj ¼ W aj Baj ¼ Baj ¼ 0;

for a ¼ 1; N and j ¼ 1; 2; . . . ; M;

s ¼ u s Uu ib ¼ U ib ¼ W ib Bib ¼ Bib ¼ 0;

for b ¼ 1; M and i ¼ 1; 2; . . . ; N: ð41Þ

Table 4 First ten frequencies for a C–C–F–C symmetric laminated catenary panel (45/0)s. Mode [Hz]

GDQ 31  31

Nastran 80  80 (4 nodes)

Abaqus 40  40 (8 nodes)

Ansys 40  40 (8 nodes)

Straus 40  40 (8 nodes)

Pro/Mechanica 21  21 (GEM)

f1 f2 f3 f4 f5 f6 f7 f8 f9 f10

271.278 320.540 330.540 365.049 384.173 400.626 417.113 457.612 468.161 507.421

271.209 320.476 330.797 365.301 384.716 401.264 417.496 458.752 469.210 508.155

271.476 320.943 330.918 365.931 385.165 402.269 418.235 460.310 470.355 509.851

270.286 318.956 329.660 363.302 381.816 398.495 414.350 454.363 464.474 501.521

271.836 321.397 330.807 365.651 385.620 401.491 418.674 459.342 469.699 511.288

270.306 319.127 329.680 363.472 382.022 398.700 414.615 454.724 464.833 502.068

Geometric characteristics: d = 2 m, Rb = 0 m, u 2 [15°, 60°], #0 = 120°, h1 = h2 = h3 = h4 = 0.025 m. Material properties: E1 = 137.9 GPa, E2 = 8.96 GPa, G12 = G13 = 7.1 GPa, G23 = 6.21 GPa, m12 = 0.3, q = 1450 Kg/m3.

Table 5 First ten frequencies for a F–C asymmetric laminated cycloidal shell (45/20/70/45). Mode [Hz]

GDQ 31  31

Nastran 40  160 (4 nodes)

Abaqus 40  120 (8 nodes)

Ansys 40  120 (8 nodes)

Straus 40  120 (8 nodes)

Pro/Mechanica 21  82 (GEM)

f1 f2 f3 f4 f5 f6 f7 f8 f9 f10

43.875 43.875 45.513 45.513 47.560 47.560 49.004 49.004 51.316 60.100

43.695 43.695 45.432 45.432 47.538 47.538 48.849 48.849 51.319 59.992

43.754 43.754 45.519 45.519 47.650 47.650 48.941 48.941 51.503 60.144

42.764 42.780 44.591 44.675 46.760 46.847 47.451 47.461 50.456 57.863

43.381 43.381 44.977 44.977 47.309 47.309 48.544 48.544 51.471 59.870

43.709 43.709 45.448 45.448 47.563 47.563 48.877 48.877 51.362 60.044

Geometric characteristics: rc = 1 m, Rb = 5 m, u 2 [70°, 5°], #0 = 360°, h1 = h2 = h3 = h4 = 0.025 m. Material properties: E1 = 137.9 GPa, E2 = 8.96 GPa, G12 = G13 = 7.1 GPa, G23 = 6.21 GPa, m12 = 0.3, q = 1450 Kg/m3.

Table 6 First ten frequencies for a C–F–C–F symmetric laminated parabolic shell (0/0/0). Mode [Hz]

GDQ 31  31

Nastran 80  160 (4 nodes)

Abaqus 40  80 (8 nodes)

Ansys 40  80 (8 nodes)

Straus 40  80 (8 nodes)

Pro/Mechanica 21  41 (GEM)

f1 f2 f3 f4 f5 f6 f7 f8 f9 f10

44.287 46.778 52.197 61.407 69.261 70.940 74.784 75.853 77.439 80.695

44.265 46.757 52.167 61.334 69.161 70.840 74.710 75.744 77.360 80.558

44.293 46.785 52.202 61.404 69.284 70.952 74.807 75.872 77.447 80.717

44.284 46.760 52.209 61.375 69.156 70.817 74.677 75.738 77.374 80.571

44.368 46.858 52.253 61.452 69.600 71.287 75.142 76.206 77.488 81.011

44.251 46.746 52.161 61.347 69.150 70.817 74.667 75.716 77.374 80.556

Geometric characteristics: a = 3 m, c = 3 m, b = 1 m, d = 0 m, Rb = 9 m, #0 = 120°, h1 = h3 = 0.02 m, h2 = 0.06 m. Orthotropic material properties (1° and 3° laminae): E1 = 137.9 GPa, E2 = 8.96 GPa, G12 = G13 = 7.1 GPa, G23 = 6.21 GPa, m12 = 0.3, q = 1450 Kg/m3. Isotropic material properties (2° lamina): E = 210 GPa, m = 0.3, q = 7800 Kg/m3.

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Free edge boundary condition (F)

8 N M N M P P P P > uð1Þ u uð1Þ s sð1Þ s A12 cos ua u ua s u > > RA11 1ak U kj þ A16 1sð1Þ U aj þ RAu16a 1ak U kj þ A12 1jm U am  A16 Rcos U aj > jm U am þ R0a > u a 0a > m¼1 m¼1 k¼1 k¼1 > > > > >   N M N M > P P P P > uð1Þ u sð1Þ u uð1Þ s ua ua u B16 cos ua s s > þ RAu11a þ A12Rsin 1ak Bkj þ B16 1jm Bam þ B12 Rcos Baj þ RBu16a 1ak Bkj þ B12 1sð1Þ Baj ¼ 0 W aj þ RBu11a > jm Bam  > R0a 0a 0a > m¼1 m¼1 > k¼1 k¼1 > > > > N M N M > P P P > A16 P uð1Þ u uð1Þ s sð1Þ s A26 cos ua u ua s u > > 1ak U kj þ A66 1sð1Þ U aj þ RAu66a 1ak U kj þ A26 1jm U am  A66 Rcos U aj > jm U am þ Rua R0a 0a > > m¼1 m¼1 k¼1 k¼1 > > > >  >  N M N M P P P P > uð1Þ u sð1Þ u uð1Þ s A26 sin ua ua u B66 cos ua s s A16 > > 1ak Bkj þ B66 1jm Bam þ B26 Rcos Baj þ RBu66a 1ak Bkj þ B26 1sð1Þ Baj ¼ 0 W aj þ RBu16a > jm Bam  R0a > þ Rua þ R0a 0a > m¼1 m¼1 k¼1 k¼1 > > > > > N M > P P > uð1Þ u A45 sin ua s s < U aj þ j RAu44a 1ak W kj þ jA45 1sð1Þ j RAu44a U u jm W am þ jA44 Baj þ jA45 Baj ¼ 0 aj  j R0a k¼1

m¼1

> > > N M N M > P P P > sð1Þ u ua u B16 cos ua s s B11 P uð1Þ u > > 1ak U kj þ B16 1jm U am þ B12 Rcos U aj þ RBu16a 1uakð1Þ U skj þ B12 1sð1Þ U aj > jm U am  Rua R0a 0a > > m¼1 m¼1 k¼1 k¼1 > > > >   > N M N M > > þ B11 þ B12 sin ua W þ D11 P 1uð1Þ Bu þ D P 1sð1Þ Bu þ D12 cos ua Bu þ D16 P 1uð1Þ Bs þ D P 1sð1Þ Bs  D16 cos ua Bs ¼ 0 > 16 12 aj > kj am aj am jm jm aj kj ak ak R R Rua R0a Rua R0a ua > 0a > m¼1 m¼1 k¼1 k¼1 > > > > > N M N M > P P P sð1Þ u > ua u B66 cos ua s s B16 P uð1Þ u > 1ak U kj þ B66 1jm U am þ B26 Rcos U aj þ RBu66a 1uakð1Þ U skj þ B26 1sð1Þ U aj > jm U am  Rua R0a > 0a > m¼1 m¼1 k¼1 k¼1 > > > > >   N M N M > P P P P > uð1Þ u sð1Þ u uð1Þ s ua ua u D66 cos ua s s > > þ RBu16a þ B26Rsin 1ak Bkj þ D66 1jm Bam þ D26 Rcos Baj þ RDu66a 1ak Bkj þ D26 1sð1Þ Baj ¼ 0; W aj þ RDu16a > jm Bam  R0a 0a 0a > > m¼1 m¼1 k¼1 k¼1 > > > : for a ¼ 1; N and j ¼ 1; 2; . . . ; M;

Table 7 First ten frequencies for a C–F–C–F symmetric laminated elliptic shell (45/0/45). Mode [Hz]

GDQ 31  31

Nastran 80  160 (4 nodes)

Abaqus 40  80 (8 nodes)

Ansys 40  80 (8 nodes)

Straus 40  80 (8 nodes)

Pro/Mechanica 21  41 (GEM)

f1 f2 f3 f4 f5 f6 f7 f8 f9 f10

108.269 141.088 228.916 290.483 290.488 341.929 349.287 402.567 462.236 475.631

108.156 139.720 226.198 289.292 290.064 339.530 346.898 401.039 458.057 473.915

108.389 139.647 225.941 289.208 289.967 339.221 347.723 402.707 457.585 476.051

108.145 139.812 226.212 288.892 289.624 339.179 346.470 400.438 457.328 472.910

110.307 141.463 227.745 298.200 298.805 341.861 356.290 410.786 460.255 483.734

108.100 139.610 225.987 288.716 289.469 339.091 346.322 400.280 457.285 472.800

Geometric characteristics: a = 1 m, b = 0.5 m, Rb = 2 m, u 2 [0°, 180°], #0 = 120°, h1 = h3 = 0.02 m, h2 = 0.06 m. Orthotropic material properties (1° and 3° laminae): E1 = 137.9 GPa, E2 = 8.96 GPa, G12 = G13 = 7.1 GPa, G23 = 6.21 GPa, m12 = 0.3, q = 1450 Kg/m3. Isotropic material properties (2° lamina): E = 210 GPa, m = 0.3, q = 7800 Kg/m3.

Table 8 First ten frequencies for a C–C asymmetric laminated elliptic (circular) shell (30/60). Mode [Hz]

GDQ 31  31

Nastran 80  160 (4 nodes)

Abaqus 40  80 (8 nodes)

Ansys 40  80 (8 nodes)

Straus 40  80 (8 nodes)

Pro/Mechanica 42  31 (GEM)

f1 f2 f3 f4 f5 f6 f7 f8 f9 f10

103.481 150.664 162.837 185.505 194.526 224.148 230.685 235.097 237.502 262.718

103.148 149.930 161.687 184.064 193.853 222.550 230.177 235.625 238.245 262.822

103.358 150.394 162.088 184.248 194.203 222.859 230.757 236.595 239.515 263.375

103.298 150.494 162.090 185.098 194.040 223.698 231.317 237.013 238.246 264.464

103.542 150.681 162.665 185.729 194.400 225.352 232.238 238.636 241.084 267.470

103.154 149.886 161.687 184.059 193.802 222.390 229.852 235.348 237.903 262.383

Geometric characteristics: a = b = 1 m, Rb = 3 m, u 2 [0°, 360°], #0 = 120°, h1 = h2 = 0.05 m. Material properties: E1 = 137.9 GPa, E2 = 8.96 GPa, G12 = G13 = 7.1 GPa, G23 = 6.21 GPa, m12 = 0.3, q = 1450 Kg/m3.

ð42Þ

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8 N M N M > P P P sð1Þ s u > A26 cos ui u A16 P uð1Þ u > 1ik U kb þ A66 1sð1Þ U ib þ AR66 1ikuð1Þ U skb þ A26 1bm U im  A66 Rcos ui U sib > bm U im þ Rui R0i > ui 0i > m¼1 m¼1 k¼1 k¼1 > > >   N M N M > P P P P > u ð1Þ sð1Þ u ð1Þ sð1Þ s u u u A sin u B cos u ui s s A B B > þ R16 þ 26R0i i W ib þ R16 1 Bkb þ B66 1 Bim þ 26 R0i i Bib þ R66 1 Bkb þ B26 1bm Bim  B66 Rcos Bib ¼ 0 > > ik bm ik u i u i u i 0i > > m¼1 m¼1 k¼1 k¼1 > > N M N M > P P P > sð1Þ s u A22 cos ui u A12 P uð1Þ u > > 1ik U kb þ A26 1sð1Þ U ib þ AR26 1ikuð1Þ U skb þ A22 1bm U im  A26 Rcos ui U sib > bm U im þ Rui R0i u i 0i > > m¼1 m¼1 k¼1 k¼1 > >  >  N M N M P P P P > sð1Þ s u A22 sin ui B22 cos ui u ui s A12 > > 1ikuð1Þ Bukb þ B26 1sð1Þ Bib þ BR26 1ikuð1Þ Bskb þ B22 1bm Bim  B26 Rcos Bib ¼ 0 W ib þ BR12 > bm Bim þ R0i > þ Rui þ R0i ui ui 0i > m¼1 m¼1 k¼1 k¼1 > > > N M > P uð1Þ P < u A55 sin ui s s j AR45 U u U ib þ j AR45 1ik W kb þ jA55 1sð1Þ ib  j bm W im þ jA45 Bib þ jA55 Bib ¼ 0 R0i ui ui m¼1 k¼1 > > N M N M > P P P > sð1Þ u ui u B66 cos ui s s B16 P uð1Þ u > > 1ik U kb þ B66 1bm U im þ B26 Rcos U ib þ BR66 1ikuð1Þ U skb þ B26 1sð1Þ U ib > Rui bm U im  R0i 0i ui > > m¼1 m¼1 k¼1 k¼1 > >  >  N M N M P P P > sð1Þ u sð1Þ s B26 sin ui B16 D66 P uð1Þ s > > W ib þ DR16 1ikuð1Þ Bukb þ D66 1bm Bim þ D26Rcos ui Bu 1ik Bkb þ D26 1bm Bim  D66 Rcos ui Bsib ¼ 0 > ib þ Rui > þ Rui þ R0i ui 0i 0i > m¼1 m¼1 k¼1 k¼1 > > > N M N M > P P P P u ð1Þ sð1Þ u ð1Þ sð1Þ ui u ui s > B > 1ik U ukb þ B26 1bm U uim þ B22 Rcos U ib þ BR26 1ik U skb þ B22 1bm U sim  B26 Rcos U ib > R12 > ui 0i ui 0i > m¼1 m¼1 k¼1 k¼1 > > >   N M N M > P uð1Þ u P sð1Þ u D22 cos ui u D26 P uð1Þ s P > sð1Þ s > þ BR12 þ B22Rsin ui W ib þ DR12 1 Bkb þ D26 1 Bim þ R Bib þ R 1 Bkb þ D22 1bm Bim  D26 Rcos ui Bsib ¼ 0; > > ik bm ik u i 0i u i 0i u i 0i > > m¼1 m¼1 k¼1 k¼1 > : for b ¼ 1; M and i ¼ 1; 2; . . . ; N:

ð43Þ

Kinematic and physical compatibility conditions along the closing meridian (s = 0, 2pR0) u s s Uu ¼ U u ; U s ¼ U siM ; W ¼i1 W ;iM Bu i1 ¼ BiM ; Bi1 ¼ BiM 8i1 N iM i1 M N M P uð1Þ u P sð1Þ u A26 cos ui u A66 P P uð1Þ s sð1Þ s A > > 1 U k1 þ A66 1 U im þ R U i1 þ R 1 U k1 þ A26 11m U im  A66 Rcos ui U si1 > R16 1m ik ik > ui 0i ui 0i > m¼1 m¼1 k¼1 k¼1 > >   > N M N M > P P P P > u ð1Þ sð1Þ u ð1Þ u u u A sin u B cos u B66 cos ui s s s A B B 26 26 > þ R16 þ R i W i1 þ R16 1ik Bk1 þ B66 11m Bim þ R0i i Bi1 þ R66ui 1ik Bk1 þ B26 1sð1Þ Bi1 > 1m Bim  R0i > u i 0i u i > m¼1 m¼1 k¼1 k¼1 > > > N M N M > P P P > A16 P sð1Þ u sð1Þ s ui u ui s > > ¼ Rui 1ikuð1Þ U ukM þ A66 1Mm U im þ A26 Rcos U iM þ AR66 1ikuð1Þ U skM þ A26 1Mm U im  A66 Rcos U iM > 0i ui 0i > > m¼1 m¼1 k¼1 k¼1 > >   > N M N M > > þ A16 þ A26 sin ui W þ B16 P 1uð1Þ Bu þ B P 1sð1Þ Bu þ B26 cos ui Bu þ B66 P 1uð1Þ Bs þ B P 1sð1Þ Bs  B66 cos ui Bs > 66 26 iM > kM iM Mm im Mm im iM kM ik ik Rui R0i Rui R0i Rui R0i > > m¼1 m¼1 k¼1 k¼1 > > > N M N M > P sð1Þ u A22 cos ui u A26 P uð1Þ s P sð1Þ s > A12 P uð1Þ u > 1 U k1 þ A26 11m U im þ R U i1 þ R 1ik U k1 þ A22 11m U im  A26 Rcos ui U si1 > > 0i ui 0i > Rui k¼1 ik m¼1 m¼1 k¼1 > > >   N M N M > P uð1Þ u P sð1Þ u B22 cos ui u B26 P uð1Þ s P > B26 cos ui s s > > þ AR12 þ A22Rsin ui W i1 þ BR12 1ik Bk1 þ B26 11m Bim þ R Bi1 þ R 1ik Bk1 þ B22 1sð1Þ Bi1 > 1m Bim  R0i ui 0i ui 0i ui > > m¼1 m¼1 k¼1 k¼1 > > > N M N M P P P P > sð1Þ u sð1Þ s ui u ui s > > ¼ AR12 1ikuð1Þ U ukM þ A26 1Mm U im þ A22 Rcos U iM þ AR26 1ikuð1Þ U skM þ A22 1Mm U im  A26 Rcos U iM > > ui 0i ui 0i > m¼1 m¼1 k¼1 k¼1 > > >   N M N M > P P P P > sð1Þ s u ui B22 cos ui u ui s > þ AR12 þ A22Rsin 1ikuð1Þ BukM þ B26 1sð1Þ BiM þ BR26 1ikuð1Þ BskM þ B22 1Mm Bim  B26 Rcos BiM W iM þ BR12 > Mm Bim þ R0i > ui 0i ui ui 0i > m¼1 m¼1 > k¼1 k¼1 > > N M > > > j A45 U u  j A55 sin ui U s þ j A45 P 1uð1Þ W þ jA55 P 1sð1Þ W im þ jA45 Bu þ jA55 Bs > k1 > i1 i1 1m i1 ik Rui i1 R0i Rui > < m¼1 k¼1 N M P P sð1Þ A55 sin ui s s > ¼ j AR45 U u U iM þ j AR45 1ikuð1Þ W kM þ jA55 1Mm W im þ jA45 Bu > iM  j iM þ jA55 BiM R0i > ui ui > m¼1 k¼1 > > > N M N M > P P P > B16 P u B26 cos ui u B66 cos ui s s > 1ikuð1Þ U uk1 þ B66 1sð1Þ U i1 þ BR66 1ikuð1Þ U sk1 þ B26 1sð1Þ U i1 > 1m U im þ 1m U im  Rui R0i R0i > ui > m¼1 m¼1 k¼1 > k¼1 > >   N M N M > > > þ B16 þ B26 sin ui W i1 þ D16 P 1uð1Þ Bu þ D66 P 1sð1Þ Bu þ D26 cos ui Bu þ D66 P 1uð1Þ Bs þ D26 P 1sð1Þ Bs  D66 cos ui Bs > > k1 i1 1m im 1m im i1 k1 ik ik Rui R0i Rui R0i Rui R0i > > m¼1 m¼1 k¼1 k¼1 > > > N M N M P sð1Þ u B26 cos ui u P uð1Þ s P > B16 P uð1Þ u B66 cos ui s s > > ¼ 1 U kM þ B66 1Mm U im þ R0i U iM þ BR66 1ik U kM þ B26 1sð1Þ U iM > Mm U im  R0i > ui > Rui k¼1 ik m¼1 m¼1 k¼1 > >   > N M N M > P P P P > u D26 cos ui u D66 cos ui s s > þ BR16 þ B26Rsin ui W iM þ DR16 1ikuð1Þ BukM þ D66 1sð1Þ BiM þ DR66 1ikuð1Þ BskM þ D26 1sð1Þ BiM > Mm Bim þ Mm Bim  R0i R0i > ui 0i ui ui > m¼1 m¼1 k¼1 k¼1 > > > N M N M > P P P P > u B22 cos ui u B26 cos ui s s > > BR12 1ikuð1Þ U uk1 þ B26 1sð1Þ U i1 þ BR26 1ikuð1Þ U sk1 þ B22 1sð1Þ U i1 > 1m U im þ 1m U im  R0i R0i ui ui > > m¼1 m¼1 k¼1 k¼1 > >   > N M N M P uð1Þ u P sð1Þ u D22 cos ui u D26 P uð1Þ s P > D26 cos ui s s B > > þ B22Rsin ui W i1 þ DR12 1ik Bk1 þ D26 11m Bim þ R Bi1 þ R 1ik Bk1 þ D22 1sð1Þ Bi1 > þ R12 1m Bim  R0i > u i 0i u i 0i u i > m¼1 m¼1 k¼1 k¼1 > > > N M N M > P P P P > sð1Þ u ui u B26 cos ui s s > ¼ BR12 1ikuð1Þ U ukM þ B26 1Mm U im þ B22 Rcos U iM þ BR26 1ikuð1Þ U skM þ B22 1sð1Þ U iM > Mm U im  R0i > u i 0i u i > m¼1 m¼1 k¼1 k¼1 > >  >  N M N M > P P P P > u D22 cos ui u D26 cos ui s s > > þ BR12 þ B22Rsin ui W iM þ DR12 1ikuð1Þ BukM þ D26 1sð1Þ BiM þ DRu26i 1ikuð1Þ BskM þ D22 1sð1Þ BiM ; > Mm Bim þ Mm Bim  R0i R0i ui 0i ui > > m¼1 m¼1 k¼1 k¼1 : for i ¼ 2; . . . ; N  1:

ð44Þ

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In the same way, by applying the GDQ methodology the discretized forms of the compatibility conditions (32) and (33) can also be obtained and are omitted for the sake of brevity. 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



db dd



¼ x2



0

0

0 Mdd



db dd

 :

ð45Þ

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 equilibrium equations, assigned on interior nodes. In order to make the computation more efficient, kinematic condensation of non-domain degrees of freedom is performed:

  Kdd  Kdb ðKbb Þ1 Kbd dd ¼ x2 Mdd dd : Fig. 10. Mode shapes for the C–C–C–C hyperbolic panel of Table 2.

ð46Þ

The natural frequencies of the structure considered can be determined by solving the standard eigenvalue problem (46). In particular,

Fig. 11. Mode shapes for the C–F hyperbolic shell of Table 3.

Fig. 12. Mode shapes for the C–C–F–C catenary panel of Table 4.

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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.

It is worth noting that, with the present approach, differing from the finite element method, no integration occurs prior to the global assembly of the linear system, and these results in a

Fig. 13. Mode shapes for the F–C cycloidal shell of Table 5.

Fig. 14. Mode shapes for the C–F–C–F parabolic panel of Table 6.

Fig. 15. Mode shapes for the C–F–C–F elliptic panel of Table 7.

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further computational cost saving in favour of the differential quadrature technique. 4. Numerical applications and results In the present paragraph, some results and considerations about the free vibration problem of laminated composite doubly-curved shells and panels of revolution are presented. The analysis has been carried out by means of numerical procedures illustrated above. One of the aims of this paper is to compare results obtained through the GDQ analysis with the ones obtained through finite element techniques and based on the same shell theory. In order to verify the accuracy of the present method, some comparisons have been performed. Extensive attempts to validate the present formulations have been made for the isotropic and anisotropic cases and can be found in the Ph.D. thesis by Tornabene [59]. In this work, the frequency parameters from the present formulations are in good agreement with the results presented in the literature and obtained with the finite element method. The geometrical boundary conditions for a shell panel (Fig. 9) are identified by the following convention. For example, symbolism

C–F–C–F shows that the edges u = u1, s = s0, u = u0, s = 0 are clamped, free, clamped and free, respectively. On the contrary, for a complete shell of revolution (Fig. 8(a)) or for a toroidal shell (Fig. 8(b)), symbolism C–F shows that the edges u = u1 and u = u0 or s = s0 and s = 0 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 = 0 and s0 = 2pR0 or at the same closing parallels for u = 0 and u = 2p, respectively. The solution procedure by means of the GDQ technique has been implemented in a MATLAB code. In order to verify the accuracy of the numerical procedure, some comparisons have also been performed. Different FEM commercial codes such as Abaqus, Ansys, Straus, Nastran and Pro/Mechanica, has been considered and Table 1 reports the finite element shell types selected in each of the commercial programs. In Tables 2–8, the results in terms of first ten frequencies obtained by the GDQ Method are compared with the FEM results obtained with commercial programs. The details regarding the geometry and the material properties of the structures analyzed are reported in Tables 2–8. For the GDQ results, the grid distributions

Fig. 16. Mode shapes for the C–C toroidal shell of Table 8.

Table 9 First ten frequencies for C–C thin isotropic elliptic (circular) shells. Mode [Hz]

GDQ 31  31

Wang et al. [18] 28  27

Nastran 1  80  40

Abaqus 1  80  40

Ansys 1  80  40

Straus 1  80  40

Pro/Mechanica AutoGEM

#0 = 90° f1 f2 f3 f4 f5 f6 f7 f8 f9 f10

3146.05 3410.26 3466.75 3596.86 3937.84 4083.55 4089.56 4254.09 4262.91 4267.02

3147.00 3411.00 3468.00 3599.00 3943.00 4080.00 4087.00 4257.00 4265.00 4270.00

3148.16 3410.23 3466.24 3596.80 3938.12 4076.55 4086.42 4251.19 4260.54 4264.43

3146.84 3408.82 3465.07 3595.27 3936.62 4076.75 4083.08 4249.70 4257.62 4262.51

3146.84 3408.82 3465.07 3595.27 3936.62 4076.75 4083.09 4249.71 4257.63 4262.51

3148.27 3410.35 3466.98 3597.76 3939.92 4079.91 4086.26 4253.25 4261.38 4266.24

3163.68 3448.38 3519.35 3674.93 4061.47 4147.91 4169.44 4380.67 4397.92 4410.18

Mode [Hz]

GDQ 31  31

Wang et al. [18] 28  27

Nastran 1  80  80

Abaqus 1  80  80

Ansys 1  80  80

Straus 1  80  80

Pro/Mechanica AutoGEM

#0 = 180° f1 f2 f3 f4 f5 f6 f7 f8 f9 f10

1278.24 2220.04 2384.86 2437.14 2450.32 2465.44 2881.78 2936.72 2986.54 3151.49

1278.00 2216.00 2384.00 2429.00 2442.00 2458.00 2877.00 2937.00 2986.00 3150.00

1279.53 2216.48 2385.35 2427.40 2442.71 2457.01 2874.77 2937.85 2984.61 3148.18

1278.66 2214.52 2384.68 2427.25 2440.37 2456.74 2874.04 2937.58 2983.57 3146.81

1278.66 2214.52 2384.68 2427.25 2440.38 2456.75 2874.04 2937.58 2983.57 3146.81

1279.60 2215.33 2385.96 2428.56 2441.79 2457.93 2875.45 2938.48 2984.60 3148.23

1283.02 2244.51 2390.89 2470.53 2493.70 2503.16 2916.02 2942.94 3008.22 3200.39

Geometric characteristics: a = b = 0.056095 m, Rb = 0.1524 m, h = 0.00211 m, u 2 [0°, 360°]. Isotropic material properties: E = 193 GPa, m = 0.291, q = 7850 Kg/m3.

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(34) with N = M = 31 has been considered, while for the commercial programs, 8-node and 4-node shell elements (Table 1) have been used. Well-converged and accurate results were obtained using different FEM meshes for shells and panels under investigation, as shown in Tables 2–8. It is noteworthy 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. As pointed out in the Femap/Nastran User Guide, for doubly-curved structures, the CQUAD4 (4-nodes) element, in general, performs better than the CQUAD8 (8-nodes) element in terms of accuracy. For this reason the CQUAD4 element has been used in Nastran code for all the laminated composite doubly-curved structures considered. Table 2 present the first ten frequencies for the C–C–C–C single layer orthotropic hyperbolic panel characterized by the orientation angle h = 30°, while Table 3 illustrates results for the F–C toroidal hyperbolic complete shell made of two orthotropic layers with different orientation angles (30/60). Geometric parameters concerns Fig. 2. It is noteworthy that the results from the GDQ methodology are very close to those obtained by the commercial programs. In Table 4 the first ten frequencies for a C–C–F–C symmetric laminated catenary panel (45/0)s made of four layers are reported, while Table 5 shows results for a F–C asymmetric laminated

cycloidal complete shell (45/20/70/45) made of four layers. Geometric parameters concerns Figs. 3 and 4, respectively. As can be seen, also for these structures numerical results show an excellent agreement. Tables 6 and 7 present the first ten frequencies for symmetric laminated composite parabolic and elliptic panels, respectively. The structures under consideration consist of three layers, two of which (1° and 3° laminae) are orthotropic layers and the middle one (2° lamina) is an isotropic ply. Geometric characteristics of these laminated composite structures are shown in Tables 6 and 7 and concern Figs. 5 and 6, respectively. Finally, as a special case of elliptic shell, a toroid with circular meridian is considered in Table 8. This table illustrates results for the C–C toroidal shell made of two orthotropic layers with different orientation angles (30/ 60). As can be seen, also for the last three structures the numerical results from the GDQ methodology are very close to those obtained by the commercial programs and show an excellent agreement. In Figs. 10–16, we have reported the first six mode shapes for all the structures 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, we have summarized the symmetrical mode shapes in one figure. The mode shapes of all the structures under discussion have been evaluated by author. By using the author’s MATLAB

Table 10 First ten frequencies for C–C moderately thick isotropic elliptic (circular) shells. Mode [Hz]

GDQ 31  31

Wang et al. [18] 28  27

Nastran 2  80  40

Abaqus 2  80  40

Ansys 2  80  40

Straus 2  80  40

Pro/Mechanica AutoGEM

#0 = 90° f1 f2 f3 f4 f5 f6 f7 f8 f9 f10

3519.15 4344.31 4891.12 5938.93 6115.07 6410.32 6909.53 7120.38 7221.64 7357.87

3547.00 4385.00 4947.00 5928.00 6164.00 6486.00 7051.00 7210.00 7278.00 7426.00

3519.05 4339.25 4884.46 5924.22 6095.13 6427.48 6915.40 7125.13 7223.41 7355.86

3523.60 4336.58 4885.24 5926.40 6095.45 6423.03 6905.08 7117.13 7227.81 7366.85

3517.98 4338.09 4884.74 5923.99 6094.26 6428.66 6918.10 7129.73 7223.88 7358.14

3518.93 4339.07 4885.05 5925.77 6095.81 6429.80 6920.52 7130.77 7225.75 7359.23

3529.42 4344.23 4895.22 5949.78 6121.82 6437.92 6943.41 7144.88 7257.04 7384.53

Mode [Hz]

GDQ 31  31

Wang et al. [18] 28  27

Nastran 2  80  80

Abaqus 2  80  80

Ansys 2  80  80

Straus 2  80  80

Pro/Mechanica ?tul=0?>AutoGEM

#0 = 180° f1 f2 f3 f4 f5 f6 f7 f8 f9 f10

1380.74 2434.16 2963.76 3139.32 3532.13 3737.22 3754.19 4045.02 4455.63 4603.66

1376.00 2418.00 2973.00 3179.00 3553.00 3754.00 3773.00 4046.00 4457.00 4607.00

1381.99 2437.45 2962.81 3131.80 3530.33 3713.36 3733.80 4032.80 4428.59 4594.09

1381.73 2437.17 2962.64 3132.25 3528.81 3713.23 3733.09 4034.01 4429.01 4593.39

1381.73 2437.17 2962.64 3132.25 3528.81 3713.23 3733.09 4034.01 4429.02 4593.40

1381.76 2437.37 2962.79 3132.43 3529.73 3713.79 3733.74 4033.62 4429.37 4594.17

1386.89 2444.87 2974.30 3138.18 3532.63 3720.70 3739.05 4044.10 4446.25 4604.12

Geometric characteristics: a = b = 0.05287 m, Rb = 0.1524 m, h = 0.008561 m, u 2 [0°, 360°]. Isotropic material properties: E = 193 GPa, m = 0.291, q = 7850 Kg/m3.

Table 11 pffiffiffiffiffiffiffiffiffi Frequency parameters k ¼ xa q=E for fully clamped (C–C–C–C) isotropic spherical shell panel (a/b = 1, Rm/a = 0.5). h/a

h/a = 0.01

h/a = 0.1

h/a = 0.2

k

GDQ 21  21

Liew et al. [74]

GDQ 21  21

Liew et al. [74]

GDQ 21  21

Liew et al. [74]

k1 k2 k3 k4 k5 k6

0.57648 – 0.59165 0.63061 0.64815 0.72685

0.57838 0.57840 0.59424 0.63399 0.65241 0.73140

1.1886 1.9122 – 2.6625 3.1095 3.1579

1.1988 1.9301 1.9340 2.6828 3.1288 3.1774

1.7360 2.8062 2.8062 3.7322 3.7322 3.8044

1.7405 2.8036 2.8091 3.7504 3.7572 3.7904

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F. Tornabene / Comput. Methods Appl. Mech. Engrg. 200 (2011) 931–952

code, these mode shapes have been reconstructed in three-dimensional view by means of considering the displacement field (17) after solving the eigenvalue problem (46). In Tables 9 and 10 the first ten frequencies for thin and thick single layer isotropic elliptic (circular a = b) toroidal shell are shown. In these tables results obtained by the GDQ method are compared with the DQ results obtained by Wang et al. [18] and

the 3D FEM results obtained with commercial programs. Regarding commercial programs, 20-node brick elements are used and the respective meshes are shown in tables. In particular, for thin shells 1  80  40 and 1  80  80 meshes have been considered. The 1  80  40 mesh is obtained using one finite element in thickness direction, 80 and 40 finite elements in meridian and circumferential directions, respectively, likewise for the 1  80  80 mesh. For

Table 12 First ten frequencies for the orthotropic hyperbolic panel of Table 2 for an increasing the number of grid points N = M of the Chebyshev–Gauss–Lobatto distribution. Mode [Hz]

N = M = 11

N = M = 15

N = M = 17

N = M = 21

N = M = 25

N = M = 29

N = M = 31

f1 f2 f3 f4 f5 f6 f7 f8 f9 f10

200.158 240.947 289.159 322.856 337.913 378.866 394.063 407.396 448.510 481.474

195.745 233.848 277.769 305.283 320.178 354.933 365.560 374.512 407.524 424.899

195.598 233.492 277.328 304.782 319.187 352.771 363.204 372.037 402.152 418.523

195.702 233.468 277.386 305.005 318.937 351.892 362.739 372.528 400.706 416.759

195.727 233.480 277.395 304.993 318.892 351.772 362.611 372.431 400.443 416.409

195.729 233.480 277.393 304.990 318.885 351.757 362.595 372.420 400.408 416.366

195.729 233.480 277.392 304.990 318.883 351.754 362.594 372.420 400.404 416.361

Fig. 17. Convergence and stability characteristics of the first ten frequencies for the C–C–C–C hyperbolic panel of Table 2.

Fig. 18. Convergence and stability characteristics of the first ten frequencies for the C–F hyperbolic shell of Table 3.

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F. Tornabene / Comput. Methods Appl. Mech. Engrg. 200 (2011) 931–952

Fig. 19. Convergence and stability characteristics of the first ten frequencies for the C–C–F–C catenary panel of Table 4.

Fig. 20. Convergence and stability characteristics of the first ten frequencies for the F–C cycloidal shell of Table 5.

Fig. 21. Convergence and stability characteristics of the first ten frequencies for the C–F–C–F parabolic panel of Table 6.

F. Tornabene / Comput. Methods Appl. Mech. Engrg. 200 (2011) 931–952

thick shells 2  80  40 and 2  80  80 meshes are used. Furthermore, Table 11 presents results obtained by the 2D GDQ method compared with the 3D results obtained by Liew et al. [74]. It is noteworthy that results obtained from the GDQ methodology are very close and in excellent agreement to those presented in literature and to those obtained by using 3D finite elements, as can be seen from Tables 9–11. In addition, Hosseini-Hashemi et al. [75] (see Table 3 of the work [75]) have compared their results obtained with a semi analytical method with those results presented in the article by Tornabene [65]. Since the code used to obtained the previous results [65] is exactly the same of the code used to obtained all the results presented in this paper, the results of the work [75] represent another proof of the validity and the accuracy of the present procedure. As shown, the exact results by Hosseini-Hashemi et al. [75] are in good agreement with those reported by Tornabene [65]. The discrepancy between results obtained from two methods is closely zero. Finally, in order to illustrate the GDQ convergence characteristic, the first ten frequencies of the single layer composite hyperbolic panel (C–C–C–C) of Table 2 is investigated by varying the

949

number of grid points. Results are collected in Table 12 when the number of points of the Chebyshev–Gauss–Lobatto grid distribution (34) is increased from N = M = 11 up to N = M = 31. It can be seen that the proposed GDQ formulation well captures the dynamic behaviour of the shell by using only 21 points in two coordinate 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 Figs. 17–23 and in the Ph.D. thesis by Tornabene [59]. 5. Conclusion remarks and summary A Generalized Differential Quadrature Method application to free vibration analysis of laminated composite doubly-curved shells and panels of revolution has been presented to illustrate the versatility and the accuracy of this methodology. Various lamination schemes with different layers have been considered. The adopted shell theory is the First-order Shear Deformation Theory. The dynamic equilibrium equations have been discretized with

Fig. 22. Convergence and stability characteristics of the first ten frequencies for the C–F–C–F elliptic panel of Table 7.

Fig. 23. Convergence and stability characteristics of the first ten frequencies for the C–C toroidal shell of Table 8.

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F. Tornabene / Comput. Methods Appl. Mech. Engrg. 200 (2011) 931–952

the present 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 so doing, complete revolution shells have been obtained as special cases of shell panels by satisfying the kinematic and physical compatibility conditions. The GDQ method provides a very simple algebraic formula to determine the weighting coefficients required by differential quadrature approximation without restricting in any way the choice of mesh grids. 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, 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 for one of the six structures considered. Furthermore, discretizing and programming procedures are quite easy. The GDQ results show to be precise and reliable. The numerical tests demonstrate and confirm the favourable precision of the differential quadrature method. Acknowledgements

L15

! B16 @ 2 B26 cos u B16 dRu @ @2 ¼ 2  þ þ B 26 Ru R0 @s2 Ru @ u 2 R3u du @ u     2 B22 cos u B66 cos u @ B12 B66 @ B16 sin u  þ þ þ þ @s R0 R0 Ru Ru @ u@s Ru R0 þ

L21 ¼

L22

L23 ¼

L24 ¼

The following are the equilibrium operators in Eq. (25):

L11 ¼

! A11 @ 2 cos u 1 dRu @ @ 2 2A16 @ 2 þ A  þ A66 2 þ 11 2 @ u2 3 Ru R0 Ru du @ u @s Ru @ u@s Ru 

L12

A12 sin u A22 cos2 u A44  j 2 ; Ru R0 R20 Ru

! A16 @ 2 A26 cos u A16 dRu @ @2 ¼ 2  þ 3 þ A26 2 2 Ru R0 @s Ru @ u Ru du @ u     2 A22 cos u A66 cos u @ A12 A66 @ A16 sin u þ  þ þ þ @s R0 R0 Ru Ru @ u@s Ru R0 þ

A26 cos2 u R20 A11

L13 ¼

R2u þ A11

L14

A45 sin u j ; Ru R0

!   A12 sin u A44 @ A16 A26 sin u A45 @ þj 2 þ þ þj Ru R0 Ru R0 Ru @s Ru @ u ! cos u 1 dRu A22 sin u cos u   ; Ru R0 R3u du R20

þ

! B11 @ 2 cos u 1 dRu @ @ 2 2B16 @ 2 ¼ 2 þ B11  3 þ B66 2 þ 2 @ @ u R R d u u @s Ru @ u@s u 0 Ru Ru 

B12 sin u B22 cos2 u A44  þj ; Ru R0 Ru R20

L25

R20

þj

A45 ; Ru

! A16 @ 2 2A16 cos u A26 cos u A16 dRu @ @2 þ þ  þ A26 2 2 @ u2 3 Ru R0 Ru R0 @s Ru Ru du @ u     2 A22 cos u A66 cos u @ A12 A66 @ þ þ þ þ @s R0 R0 Ru Ru @ u@s ! cos2 u sin u A45 sin u þ A26  ; j Ru R0 Ru R0 R20

! A66 @ 2 cos u 1 dRu @ @ 2 2A26 @ 2 ¼ 2 þ A66  3 þ A22 2 þ 2 Ru R0 Ru du @ u @s Ru @ u@s Ru @ u ! 2 cos2 u sin u A55 sin u  A66  ; j 2 2 Ru R0 R0 R0

This research was supported by the Italian Ministry for University and Scientific, Technological Research MIUR (40% and 60%). The research topic is one of the subjects of the Centre of Study and Research for the Identification of Materials and Structures (CIMEST)-‘‘M. Capurso’’ of the University of Bologna (Italy). Appendix A

B26 cos2 u

! A26 sin u A45 sin u @ þ j @u Ru R0 Ru R0 R2u   A12 A22 sin u A55 sin u @ þ þ þj @s Ru R0 R0 ! ! sin u cos u cos u 2 cos u 1 dRu þ A26 þ A16 ; þ  3 Ru R0 Ru R0 Ru du R20 A16

þ

!! B16 @ 2 B26 cos u 2 cos u 1 dRu @ þ þ B  þ B26 16 @u Ru R0 Ru R0 R3u du R2u @ u2     2 @2 B22 cos u B66 cos u @ B12 B66 @ þ  2þ þ þ @s @s R0 R0 Ru Ru @ u@s ! cos2 u sin u A45 sin u þj þ B26  ; Ru R0 R0 R20

! B66 @ 2 cos u 1 dRu @ @ 2 2B26 @ 2 ¼ 2 þ B66  3 þ B22 2 þ 2 Ru R0 Ru du @ u @s Ru @ u@s Ru @ u ! cos2 u sin u A55 sin u þj  B66  ; 2 R R R0 u 0 R0 ! A12 sin u A44 @ þ j Ru R0 R2u R2u @ u   A16 A26 sin u A45 @ A12 cos u A22 sin u cos u   þ þj  Ru R0 Ru @s Ru R0 R20 ! cos u 1 dRu ;  jA44  Ru R0 R3u du

L31 ¼ 

A11

þ

! A26 sin u A45 sin u @ þ j @u Ru R0 Ru R0 R2u   A12 A22 sin u A55 sin u @ A16 cos u  þ þj þ @s Ru R0 R0 Ru R0

L32 ¼ 

þ

A16

þ

A26 sin u cos u R20

j

A45 cos u ; Ru R0

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F. Tornabene / Comput. Methods Appl. Mech. Engrg. 200 (2011) 931–952

!

L33 ¼ j

A44 @ 2 cos u 1 dRu @ @2 2A45 þ j A  þ j A þj 44 55 2 @ u2 3 du 2 @ R R u @s Ru u 0 Ru Ru

!

L52 ¼

2



L34

@2 A11 2A12 sin u A22 sin u   ;  @ u@s R2u Ru R0 R20

! B12 sin u A44 @ ¼  2  þj R0 Ru Ru @ u Ru   B16 B26 sin u @ B12 cos u B22 sin u cos u  þ   þ jA45  @s Ru R0 Ru R0 R20 B11

þj

L35

A44 cos u ; R0

! B26 sin u A45 @ ¼  2  þj R0 Ru Ru @ u Ru   B12 B22 sin u @ B16 cos u B26 sin u cos u þ þ   þ jA55 þ @s Ru R0 Ru R0 R20

L41 ¼

!

2

2

B11 @ cos u 1 dRu @ @ 2B16 @ þ B11  þ B66 2 þ Ru R0 R3u du @ u @s Ru @ u@s R2u @ u2

B12 sin u B22 cos2 u A44  þj ; Ru R0 Ru R20 ! B16 @ 2 B26 cos u B16 dRu @ @2 ¼ 2  þ 3 þ B26 2 2 Ru R0 @s Ru @ u Ru du @ u     2 B22 cos u B66 cos u @ B12 B66 @ B16 sin u þ  þ þ þ @s R0 R0 Ru Ru @ u@s Ru R0 

2

þ

B26 cos u R20 B11

L43 ¼

R2u þ B11

L44

A45 sin u ; R0

!   B12 sin u A44 @ B16 B26 sin u @ j þ þ  jA45 @s Ru R0 Ru @ u Ru R0 ! cos u 1 dRu B22 sin u cos u   ; Ru R0 R3u du R20

þ

D12 sin u D22 cos2 u   jA44 ; Ru R0 R20

! D16 @ 2 D26 cos u D16 dRu @ @2 ¼ 2  þ þ D 26 3 Ru R0 @s2 Ru @ u2 Ru du @ u     2 D22 cos u D66 cos u @ D12 D66 @ þ  þ þ @s R0 R0 Ru Ru @ u@s þ

L51

þj

! D11 @ 2 cos u 1 dRu @ @ 2 2D16 @ 2 ¼ 2 þ D11  3 þ D66 2 þ 2 @ @ u R R d u u @s Ru @ u@s u 0 Ru Ru 

L45

L54 ¼

A45 cos u ; R0 2

D16 sin u D26 cos2 u þ  jA45 ; Ru R0 R20

!! B16 @ 2 B26 cos u 2 cos u 1 dRu @ ¼ 2 þ þ B16  3 þ B26 @u Ru R0 Ru R0 Ru du Ru @ u 2     2 @2 B22 cos u B66 cos u @ B12 B66 @  2þ þ þ þ @s @s R0 R0 Ru Ru @ u@s ! cos2 u sin u A45 þj þ B26  ; 2 R R Ru u 0 R0

B16 R2u

þ

þ B26

B16

þj

L42

L53 ¼

B66 @ 2 cos u 1 dRu @ @ 2 2B26 @ 2 þ B  þ B þ 66 22 Ru R0 R3u du @ u @s2 Ru @ u@s R2u @ u2 ! cos2 u sin u A55 sin u þj  ;  B66 Ru R0 R0 R20

L55

!   B26 sin u A45 @ B12 B22 sin u @ j þ þ  jA55 @s Ru R0 Ru @ u Ru R0 ! ! sin u cos u cos u 2 cos u 1 dRu þ  3 þ B16 ; Ru R0 Ru R0 Ru du R20

!! D16 @ 2 D26 cos u 2 cos u 1 dRu @ þ þ D  þ D26 16 @u Ru R0 Ru R0 R2u @ u2 R3u du     2 @2 D22 cos u D66 cos u @ D12 D66 @ þ  2þ þ þ @s @s R0 R0 Ru Ru @ u@s ! cos2 u sin u  jA45 ; þ D26  Ru R0 R20

! D66 @ 2 cos u 1 dRu @ @ 2 2D26 @ 2 ¼ 2 þ D66  3 þ D22 2 þ 2 Ru R0 Ru du @ u @s Ru @ u@s Ru @ u ! cos2 u sin u  jA55 :   D66 Ru R0 R20

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