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Static and free vibration analysis of composite shells by radial basis functions
ARTICLE IN PRESS
Engineering Analysis with Boundary Elements 30 (2006) 719–733 www.elsevier.com/locate/enganabound
Static and free vibration analysis of composite shells by radial basis functions A.J.M. Ferreira, C.M.C. Roque, R.M.N. Jorge Departamento de Engenharia Mecaˆnica e Gesta˜o Industrial, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal Received 27 January 2006; accepted 11 May 2006 Available online 3 July 2006
Abstract The higher-order shear deformation theory of laminated orthotropic elastic shells of Reddy accounts for parabolic distribution of the transverse shear strains through the thickness of the shell. The Reddy shell theory allows the fulfillment of homogeneous conditions (zero values) at the top and bottom surfaces of the shell. This paper deals with a meshless solution of the Reddy higher order shell theory in static and free vibration analysis. The meshless technique is based on the asymmetric global multiquadric radial basis function method proposed by Kansa. This paper demonstrates that this truly meshless method is very successful in the static and free vibration analysis of laminated composite shells. r 2006 Elsevier Ltd. All rights reserved. Keywords: Radial basis functions; Composite materials; Shells; Collocation methods; Free vibrations
1. Introduction Classical theories developed for thin elastic shells are mostly based on the Love–Kirchhoff assumptions. This theory considers that straight lines normal to the undeformed middle surface remain straight and normal to the deformed middle surface; that the normal stress perpendicular to the middle surface can be neglected in the stress–strain relations and the transverse displacement is independent of the thickness coordinate. Therefore, transverse shear strains are neglected as reported in surveys of classical shell theories by Naghdi [1] and Bert [2,3]. These theories are expected to produce accurate results when the side-to-thickness ratio ða=hÞ is large or when material anisotropy is low. The application of such theories to thick or moderately thick or laminated composite shells can lead to serious errors in terms of deflection or stress. The introduction of transverse shear and normal stress represents an improvement to classical theories. However, for a=Ro10 (side to radius ratio) the transverse normal Corresponding author. Tel.: +351 225 081 705; fax: +351 225 081 445.
E-mail address: [email protected] (A.J.M. Ferreira). 0955-7997/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2006.05.002
stress are negligible when compared to the transverse shear stress. The effect of transverse shear and normal stress in shells was studied by Reissner [4]. The effect of transverse shear deformation was also considered by Vinson [5], Dong et al. [6,7] and Whitney and Sun [8,9]. The higher-order theory of Reddy [10] for composite shells is based on five degrees of freedom (same number as in a first-order shear deformation theory by Reddy [11]). This theory assumes a constant transverse deflection through the thickness and the displacements of the middle surface are expanded as cubic functions of the thickness coordinate. The displacement field leads to parabolic distribution of the transverse shear stress and zero transverse normal strain. Therefore, no shear correction factors are used. The analysis of laminated composite shells by meshless methods is still starting. The purpose of the present research is to demonstrate that a truly meshless method such as the multiquadric radial basis function (RBF) method can be successful in the static and free vibration analysis of laminated composite shells. This paper addresses the use of third-order theory in combination with a
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recent meshless technique based on RBFs in a strong form. This approach was not yet tried as far as the author’s knowledge. The multiquadric RBF method was introduced by Hardy [12] for the interpolation of scattered geographical data. The use of multiquadrics for the solution of partial differential equations was first proposed by Kansa [13,14], followed by others in the analysis of boundary-value problems. Here we show for the first time that the RBFs method can be very successful in the static and free vibration analysis of doubly curved laminated elastic shells. We show that the static displacements and stress and the natural frequencies obtained from present method are in excellent agreement with analytical solutions. We prove that this technique is as good as the exact solutions by [10], with a simple yet accurate formulation.
ð0Þ 2 ð2Þ 1 ¼ ð0Þ 1 þ zðk 1 þ z k1 Þ; 2 ð1Þ 4 ¼ ð0Þ 4 þ z k4 ;
The third-order theory of Reddy for the analysis of laminated composite elastic shells is presented. The shell is composed of a finite number of orthotropic layers of uniform thickness (h). Let ðx1 ; x2 ; zÞ denote the orthogonal curvilinear coordinates (or shell coordinates) such that the x1 - and x2 -curves are lines of curvature on the middle surface z ¼ 0, and z-curves are straight lines perpendicular to the middle surface z ¼ 0. For cylindrical and spherical shells the lines of principal curvature coincide with the coordinate lines. The values of the principal radii of curvature are denoted by R1 and R2 . The displacement field is obtained as [10] z 4 1 qw0 u¼ 1þ u0 þ zf1 2 z3 f1 þ , R1 a1 qx1 3h z 4 1 qw0 v¼ 1þ v0 þ zf2 2 z3 f2 þ , R2 a2 qx2 3h (1)
where ai are the surface metrics, u0 , v0 and w0 are the displacements of the middle surface of the shell and f1 and f2 the rotations of the normals to the x1 - and x2 -axes, respectively. This displacement field is used to compute strains and stress and the equations of motion are then obtained by the dynamic version of the principle of virtual work. The first-order displacements can be readily obtained by deleting the z3 terms in (1). If we assume the term z=R in the definition of stress resultants and assume constant radii of curvatures, we can simplify the shell theory. If we consider thin shallow shells we can therefore have z z 1þ 1; 1þ 1 (2) R1 R2 and we can simplify the calculation of stress resultants as mentioned before. As will be seen further in numerical examples these assumptions still produce quite good results for a large range of shell cases.
ð0Þ 2 ð2Þ 2 ¼ ð0Þ 2 þ zðk 2 þ z k2 Þ,
2 ð1Þ 5 ¼ ð0Þ 5 þ z k5 ,
ð0Þ 2 ð2Þ 6 ¼ ð0Þ 6 þ zðk 6 þ z k6 Þ,
ð3Þ
where ð0Þ 1 ¼
qu0 w þ ; qx1 R1
ð0Þ 2 ¼
qv0 w þ ; qx2 R2
ð0Þ 4 ¼
qw0 þ f2 ; qx2
ð0Þ 5 ¼
qw0 þ f1 , qx1
kð1Þ 4
qf1 ; qx1
ð0Þ 6 ¼
qu0 qv0 þ , qx2 qx1
qf2 qf2 qf1 ; kð0Þ þ , 6 ¼ qx2 qx1 qx2 4 qw0 4 qw0 ¼ 2 f2 þ ¼ f þ ; kð1Þ , 1 5 qx2 qx1 h h2
kð0Þ 1 ¼
2. Third-order shear theory
w ¼ w0 ,
The strain–displacement relations referred to an orthogonal curvilinear coordinate system lead to the following deformation field, where xi denote the cartesian coordinates ðdx1 ¼ ai dxi Þ; i ¼ 1; 2 [10].
kð0Þ 2 ¼
4 qf1 q2 w0 þ , qx21 3h2 qx1 4 qf2 q2 w0 ¼ 2 þ , qx22 3h qx2
kð2Þ 1 ¼ kð2Þ 2 kð2Þ 6
4 qf2 qf1 q2 w0 ¼ 2 þ þ2 . qx1 qx2 3h qx1 qx2
(4)
The stress–strain relation for the kth layer are given by rk ¼ Dk ek ,
(5)
where ðkÞ ðkÞ ðkÞ ðkÞ T rk ¼ ½sðkÞ 1 s2 s6 s4 s5 , ðkÞ ðkÞ ðkÞ ðkÞ T ek ¼ ½ðkÞ 1 2 6 4 5
and
2
QðkÞ 11
6 6 6 6 Dk ¼ 6 6 symm: 6 6 4
ð6Þ
QðkÞ 12
QðkÞ 16
0
0
QðkÞ 22
QðkÞ 26
0
0
QðkÞ 66
0
0
QðkÞ 44
0 QðkÞ 55
3 7 7 7 7 7, 7 7 7 5
(7)
where QðkÞ ij are the material constants of the kth layer in the laminate coordinate system [15]. In the following we substitute x1 ; x2 by x; y. Using Hamilton’s principle, the equations of motion of the third-order shell theory are obtained as [15] qN xx qN xy € c1 I 3 qw€ 0 , þ ¼ I 0 u€ 0 þ J 1 f x qx qy qx
(8)
qN xy qN yy qw€ 0 þ ¼ I 0 v€0 þ J 1 f€ y c1 I 3 , qx qy qy
(9)
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qQx qQy 4 qK x qK y þ 2 þ qx qy qx qy h
ð0Þ ð2Þ M xy ¼ B6j ð0Þ j þ D6j kj þ F 6j kj
! q2 Pxy q2 Pxx q2 Pyy N xx N yy þ þ2 þq qx2 qy2 qx qy R1 R2 2 q w€ 0 q2 w€ 0 ¼ I 0 w€ 0 c21 I 6 þ qx2 qy2 " !# qu€ 0 q€v0 qf€ x qf€ y þ c1 I 3 þ þ þ J4 , ð10Þ qx qy qx qy
4 þ 2 3h
qM xx qM xy 4 4 qPxx qPxy þ Qx þ 2 K x 2 þ qx qy qx qy h 3h qw€ 0 , ¼ J 1 u€ 0 þ K 2 f€ x c1 J 4 qx qM xy qM yy 4 4 qPxy qPyy þ Qy þ 2 K y 2 þ qx qy qx qy h 3h qw€ 0 , ¼ J 1 v€0 þ K 2 f€ y c1 J 4 qy
ðQx ; K x Þ ¼
NL Z X k¼1
ðQy ; K y Þ ¼
zk1
NL Z X k¼1
zk
zk
zk1
sðkÞ xz ð1; z
(22)
ð0Þ ð2Þ Pyy ¼ E 2j ð0Þ j þ F 2j kj þ H 2j kj ,
(23)
ð0Þ ð2Þ Pxy ¼ E 6j ð0Þ j þ F 6j kj þ H 6j kj
ðj ¼ 1; 2; 6Þ,
(24)
(25)
ð11Þ ð1Þ Qx ¼ A5j ð0Þ j þ D5j kj
ð12Þ
ðj ¼ 4; 5Þ,
Þ dz,
(26)
ð1Þ K y ¼ D4j ð0Þ j þ F 4j kj ,
ð1Þ K x ¼ D5j ð0Þ j þ F 5j kj
(27)
ðj ¼ 4; 5Þ,
(28)
where Aij , Bij , etc. are the laminate stiffness components obtained by ðAij ; Bij ; Dij ; E ij ; F ij ; H ij Þ ¼
NL Z X k¼1
zk
zk1
2 3 4 6 QðkÞ ij ð1; z; z ; z ; z ; z Þ dz.
(29) 2 sðkÞ yz ð1; z Þ dz,
ð13Þ
where NL denotes the number of the orthotropic layers of the shell and zk , zk1 the top and bottom z-coordinates of the kth lamina. The inertia terms are given by NL Z zk X rðkÞ zi dz ði ¼ 0; 1; 2; . . . ; 6Þ, (14) Ii ¼ k¼1
(21)
ð0Þ ð2Þ Pxx ¼ E 1j ð0Þ j þ F 1j kj þ H 1j kj ,
zk1
2
ðj ¼ 1; 2; 6Þ,
ð1Þ Qy ¼ A4j ð0Þ j þ D4j kj ,
where q is the distributed transverse load, N i , M i , etc. are the stress resultants, given by NL Z zk X 3 sðkÞ ðN i ; M i ; Pi Þ ¼ i ð1; z; z Þ dz ði ¼ xx; yy; xyÞ, k¼1
721
zk1
J i ¼ I i c1 I iþ2 ; K 2 ¼ I 2 2c1 I 4 þ c21 I 6 , 4 ð15Þ c1 ¼ 2 ; c2 ¼ 3c1 3h being rðkÞ the density of the material of layer k. The stress resultants can also be obtained in terms of the strain components by (13) ð0Þ ð2Þ N xx ¼ A1j ð0Þ j þ B1j kj þ E 1j kj ,
(16)
ð0Þ ð2Þ N yy ¼ A2j ð0Þ j þ B2j kj þ E 2j kj ,
(17)
ð0Þ ð2Þ N xy ¼ A6j ð0Þ j þ B6j kj þ E 6j kj
ðj ¼ 1; 2; 6Þ,
(18)
ð0Þ ð2Þ M xx ¼ B1j ð0Þ j þ D1j kj þ F 1j kj ,
(19)
ð0Þ ð2Þ M yy ¼ B2j ð0Þ j þ D2j kj þ F 2j kj ,
(20)
The equations of motion [15] can now be expressed in terms of the displacements u0 , v0 , w0 and the rotations fx , fy , as
2 q2 u0 1 qw0 q v0 1 qw0 þ þ þ A 12 qx2 R1 qx qx qy R2 qx 2 2 q v0 q u0 qw€ 0 þ , ¼ I 0 u€ 0 þ J 1 f€ x c1 I 3 þ A66 qx qy qy2 qx
A11
2 q2 v 0 1 qw0 q u0 A12 qw0 þ þ þ A 12 qy2 R2 qy qx qy R1 qy 2 2 q v 0 q u0 qw€ 0 , þ þ A66 ¼ I 0 v€0 þ J 1 f€ y c1 I 3 qx2 qx qy qy
ð30Þ
A22
q3 fy 32 q3 f q4 w0 2 x 2 2 2 H 66 4 2 qx qy qy qx qy qx 9h 4 3 16 q w0 q fx þ 4 H 11 4 qx qx3 9h
!
! q3 fy q4 w0 q3 fy 4 q3 fx 16 4 þ 4 H 22 3 þ 2 F 22 þ 2 F 11 3 4 qx qy qy qy3 3h 9h 3h ! q3 fy qfy q2 w0 4 q3 fx 16 þ þ þ þ 2 F 12 F 44 qy2 qx qx2 qy qy qy2 3h h4
ð31Þ
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722
! q3 fy 16 q3 fx q4 w0 þ 4 H 12 2 2 2 2 2 qx qy qy qx qy qx 9h 2 8 q w0 qf þ 2 D55 2 x qx qx h 2 2 8 q w0 qfy 16 q w0 qfx þ 4 F 55 þ þ 2 D44 2 qy qy qx2 qx h h ! 3 3 q fy 8 q fx qfx q2 w0 þ þ A þ þ 2 F 66 55 qx2 qy qy2 qx qx qx2 3h qfy q2 w0 2 1 qv0 1 qu0 þ þ A w þ A44 12 0 R1 R2 R1 qy R2 qx qy qy2
þ
A22 qv0 1 A11 qu 1 w0 þ w0 þ q qx R1 R2 qy R2 R1 " 2 2 € € q q qu€ 0 q€v0 w w 0 0 2 þ þ þ c1 I 3 ¼ I 0 w€ 0 c1 I 6 qx2 qy2 qx qy !# qf€ x qf€ y þ þJ 4 , ð32Þ qx qy 2
3
2
!
3
2
q fy q fy 16 q w0 q fx q w0 4 þ 2 þ þ 2 H 12 2 F 22 4 qx qy qx qy qx qy qy qx qy2 9h 3h ! 3 2 4 q w0 q2 f x 16 q3 w0 q fy þ 2 F 11 3 2 2 þ 4 H 22 þ qx qx qy3 qy2 3h 9h 3 16 q w 0 q2 f x 4 þ þ 4 H 11 þ 2 3 2 qx qx 9h 3h ! 2 3 2 q fy q w0 q fx 2 F 12 2 qy qx qx qy qx qy ! 2 q2 fy qw0 q2 fx q fy fx þ D66 þ A55 þ þ D 12 qx qy2 qx qy qx qy ! 2 q fy q2 fx 4 q3 w0 þ 2 F 66 2 2 2 qx qy qy qy qx 3h 8 qw0 þ 2 D55 fx þ qx h 2 q fx 16 qw0 f þ F þ D11 55 x qx2 qx h4 € c1 J 4 qw€ 0 , ð33Þ ¼ J 1 u€ 0 þ K 2 f x qx q2 fy 8 qw0 qw0 þ f f D þ A þ D 44 44 22 y y qy qy qy2 h2 ! q2 fy 4 q3 w 0 4 þ 2 F 22 3 2 2 þ 2 qy qy 3h 3h 3 2 q w0 q fx F 12 2 2 qx qy qx qy 2 8 q3 w 0 q2 fx q fy þ 2 F 66 2 qx qy qx qy qx2 3h
!
þ D66
2 q2 fx q fy þ qxqy qx2
!
q2 fx 16 qw0 þ F 44 fy þ D12 qx qy h4 qy 2 3 16 q fx q w0 þ þ 4 H 12 qx qy qx2 qy 9h q2 f y q2 f x 16 q3 w 0 þ 2 þ 4 H 66 þ qx2 qx qy qx2 qy 9h ! q2 fy q3 w0 16 þ 4 H 22 þ qy2 qy3 9h
!
€ c1 J 4 qw€ 0 . ¼ J 1 v€0 þ K 2 f y qy
ð34Þ
3. Free vibration analysis For free vibration problems we assume harmonic solution in terms of displacements and rotations u0 ; v0 ; w0 ; fx ; fy in the form u0 ðx; y; tÞ ¼ u0 ðw; yÞeiot ,
(35)
v0 ðx; y; tÞ ¼ v0 ðw; yÞeiot ,
(36)
w0 ðx; y; tÞ ¼ w0 ðw; yÞeiot ,
(37)
yx ðx; y; tÞ ¼ yx ðw; yÞeiot ,
(38)
yy ðx; y; tÞ ¼ yy ðw; yÞeiot ,
(39)
where o is the frequency of natural vibration. Removing the external force q and substituting the harmonic expansions into equations of motion we obtain equations of motion in terms of the amplitudes as
2 q2 u0 1 qw0 q v0 1 qw0 þ A11 þ þ A12 qx2 R1 qx qx qy R2 qx 2 q v0 q2 u 0 þ 2 þ A66 qx qy qy qw0 ¼ I 0 o2 u0 J 1 o2 yx þ c1 I 3 o2 , qx 2 q2 v 0 1 qw0 q u0 A12 qw0 þ A22 þ þ A12 qy2 R2 qy qx qy R1 qy 2 q v 0 q2 u0 þ þ A66 qx2 qx qy qw0 , ¼ I 0 o2 v0 J 1 o2 yy þ c1 I 3 o2 qy
ð40Þ
ð41Þ
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q3 fy 32 q3 fx q4 w0 2 H 66 qx2 qy qy2 qx qy2 qx2 9h4 4 16 q w0 q 3 fx þ 4 H 11 4 qx qx3 9h
!
! q3 fy q4 w0 q3 fy 4 q3 fx 16 4 þ 4 H 22 3 þ 2 F 22 þ 2 F 11 3 4 qx qy qy qy3 3h 9h 3h ! q3 fy qfy q2 w0 4 q 3 fx 16 þ þ þ þ 2 F 12 F 44 qy2 qx qx2 qy qy qy2 3h h4 ! q3 fy 16 q3 f q4 w0 þ 4 H 12 2 2 x 2 2 2 qx qy qy qx qy qx 9h 2 8 q w0 qf þ 2 D55 2 x qx qx h 2 2 8 q w0 qfy 16 q w0 qfx þ 4 F 55 þ 2 D44 2 þ qy qy qx2 qx h h ! 3 3 2 q fy 8 q fx qfx q w0 þ þ A þ þ 2 F 66 55 qx2 qy qy2 qx qx qx2 3h qfy q2 w0 2 1 qv0 1 qu0 þ w þ A þ A44 12 0 R1 R2 R1 qy R2 qx qy qy2
A22 qv0 1 A11 qu 1 þ w0 þ w0 qx R1 R2 qy R2 R1 2 q w0 q2 w0 þ ¼ I 0 o2 w0 þ c21 I 6 o2 qx2 qy2 qu0 qv0 qyx qyy c1 o 2 I 3 þ þ þ J4 , qx qy qx qy
ð42Þ
! q2 fy q2 fy 16 q3 w 0 q2 f x q3 w0 4 þ þ þ H F 12 22 qx qy qx2 qy qx qy qy2 qx qy2 9h4 3h2 ! 3 2 4 q w0 q2 f x 16 q3 w0 q fy þ 2 F 11 3 2 2 þ 4 H 22 þ qx qx qy3 qy2 3h 9h 3 16 q w 0 q2 f x þ þ 4 H 11 qx3 qx2 9h ! q2 fy 4 q3 w 0 q2 f x þ 2 F 12 2 2 qy qx qx qy qx qy 3h ! 2 q2 f y qw0 q2 fx q fy fx þ D66 þ A55 þ þ D 12 qx qy2 qx qy qx qy ! 2 q fy q2 fx 4 q3 w0 þ 2 F 66 2 2 2 qx qy qy qy qx 3h 8 qw0 þ 2 D55 fx þ qx h 2 q fx 16 qw0 f þ F þ D11 55 x qx2 qx h4 qw0 , ð43Þ ¼ J 1 o2 u0 K 2 o2 yx þ c1 J 4 o2 qx q2 fy 8 qw0 qw0 þ fy þ A44 fy þ D22 D 2 44 qy qy qy2 h
! q2 fy 4 q3 w0 þ 2 F 22 3 2 2 qy qy 3h 3 4 q w0 q2 fx þ 2 F 12 2 2 qx qy qx qy 3h 2 8 q3 w0 q2 f x q f y þ 2 F 66 2 qx qy qx qy qx2 3h
723
! þ D66
q2 fx 16 qw0 þ 4 F 44 fy qx qy h qy 2 3 16 q fx q w0 þ þ 4 H 12 qx qy qx2 qy 9h
2 q2 fx q fy þ qxqy qx2
!
þ D12
q2 fy q2 fx 16 q3 w 0 þ 2 þ 4 H 66 þ qx2 qx qy qx2 qy 9h ! q2 f y q3 w 0 16 þ 4 H 22 þ qy2 qy3 9h ¼ J 1 o2 v0 K 2 o2 yy þ c1 J 4 o2
!
qw0 . qy
ð44Þ
This set of equations of motion can now be interpolated by multiquadrics RBFs by a global unsymmetric collocation scheme [13,14]. The resulting linear system of equations is later modified by the introduction of boundary conditions, by introducing new lines with the new interpolation equations. The multiquadric RBF method is explained in the next section. 4. The multiquadric radial basis method 4.1. Kansa’s unsymmetric collocation method The discretization is based on the multiquadrics radial basis method [13,14] which is here resumed. Consider a set of nodes x1 ; x2 ; . . . ; xN 2 O Rn . The RBFs centered at xj are defined as gj ðxÞ gðkx xj kÞ 2 Rn ; j ¼ 1; . . . ; N where kx xj k is the Euclidian norm. In this work we use multiquadrics, in the form gj ðxÞ ¼ ðkx xj k2 þ c2 Þ1=2 where c is a shape (user-defined) parameter. In this paper we used c ¼ 2=N 0:5 , N being the total number of nodes, as proposed by Fasshauer [16]. One of the main advantages of RBFs is the insensitivity to spatial dimension, making the implementation of this method much easier than, e.g., finite elements [13,14]. The method does not require a grid or mesh, the pairwise distances between the points being the only geometric properties required by the method. In this paper it is proposed to use Kansa’s unsymmetric collocation method [13,14]. Consider a boundary-valued problem with a domain O Rn and a linear elliptic partial differential equation of the form LuðxÞ ¼ xðxÞ 2 O Rn , BuðxÞjqO ¼ f ðxÞ;
qO Rn ,
(45) (46)
ARTICLE IN PRESS A.J.M. Ferreira et al. / Engineering Analysis with Boundary Elements 30 (2006) 719–733
724
where qO represents the boundary of the problem. We use points along the boundary ðxj ; j ¼ 1; . . . ; N B Þ and in the interior ðxj ; j ¼ N B þ 1; . . . ; NÞ. Let the RBF interpolant to the solution uðxÞ be xðx; cÞ ¼
N X
4.3. Solution of the eigenproblem aj gðkx xj kÞ.
(47)
j¼1
Collocation with the boundary data at the boundary points and with PDE at the interior points leads to equations xB ðx; cÞ
eigenvectors of this matrix are then evaluated by standard techniques.
N X
aj Bgðkxi xj kÞ ¼ lðxi Þ;
i ¼ 1; . . . ; N B ,
j¼1
We follow a simple scheme, as proposed also by Platte and Driscoll [24] for the solution of the eigenproblem (51) and (52). We consider N I nodes in the interior of the domain and N B nodes on the boundary, with N ¼ N I þ N B. We denote interpolation points by xi 2 O; i ¼ 1; . . . ; N I and xi 2 qO; i ¼ N I þ 1; . . . ; N. For the interior points we have that
(48) xL ðx; cÞ
N X
N X
aj Lgðkxi xj kÞ ¼ Fðxi Þ;
i ¼ N B þ 1; . . . ; N,
j¼1
(49) where lðxi Þ, Fðxi Þ are the prescribed values at the boundary nodes and the function values at the interior nodes, respectively. This corresponds to a system of equations with an unsymmetric coefficient matrix, structured in matrix form as " # Bg l ½a ¼ . (50) Lg U The use of globally supported RBFs for large problems can bring problems due to the full populated matrices. To solve this drawback a localization scheme is advisable. Domain decomposition methods [17,18], localization of the basis functions [19,18] claim to be able to deal with tens of thousands of nodes. These techniques are not used in this paper. The discretization method applied in the present work follows closely the work of the authors for composite plates in [20–23] for first-order and third-order shear deformation theories.
Consider a linear elliptic partial differential operator L and a bounded region O in Rn with some boundary qO. The eigenproblem seeks eigenvalues (l) and eigenvectors ðuÞ that satisfy
LB u ¼ 0
in O,
on qO,
j ¼ 1; 2; . . . ; N I
(53)
or LI a ¼ le uI ,
(54)
where LI ¼ ½Lgðkx xj kÞN I N .
(55)
For the boundary conditions we have N X
ai LBg ðkx xj kÞ ¼ 0;
j ¼ N I þ 1; . . . ; N
(56)
i¼1
or B a ¼ 0.
(57)
Therefore we can write a finite-dimensional problem as a generalized eigenvalue problem " # " # LI AI a¼l a, (58) B 0 where AI ¼ g½ðkx xj kÞN I N ;
4.2. The eigenproblem
Lu lu ¼ 0
ai Lgðkx xj kÞ ¼ le uðxj Þ;
i¼1
BI ¼ LBg ½ðkx xj kÞN B N .
We seek the generalized eigenvalues and eigenvectors of these matrices.
5. Multiquadrics interpolation of equations of motion
(51) (52)
where LB is a linear boundary operator. The eigenproblem of (51) and (52) is replaced by a finite-dimensional eigenvalue problem, based on RBF approximations. The operator L is then approximated by a matrix that incorporates the boundary conditions. The eigenvalues and
Applying the multiquadrics method previously explained, the equations of motion are now interpolated, for each node. In the following equations, a is equivalent to a. " # " # 2 N N X X q2 gj q2 gj u0 v 0 q gj ðA12 þ A66 Þ aj A11 2 þ A66 2 þ aj qx qy qx qy j¼1 j¼1
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þ
N X
awj 0
j¼1
"
qgj A11 A12 þ qx R1 R2
¼ I 0 o2
N X
auj 0 gj J 1 o2
j¼1
þ c1 I 3 o2
N qgj X qgj þ avj 0 c1 o I 3 qx j¼1 qy j¼1 !# N N X qgj X y qgj þ ayj x aj y þJ 4 , qx j¼1 qy j¼1
ayj x gj
j¼1
N X
awj 0
j¼1
qgj , qx
ð59Þ
N X
awj 0
j¼1 N X
"
#
"
N X q2 gj q2 gj q2 gj ðA66 þ A12 Þ þ avj 0 A66 2 þ A22 2 qx qy qx qy j¼1 j¼1 N X qgj A12 A22 þ awj 0 þ qy R1 R2 j¼1
#
auj 0
¼ I 0 o2
N X
avj 0 gj J 1 o2
j¼1
þ c1 I 3 o2
N X
y
aj y gj
j¼1
N X
awj 0
j¼1
qgj , qy
ð60Þ
X N qgj A12 A11 qgj A22 A12 v0 þ aj þ þ qx R2 R1 qy R2 R1 j¼1 j¼1 " N X A12 A22 A11 awj 0 gj 2 2 2 þ R1 R2 R2 R1 j¼1 q2 gj 16 8 þ 2 þ 4 F 55 2 D55 þ A55 qx h h 2 q gj 16 q4 gj 8 16 þ 2 F D þ A H 44 44 44 11 qy h4 qx4 h2 9h4 # q4 gj q4 gj 16 64 32 4 H 22 4 2 2 H 66 þ 4 H 12 qy qy qx 9h4 9h 9h " N X 16 8 fx qgj þ A55 þ 4 F 55 2 D55 aj qx h h j¼1 3 q gj 4 16 F 11 4 H 11 þ 3 2 qx 3h 9h # 3 q gj 4 32 8 16 þ 2 F 12 4 H 66 þ 2 F 66 4 H 12 qy qx 3h2 9h 3h 9h
N X
auj 0
"
qgj 16 8 F D þ A 44 44 44 qy h4 h2 j¼1 q3 gj 4 16 F 22 4 H 22 þ 3 qy 3h2 9h # 3 q gj 8 4 32 16 þ 2 F 66 þ 2 F 12 4 H 66 4 H 12 qx qy 3h2 3h 9h 9h þ
N X
f aj y
2
¼ I 0 o
N X j¼1
awj 0 gj
þ
N X
2
N X
c21 I 6 o2
N X j¼1
awj 0
725
q2 gj q2 gj þ 2 qx2 qy
!
!
auj 0
ð61Þ
qgj 8 16 D F A 55 55 55 qx h2 h4
q3 gj q3 gj 16 16 H þ H 22 12 qy3 9h4 qx2 qy 9h4 3 q gj 16 q3 gj 4 H F þ 3 þ 11 11 qx 9h4 qy2 qx 3h2 16 4 8 H F F 12 12 66 9h4 3h2 3h2 N X 8 16 fx aj gj 2 D55 A55 4 F 55 þ h h j¼1 2 q gj 8 16 þ 2 D11 2 F 11 þ 4 H 11 qx 3h 9h # 2 q gj q2 gj 16 4 4 þ 2 D66 2 F 66 þ H 12 2 F 12 qy qx qy 9h4 3h 3h " 2 N X f q gj 16 4 þ aj y H 22 2 F 22 4 2 qy 9h 3h j¼1 # q2 gj 8 4 16 D12 2 F 12 2 F 66 þ D66 þ 4 H 12 þ qx qy 3h 3h 9h þ
¼ J 1 o2
N X
auj 0 gj K 2 o2
j¼1
þ c1 J 4 o 2
N X j¼1
"
N X
ayj x gj
j¼1
awj 0
qgj , qx
ð62Þ
qgj 8 16 D A F 44 44 44 qy h2 h4 j¼1 q3 gj 16 4 H 22 2 F 22 þ 3 qy 9h4 3h # 3 q gj 16 32 8 4 þ 2 H 12 þ 4 H 66 2 F 66 2 F 12 qx qy 9h4 9h 3h 3h " N X q2 gj 8 8 f þ D66 D12 2 F 66 2 F 12 aj x qxqy 3h 3h j¼1 # 16 16 þ 4 H 66 þ 4 H 12 9h 9h " N X 8 16 f þ aj y gj 2 D44 A44 4 F 44 h h j¼1 q2 gj 8 16 þ 2 D66 2 F 66 þ 4 H 66 qx 3h 9h N X
awj 0
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# q 2 gj 8 16 þ 2 D22 2 F 22 þ 4 H 22 qy 3h 9h ¼ J 1 o2
N X
avj 0 gj K 2 o2
j¼1
þ c1 J 4 o 2
N X
N X
the model is validated for a composite plate. The second example consists of a spherical shell in bending by distributed loading. All examples consider laminated composites and are subjected to bending loads. The material properties for each layer are the following:
y
aj y gj
j¼1
awj 0
j¼1
E 1 ¼ 25:0E 2 ; G 12 ¼ G 13 ¼ 0:5E 2 ; n12 ¼ n13 ¼ 0:25,
qgj . qy
wðx ¼ aÞ ¼ 0,
(63)
uðx ¼ aÞ ¼ 0,
(64)
fx ðx ¼ aÞ ¼ 0,
(65)
M x ðx ¼ aÞ ¼ 0,
(66)
N x ðx ¼ aÞ ¼ 0,
(67)
can be interpolated by RBF, respectively, by awj 0 gj ¼ 0,
(68)
auj 0 gj ¼ 0,
(69)
j¼1 N X j¼1 N X
f
aj x gj ¼ 0,
(70)
j¼1
"
q2 gj 16 4 H F 11 11 qx2 9h4 3h2 j¼1 # q2 gj 16 4 H 12 2 F 12 þ 2 qy 9h4 3h N X 8 16 f qgj D11 2 F 11 þ 4 H 11 þ aj x qx 3h 9h j¼1 N X 8 16 f qgj D12 2 F 12 þ 4 H 12 þ aj y ¼ 0, qy 3h 9h j¼1
N X
N X j¼1
ð71Þ
ð72Þ
6. Numerical problems 6.1. Static problems The accuracy of the present model for static analysis is examined in two numerical examples. In the first example
6.1.1. Composite plate in bending A square simply supported laminated plate under sinusoidal pressure is considered. The sinusoidal force is given by px py sin (74) P ¼ P0 sin a a acting on a square plate of side a, where P0 is the uniform pressure. In order to model the plate (infinite radii), the shell radii are set equal to R1 =a ¼ R2 =a ¼ 109 . In this example the present model is compared in Tables 1 and 2 with other models. The finite strip formulation of Akhras [25], the finite element formulation of Reddy [26], both considering a third-order theory, the finite strip model of Akhras et al. [27] considering a first-order theory and the work of Ferreira et al. [22,28] using a third-order shear deformation theory and also a layerwise theory and multiquadrics for plates are considered for comparison. The exact (elasticity) solution of Pagano [29] is also presented for comparison. Results are presented in Tables 1 and 2, in normalized form as
txz
N N qgj X qgj X A11 A12 þ þ avj 0 A12 awj 0 gj þ qx j¼1 qy R1 R2 j¼1
¼ 0.
Two cross-ply laminates are considered: ½0 =90 =90 =0 and ½0 =90 =0 , with the origin of the coordinate system being located at the lower left corner on the midplane.
102 E 2 h3 ; P0 a4 h . ¼ txz P0 a
w¼w
awj 0
auj 0 A11
ð73Þ
The interpolation of boundary conditions follows a similar procedure as in the interpolation equilibrium equations. A typical simply supported boundary condition in the form
N X
G 23 ¼ 0:2E 2 ,
si ¼ si
h2 ; i ¼ x; y; P0 a2
txy ¼ txy
h2 , P0 a2 ð75Þ
Here we consider a uniform grid with N being the number of points in x-direction and y-direction. Therefore 11 11, 15 15, and 21 21 uniform grids are used. Thick (a=h ¼ 4) to thin (a=h ¼ 100) plates are considered. In all cases our shell model can predict with good accuracy the transverse deflections, the normal stress and the transverse shear stress. Even in thicker plates our results are in good agreement with other higher-order formulations. 6.1.2. Spherical shell in bending A laminated composite spherical shell is here considered. The shell is loaded with the same sinusoidal load as in the plate example and is simply supported in all edges. In Table 3 we compare the present shell model with results of Reddy shell formulation using first-order and third-order shear-deformation theories. We consider various values of R=a and two values of a=h. Results are in good agreement for various a=h ratios, in particular with
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Table 1 Non-dimensional deflection and stress results for different methods and shear deformation theories, for a ½0 =90 =90 =0 shell, with R=a ¼ 109 a=h
Method
w
sx ða=2; a=2; h=2Þ
sy ða=2; a=2; h=4Þ
tzx ð0; a=2; 0Þ
txy ð0; 0; h=2Þ
4
3 strip [25] HSDT [26] FSDT [27] Elasticity [29] Ferreira et al. [22] ðN ¼ 21Þ Ferreira (layerwise) [28] ðN ¼ 21Þ Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ
1.8939 1.8937 1.7100 1.954 1.8864 1.9075 1.8848 1.8885 1.8900
0.6806 0.6651 0.4059 0.720 0.6659 0.6432 0.6616 0.6604 0.6599
0.6463 0.6322 0.5765 0.666 0.6313 0.6228 0.6293 0.6307 0.6312
0.2109 0.2064 0.1398 0.270 0.1352 0.2166 0.2185 0.2148 0.2128
0.0450 0.0440 0.0308 0.0467 0.0433 0.0441 0.0458 0.0463 0.0466
10
3 strip [25] HSDT [26] FSDT [27] Elasticity [29] Ferreira et al. [22] ðN ¼ 21Þ Ferreira (layerwise) [28] ðN ¼ 21Þ Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ
0.7149 0.7147 0.6628 0.743 0.7153 0.7309 0.7115 0.7120 0.7121
0.5589 0.5456 0.4989 0.559 0.5466 0.5496 0.5413 0.5411 0.5411
0.3974 0.3888 0.3615 0.403 0.4383 0.3956 0.3873 0.3874 0.3874
0.2697 0.2640 0.1667 0.301 0.3347 0.2888 0.2981 0.2911 0.2921
0.0273 0.0268 0.0241 0.0276 0.0267 0.0273 0.0269 0.0271 0.0272
Table 2 Non-dimensional deflection and stress results for different methods and shear deformation theories, for a ½0 =90 =90 =0 shell, with R=a ¼ 109 a=h
Method
w
sx ða=2; a=2; h=2Þ
sy ða=2; a=2; h=4Þ
tzx ð0; a=2; 0Þ
txy ð0; 0; h=2Þ
20
3 strip [25] HSDT [26] FSDT [27] Elasticity [29] Ferreira et al. [22] ðN ¼ 21Þ Ferreira (layerwise) [28] ðN ¼ 21Þ Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ
0.5061 0.5060 0.4912 0.517 0.5070 0.5121 0.5039 0.5035 0.5032
0.5523 0.5393 0.5273 0.543 0.5405 0.5417 0.5365 0.5358 0.5354
0.3110 0.3043 0.2957 0.309 0.3648 0.3056 0.3028 0.3027 0.3026
0.2883 0.2825 0.1749 0.328 0.3818 0.3248 0.3238 0.3259 0.3262
0.0233 0.0228 0.0221 0.0230 0.0228 0.0230 0.0226 0.0227 0.0228
100
3 strip [25] HSDT [26] FSDT [27] Elasticity [29] Ferreira et al. [22] ðN ¼ 21Þ Ferreira (layerwise) [28] ðN ¼ 21Þ Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ
0.4343 0.4343 0.4337 0.4347 0.4365 0.4374 0.4496 0.4367 0.4327
0.5507 0.5387 0.5382 0.539 0.5413 0.5420 0.5546 0.5408 0.5364
0.2769 0.2708 0.2705 0.271 0.3359 0.2697 0.2739 0.2706 0.2694
0.2948 0.2897 0.1780 0.339 0.4106 0.3232 0.3737 0.3466 0.3454
0.0217 0.0213 0.0213 0.0214 0.0215 0.0216 0.0226 0.0216 0.0213
the higher-order results of Reddy [27]. It is important to notice that our present approach agrees quite well with Reddy third-order theory for shells, but can present small differences when compared with Reddy first-order shell theory for thicker shells. This shows that the formulation is consistent. This remark is relevant to understand the results of Tables 4 and 5. In Tables 4 and 5 we compare our solution with an exact solution and various finite element solutions [11] for a ½0 =90 =90 =0 simply supported spherical shell, under sinusoidal pressure. Here we set h ¼ 0:1 and we consider
various R=h ratios. The transverse deflection and stress are normalized as 102 E 2 h3 h2 ; s ¼ s ; i ¼ x; y, i i P0 a4 P0 a2 h2 tij ¼ tij ð1 dij Þ; i; j ¼ x; y; z. P 0 a2 w¼w
ð76Þ
Our results are in reasonable agreement with CFS and finite element results of Reddy [28] using a first-order shear deformation theory for shells. The agreement is better for
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Table 3 Non-dimensional central deflection, w ¼ wð102 E 2 h3 =P0 a4 Þ variation with various number of grid points per unit length, N for different R=a ratios, for R1 ¼ R2 a=h
½0 =90 =0
½0 =90 =90 =0
Method
R=a 5
10
20
50
100
109
10 10 10 10 10
Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ HSDT [10] FSDT [10]
6.7047 6.7308 6.7396 6.7688 6.4253
6.9900 6.9994 7.0028 7.0325 6.6247
7.0652 7.0700 7.0718 7.1016 6.6756
7.0865 7.0900 7.0914 7.1212 6.6902
7.0896 7.0928 7.0942 7.1240 6.6923
7.0906 7.0938 7.0951 7.125 6.6939
100 100 100 100 100
Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ HSDT [10] FSDT [10]
0.9608 1.0084 1.0253 1.0321 1.0337
2.3356 2.3810 2.3964 2.4099 2.4109
3.6423 3.6104 3.6003 3.617 3.6150
4.3199 4.2209 4.1897 4.2071 4.2027
4.4379 4.3254 4.2900 4.3074 4.3026
4.4786 4.3614 4.3244 4.3420 4.3370
10 10 10 10 10
Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ HSDT [10] FSDT [10]
6.7237 6.7515 6.7608 6.7865 6.3623
7.0127 7.0239 7.0277 7.0536 6.5595
7.0889 7.0955 7.0978 7.1237 6.6099
7.1105 7.1158 7.1176 7.1436 6.6244
7.1136 7.1187 7.1205 7.1464 6.6264
7.1146 7.1196 7.1214 7.1474 6.6280
100 100 100 100 100
Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ HSDT [10] FSDT [10]
0.9572 1.0034 1.0198 1.0264 1.0279
2.3335 2.3752 2.3894 2.4024 2.4030
3.6495 3.6101 3.5974 3.6133 3.6104
4.3349 4.2254 4.1908 4.2071 4.2015
4.4544 4.3309 4.2919 4.3082 4.3021
4.4957 4.3672 4.3284 4.3430 4.3368
Table 4 Non-dimensional deflection and stress results for different methods, for a ½0 =90 =90 =0 shell, with R1 ¼ R2 , a=h ¼ 10 R=h
Method
w
sx ða=2; a=2; h=2Þ
sy ða=2; a=2; h=4Þ
txy ð0; 0; h=2Þ
txz ð0; a=2; 0Þ
10
CFS FEM Q4-R [11] FEM Q2-F9 [11] FEM L2-F [11] Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ
0.3229 0.3229 0.3241 0.2337 0.2912 0.3015 0.3052
0.1678 0.1678 0.1632 0.1459 0.1663 0.1746 0.1775
0.1032 0.1032 0.1000 0.1165 0.1034 0.1095 0.1117
0.04055 0.04055 0.03947 0.02184 0.03753 0.03934 0.04001
0.01824 0.01824 0.01772 0.01153 0.00883 0.01112 0.01235
20
CFS FEM Q4-R [11] FEM Q2-F9 [11] FEM L2-F [11] Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ
0.5254 0.5254
0.3426 0.3426
0.2331 0.2330
0.04295 0.04294
0.03201 0.03201
0.4276 0.5221 0.5310 0.5341
0.2597 0.3475 0.3555 0.3582
0.1994 0.2343 0.2408 0.2430
0.02615 0.04367 0.04480 0.04522
0.02245 0.01883 0.02097 0.02207
lower R=h ratios which proves the flexibility of the theory for analyzing both plates and shells of higher curvature. In Tables 6 and 7 the present approach is compared with third-order theory of Reddy and first-order theory results by Reddy [10], for various values of R=a and various laminates for spherical shells, under uniform load. The results clearly show an excellent agreement with Reddy third-order theory and some expected deviation from firstorder theory as mentioned before. For all laminates and for all R=a ratios results for central deflection are very good.
6.2. Free vibration problems In free vibration problems we consider nodal regular grids with 7 7 up to 17 17 points. Tables 8–10 contain nondimensionalized natural frequencies obtained using the first- and higher-order theories for various cross-ply spherical shells. Present results are compared with analytical solutions by Reddy and Liu [10] who considered both the first-order (FSDT) and the third-order (HSDT) theories. Analogous to cylindrical shells, the first-order
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Table 5 Non-dimensional deflection and stress results for different methods, for a ½0 =90 =90 =0 shell, with R1 ¼ R2 , a=h ¼ 10 Method
w
sx
sy
txy
txz
CFS FEM Q4-R [11] FEM Q2-F9 [11] FEM L2-F [11] Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ
0.6362 0.6361 0.6357
0.4541 0.4540 0.4427
0.3216 0.3215 0.3128
0.03474 0.03473 0.03391
0.03955 0.03955 0.03865
0.6724 0.6752 0.6761
0.4859 0.4887 0.4896
0.3402 0.3428 0.3437
0.03777 0.03822 0.03840
0.02661 0.02771 0.02829
100
CFS FEM Q4-R [11] FEM Q2-F9 [11] FEM L2-F [11] Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ
0.6559 0.6558 0.6357 0.5813 0.7013 0.7024 0.7028
0.4797 0.4796 0.4678 0.3349 0.5202 0.5211 0.5214
0.3437 0.3437 0.3344 0.2484 0.3682 0.3694 0.3698
0.02979 0.02978 0.02909 0.02000 0.03295 0.03326 0.03339
0.04090 0.04089 0.03997 0.03111 0.02857 0.02918 0.02953
109 (Plate)
CFS FEM Q4-R [11] FEM Q2-F9 [11] FEM L2-F [11] Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ
0.6627 0.6627 0.6623 0.5901 0.7115 0.7120 0.7121
0.4995 0.4954 0.4832 0.3339 0.5413 0.5211 0.5214
0.3589 0.3589 0.3492 0.2454 0.3873 0.3694 0.3698
0.02397 0.02396 0.02340 0.01629 0.02688 0.03326 0.03339
0.04136 0.04136 0.04043 0.03161 0.02981 0.02994 0.03005
R=h 50
Table 6 Non-dimensional central deflection, w ¼ wð102 E 2 h3 =P0 a4 Þ variation with various number of grid points per unit length, N for different R=a ratios, for R1 ¼ R2 , with uniform load (½0 =90 , ½0 =90 =0 ) a=h
½0 =90
½0 =90 =0
Method
R=a 5
10
20
50
100
109
10 10 10 10 10
Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ HSDT [10] FSDT [10]
17.1112 17.4314 17.5408 17.566 19.944
18.4090 18.6475 18.7296 18.744 19.065
18.7638 18.9780 19.0518 19.064 19.365
18.8659 19.0730 19.1442 19.155 19.452
18.8807 19.0867 19.1578 19.168 19.464
18.8858 19.0914 19.1623 19.172 19.469
100 100 100 100 100
Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ HSDT [10] FSDT [10]
1.5323 1.6849 1.7370 1.7519 1.7535
5.0401 5.3711 5.5008 5.5388 5.5428
10.9689 11.1385 11.2346 11.268 11.273
16.1840 15.7804 15.7221 15.711 15.714
17.3556 16.7727 16.6678 16.642 16.645
17.7844 17.1314 17.008 16.977 16.980
10 10 10 10 10
Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ HSDT [10] FSDT [10]
10.132 10.285 10.339 10.332 9.7937
10.741 10.841 10.876 10.862 10.191
10.777 10.874 10.908 10.893 10.214
10.783 10.878 10.913 10.898 10.218
10.784 10.88 10.914 10.899 10.220
100 100 100 100 100
Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ HSDT [10] FSDT [10]
1.3436 1.4589 1.499 1.5092 1.5118
theory underpredicts fundamental natural frequencies of antisymmetric cross-ply shells; for symmetric thick shells and symmetric shallow thin shells the trend reverses. The present radial basis function method is compared with analytical results by Reddy [10] and shows excellent
10.614 10.725 10.765 10.752 10.110 3.3984 3.562 3.6235 3.6426 3.6445
5.3948 5.4899 5.5358 5.5503 5.5473
6.4392 6.4562 6.4791 6.4895 6.4827
6.6216 6.6221 6.6421 6.6496 6.6421
6.6846 6.6793 6.6975 6.7047 6.6970
agreement. In fact for both first- and third-order approaches differences are lower than 1% for all ratios and laminates. Tables 11 and 12 contain nondimensionalized natural frequencies obtained using the higher-order theory for
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Table 7 Non-dimensional central deflection, w ¼ wð102 E 2 h3 =P0 a4 Þ variation with various number of grid points per unit length, N for different R=a ratios, for R1 ¼ R2 , with uniform load (½0 =90 =90 =0 ) a=h
½0 =90 =90 =0
R=a
Method
5
10
20
50
100
109
10.761 10.876 10.917 10.904 10.141
10.891 10.994 11.031 11.017 10.222
10.928 11.028 11.063 11.049 10.245
10.933 11.033 11.068 11.053 10.249
10.935 11.034 11.069 11.055 10.251
10 10 10 10 10
Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ HSDT [10] FSDT [10]
10.269 10.427 10.482 10.476 9.8249
100 100 100 100 100
Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ HSDT [10] FSDT [10]
1.3666 1.482 1.5225 1.5332 1.5358
3.4735 3.6379 3.6997 3.7195 3.7208
5.5172 5.608 5.6511 5.666 5.6618
6.5875 6.5953 6.6153 6.6234 6.6148
6.7745 6.7649 6.7785 6.7866 6.7772
6.8392 6.8233 6.8366 6.8427 6.8331
Table 8 pffiffiffiffiffiffiffiffiffiffiffi Nondimensionalized fundamental frequencies of cross-ply laminated spherical shells, o ¼ oða2 =hÞ r=E 2 , laminate (½0 =90 =90 =0 ) a=h
½0 =90 =90 =0
R=a
Method
5
10
20
50
100
109
10
Present ðN ¼ 7Þ Present ðN ¼ 9Þ Present ðN ¼ 11Þ Present ðN ¼ 13Þ Present ðN ¼ 15Þ Present ðN ¼ 17Þ HSDT [10]
12.239 12.127 12.085 12.066 12.056 12.051 12.040
11.933 11.881 11.862 11.854 11.850 11.848 11.840
11.854 11.818 11.806 11.800 11.798 11.796 11.790
11.832 11.800 11.790 11.785 11.783 11.782 11.780
11.829 11.798 11.787 11.783 11.781 11.780 11.780
11.828 11.797 11.787 11.782 11.780 11.779 11.780
100
Present ðN ¼ 7Þ Present ðN ¼ 9Þ Present ðN ¼ 11Þ Present ðN ¼ 13Þ Present ðN ¼ 15Þ Present ðN ¼ 17Þ HSDT [10]
35.544 33.213 32.176 31.681 31.430 31.296 31.100
21.118 20.816 20.612 20.506 20.457 20.422 20.380
15.436 16.260 16.467 16.545 16.579 16.595 16.630
13.412 14.730 15.102 15.250 15.318 15.359 15.420
13.097 14.498 14.896 15.055 15.130 15.170 15.230
12.990 14.420 14.827 14.990 15.067 15.109 15.170
Table 9 pffiffiffiffiffiffiffiffiffiffiffi Nondimensionalized fundamental frequencies of cross-ply laminated spherical shells, o ¼ oða2 =hÞ r=E 2 , laminate (½0 =90 =0 ) a=h
Method
R=a 5
10
20
50
100
109
10
Present ðN ¼ 7Þ Present ðN ¼ 9Þ Present ðN ¼ 11Þ Present ðN ¼ 13Þ Present ðN ¼ 15Þ Present ðN ¼ 17Þ HSDT [10]
12.242 12.134 12.094 12.076 12.068 12.063 12.060
11.935 11.889 11.873 11.867 11.863 11.861 11.860
11.856 11.826 11.817 11.813 11.811 11.810 11.810
11.834 11.809 11.801 11.798 11.797 11.796 11.790
11.831 11.806 11.799 11.796 11.794 11.794 11.790
11.830 11.805 11.798 11.795 11.794 11.793 11.790
100
Present ðN ¼ 7Þ Present ðN ¼ 9Þ Present ðN ¼ 11Þ Present ðN ¼ 13Þ Present ðN ¼ 15Þ Present ðN ¼ 17Þ HSDT [10]
35.590 33.177 32.114 31.609 31.353 31.216 31.020
21.221 20.828 20.600 20.484 20.425 20.394 20.350
15.587 16.297 16.480 16.547 16.578 16.595 16.620
13.588 14.778 15.124 15.261 15.326 15.361 15.420
13.277 14.548 14.920 15.068 15.139 15.176 15.240
13.172 14.471 14.851 15.004 15.077 15.115 15.170
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Table 10 pffiffiffiffiffiffiffiffiffiffiffi Nondimensionalized fundamental frequencies of cross-ply laminated spherical shells, o ¼ oða2 =hÞ r=E 2 , laminate (½0 =90 ) a=h
Method
R=a 5
10
Present ðN ¼ 7Þ Present ðN ¼ 9Þ Present ðN ¼ 11Þ Present ðN ¼ 13Þ Present ðN ¼ 15Þ Present ðN ¼ 17Þ HSDT [10]
100
Present ðN ¼ 7Þ Present ðN ¼ 9Þ Present ðN ¼ 11Þ Present ðN ¼ 13Þ Present ðN ¼ 15Þ Present ðN ¼ 17Þ HSDT [10]
9.5185 9.4085 9.3697 9.3536 9.3460 9.3420 9.3370 33.721 31.149 30.011 29.470 29.195 29.049 28.840
10 9.1233 9.0859 9.0754 9.0717 9.0702 9.0694 9.0680 17.698 17.287 17.014 16.874 16.805 16.762 16.710
20 9.0213 9.0031 8.9999 8.9995 8.9995 8.9995 8.9990 10.169 11.379 11.647 11.740 11.785 11.806 11.840
100
109
8.9926 8.9798 8.9787 8.9792 8.9796 8.9798 8.9800
8.9886 8.9765 8.9757 8.9763 8.9767 8.9770 8.9770
8.9873 8.9755 8.9747 8.9753 8.9758 8.9761 8.9760
6.6587 9.0525 9.6176 9.8312 9.9302 9.9822 10.060
5.9895 8.6690 9.2917 9.5167 9.6389 9.6929 9.7840
5.7476 8.5372 9.1806 9.4196 9.5375 9.5905 9.6880
50
Table 11 pffiffiffiffiffiffiffiffiffiffiffi Nondimensionalized fundamental frequencies of cross-ply cylindrical shells, o ¼ oða2 =hÞ r=E 2 R=a
Method
[0/90]
[0/90/0]
[0/90/90/0]
a=h ¼ 100
a=h ¼ 10
a=h ¼ 100
a=h ¼ 10
a=h ¼ 100
a=h ¼ 10
5
Present ðN ¼ 7Þ Present ðN ¼ 9Þ Present ðN ¼ 11Þ Present ðN ¼ 13Þ Present ðN ¼ 15Þ Present ðN ¼ 17Þ FSDT [10] HSDT [10]
20.2936 18.4646 17.5967 17.1762 16.9639 16.8523 16.668 16.690
9.1302 9.0993 9.0943 9.0941 9.0945 9.0949 8.9082 9.0230
24.7600 22.5379 21.4708 20.9488 20.6803 20.5357 20.332 20.330
11.9952 11.9068 11.8754 11.8617 11.8550 11.8513 12.207 11.850
23.3888 21.8946 21.1533 20.7891 20.6023 20.5033 20.361 20.360
11.9661 11.8873 11.8592 11.8469 11.8408 11.8376 12.267 11.830
10
Present ðN ¼ 7Þ Present ðN ¼ 9Þ Present ðN ¼ 11Þ Present ðN ¼ 13Þ Present ðN ¼ 15Þ Present ðN ¼ 17Þ FSDT [10] HSDT [10]
11.2950 11.8050 11.8397 11.8349 11.8251 11.8250 11.831 11.840
9.0109 9.0053 9.0084 9.0112 9.0129 9.0139 8.8879 8.9790
16.8901 16.8697 16.7605 16.6947 16.6500 16.6417 16.625 16.620
11.8717 11.8310 11.8177 11.8120 11.8092 11.8076 12.173 11.800
16.2680 16.6195 16.6423 16.6353 16.6293 16.6289 16.634 16.630
11.8626 11.8196 11.8050 11.7986 11.7954 11.7937 12.236 11.790
20
Present ðN ¼ 7Þ Present ðN ¼ 9Þ Present ðN ¼ 11Þ Present ðN ¼ 13Þ Present ðN ¼ 15Þ Present ðN ¼ 17Þ FSDT [10] HSDT [10]
7.4668 9.4336 9.8965 10.0695 10.1436 10.1955 10.265 10.27
8.9872 8.9824 8.9850 8.9874 8.9888 8.9895 8.8900 8.972
14.1975 15.1074 15.3513 15.4441 15.4915 15.5106 15.556 15.55
11.8402 11.8118 11.8032 11.7995 11.7976 11.7966 12.166 11.79
13.8853 15.0009 15.3017 15.4182 15.4727 15.5022 15.559 15.55
11.8364 11.8025 11.7913 11.7865 11.7840 11.7827 12.230 11.78
cross-ply cylindrical shells with lamination schemes [0/90], [0/90/0], [0/90/90/0]. Present results are compared with analytical solutions by Reddy and Liu [10] who considered both the first-order (FSDT) and the third-order (HSDT) theories. The first-order theory underpredicts fundamental natural frequencies of antisymmetric cross-ply shells; for
symmetric thick shells and symmetric shallow thin shells the trend reverses. The present radial basis function method is compared with analytical results by Reddy [10] and shows excellent agreement. In fact for both first- and third-order approaches differences are lower than 1% for all ratios and laminates.
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Table 12 pffiffiffiffiffiffiffiffiffiffiffi Nondimensionalized fundamental frequencies of cross-ply cylindrical shells, o ¼ oða2 =hÞ r=E 2 R=a
Method
[0/90]
[0/90/0]
[0/90/90/0]
a=h ¼ 100
a=h ¼ 10
a=h ¼ 100
a=h ¼ 10
a=h ¼ 100
a=h ¼ 10
50
Present ðN ¼ 7Þ Present ðN ¼ 9Þ Present ðN ¼ 11Þ Present ðN ¼ 13Þ Present ðN ¼ 15Þ Present ðN ¼ 17Þ FSDT [10] HSDT [10]
6.0147 8.6719 9.2903 9.5239 9.6417 9.7022 9.7816 9.783
8.9847 8.9763 8.9773 8.9787 8.9796 8.9802 8.8951 8.973
13.3418 14.5747 14.9324 15.0764 15.1422 15.1789 15.244 15.24
11.8313 11.8065 11.7991 11.7960 11.7944 11.7935 12.163 11.79
13.1374 14.5144 14.9043 15.0616 15.1326 15.1726 15.245 15.23
11.8290 11.7977 11.7875 11.7831 11.7808 11.7796 12.228 11.78
100
Present ðN ¼ 7Þ Present ðN ¼ 9Þ Present ðN ¼ 11Þ Present ðN ¼ 13Þ Present ðN ¼ 15Þ Present ðN ¼ 17Þ FSDT [10] HSDT [10]
5.7976 8.5648 9.2045 9.4466 9.5542 9.6201 9.7108 9.712
8.9854 8.9756 8.9757 8.9768 8.9775 8.9602 8.8974 8.975
13.2148 14.4969 14.8716 15.0216 15.0932 15.1306 15.198 15.19
11.8301 11.8057 11.7985 11.7955 11.7940 11.7930 12.163 11.79
13.0270 14.4435 14.8467 15.0076 15.0840 15.1261 15.199 15.19
11.8279 11.7970 11.7869 11.7826 11.7804 11.7792 12.227 11.78
Plate
Present ðN ¼ 7Þ Present ðN ¼ 9Þ Present ðN ¼ 11Þ Present ðN ¼ 13Þ Present ðN ¼ 15Þ Present ðN ¼ 17Þ FSDT [10] HSDT [10]
5.7476 8.5372 9.1806 9.4230 9.5364 9.5985 9.6873 9.6880
8.9873 8.9755 8.9747 8.9753 8.9758 8.9755 8.8998 8.9760
13.1722 14.4709 14.8513 15.0039 15.0737 15.1157 15.183 15.170
11.8296 11.8054 11.7983 11.7953 11.7938 11.7929 12.162 11.790
12.9899 14.4198 14.8274 14.9902 15.0672 15.1087 15.184 15.170
11.8276 11.7968 11.7867 11.7824 11.7802 11.7790 12.226 11.780
7. Concluding remarks
References
In this paper the Reddy higher-order shear deformation theory of laminated orthotropic elastic shells was implemented for the first time through a multiquadrics discretization of equations of motion and boundary conditions. The theory accounts for parabolic distribution of the transverse shear strains through the thickness of the shell and tangential stress-free boundary conditions on the boundary surfaces of the shell. The multiquadric RBF method for the solution of shell bending and free vibration problems was presented. Results for static deformations and natural frequencies were obtained and compared with other sources. This meshless approach demonstrated that is very successful in the static deformations and free vibration analysis of laminated composite shells or plates. Advantages of RBFs are absence of mesh, ease of discretization of boundary conditions and equations of equilibrium or motion and very easy coding. We show that the static displacements and stress and the natural frequencies obtained from present method are in excellent agreement with analytical solutions. We illustrate that this combination of a thirdorder theory for shells and the discretization by multiquadrics produces results for static and free vibrations as good as the exact solutions from [10], with a simple yet accurate formulation.
[1] Naghdi PM. A survey of recent progress in the theory of elastic shells. Appl Mech Rev 1956;9:365–8. [2] Bert CW. Dynamics of composite and sandwich panels—parts i and ii. Shock Vib Dig 1976;8:37–48 15–24. [3] Bert CW, Analysis and performance of composites. Analysis of shells vol. 8, 1980. p. 37–48, 15–24. [4] Reissner E. Stress-strain relations in the theory of thin elastic shells. J Math Phys 1952;31:109–19. [5] Zukas JA, Vinson JR. Laminated transversely isotropic cylindrical shells. J Appl Mech 1971;38:400–7. [6] Dong SB, Pister KS, Taylor RL. On the theory of laminated anisotropic shells and plates. J Aerosp Sci 1962;29. [7] Dong SB, Tso FKW. On a laminated orthotropic shell theory including transverse shear deformation. J Appl Mech 1972;39:1091–6. [8] Whitney JM, Sun CT. A higher order theory for extensional motion of laminated anisotropic shells and plates. J Sound Vib 1973;30. [9] Whitney JM, Sun CT. A refined theory for laminated anisotropic cylindrical shells. J Appl Mech 1974;41. [10] Reddy JN, Liu CF. A higher-order shear deformation theory of laminated elastic shells. Int J Eng Sci 1985;23:319–30. [11] Reddy JN. Bending of laminated anisotropic shells by a shear deformable finite element. Fibre Sci Technol 1982;17:9–24. [12] Hardy RL. Multiquadric equations of topography and other irregular surfaces. Geophys Res 1971;176:1905–15. [13] Kansa EJ. Multiquadrics—a scattered data approximation scheme with applications to computational fluid dynamics. i: Surface approximations and partial derivative estimates. Comput Math Appl 1990;19(8/9):127–45. [14] Kansa EJ. Multiquadrics—a scattered data approximation scheme with applications to computational fluid dynamics. ii: Solutions, to
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[17]
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[21]
parabolic, hyperbolic and elliptic partial differential equations. Comput Math Appl 1990;19(8/9):147–61. Reddy JN. Mechanics of laminated composite plates. New York: CRC Press; 1997. Fasshauer GE. Solving partial differential equations by collocation with radial basis functions. Surface fitting and multiresolution methods. In: Proceedings of the third international conference on curves and surfaces, vol. 2; 1997. p. 131–8. Dubal MR, Oliveira SR, Matzner RA. Domain decomposition and local refinement for multiquadric approximations. i: second-order equations in one-dimension. J Appl Sci Comput 1994;1:146–71. Kansa EJ, Hon YC. Circumvecting the ill-conditioning problem with multiquadric radial basis functions. Comput Math Appl 2000;39(78):123–37. Beatson RK, Light WA. Fast evaluation of radial basis functions: Methods for 2-dimensional polyharmonic splines. IMA J Numer Anal 1997;17:343–72. Ferreira AJM. A formulation of the multiquadric radial basis function method for the analysis of laminated composite plates. Compos Struct 2003;59:385–92. Ferreira AJM. Thick composite beam analysis using a global meshless approximation based on radial basis functions. Mech Adv Mater Struct 2003;10:271–84.
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[22] Ferreira AJM, Roque CMC, Martins PALS. Analysis of composite plates using higher-order shear deformation theory and a finite point formulation based on the multiquadric radial basis function method. Compos Part B 2003;34:627–36. [23] Ferreira AJM. Polyharmonic thin-plate splines in the analysis of composite plates. Int J Mech Sci 2004;46(10):1549–69. [24] Platte RB, Driscoll TA. Computing eigenmodes of elliptic operators using radial basis functions. Comput Math Appl 2004;48: 561–76. [25] Akhras G, Cheung MS, Li W. Finite strip analysis for anisotropic laminated composite plates using higher-order deformation theory. Comput Struct 1994;52(3):471–7. [26] Reddy JN. A simple higher-order theory for laminated composite plates. J Appl Mech 1984;51:745–52. [27] Akhras G, Cheung MS, Li W. Static and vibrations analysis of anisotropic laminated plates by finite strip method. Int J Solids Struct 1993;30(22):3129–37. [28] Ferreira AJM. Analysis of composite plates using a layerwise deformation theory and multiquadrics discretization. Mech Adv Mater Struct 2005;12(2):99–112. [29] Pagano NJ. Exact solutions for rectangular bidirectional composites and sandwich plates. J Compos Mater 1970;4:20–34.
Engineering Analysis with Boundary Elements 30 (2006) 719–733 www.elsevier.com/locate/enganabound
Static and free vibration analysis of composite shells by radial basis functions A.J.M. Ferreira, C.M.C. Roque, R.M.N. Jorge Departamento de Engenharia Mecaˆnica e Gesta˜o Industrial, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal Received 27 January 2006; accepted 11 May 2006 Available online 3 July 2006
Abstract The higher-order shear deformation theory of laminated orthotropic elastic shells of Reddy accounts for parabolic distribution of the transverse shear strains through the thickness of the shell. The Reddy shell theory allows the fulfillment of homogeneous conditions (zero values) at the top and bottom surfaces of the shell. This paper deals with a meshless solution of the Reddy higher order shell theory in static and free vibration analysis. The meshless technique is based on the asymmetric global multiquadric radial basis function method proposed by Kansa. This paper demonstrates that this truly meshless method is very successful in the static and free vibration analysis of laminated composite shells. r 2006 Elsevier Ltd. All rights reserved. Keywords: Radial basis functions; Composite materials; Shells; Collocation methods; Free vibrations
1. Introduction Classical theories developed for thin elastic shells are mostly based on the Love–Kirchhoff assumptions. This theory considers that straight lines normal to the undeformed middle surface remain straight and normal to the deformed middle surface; that the normal stress perpendicular to the middle surface can be neglected in the stress–strain relations and the transverse displacement is independent of the thickness coordinate. Therefore, transverse shear strains are neglected as reported in surveys of classical shell theories by Naghdi [1] and Bert [2,3]. These theories are expected to produce accurate results when the side-to-thickness ratio ða=hÞ is large or when material anisotropy is low. The application of such theories to thick or moderately thick or laminated composite shells can lead to serious errors in terms of deflection or stress. The introduction of transverse shear and normal stress represents an improvement to classical theories. However, for a=Ro10 (side to radius ratio) the transverse normal Corresponding author. Tel.: +351 225 081 705; fax: +351 225 081 445.
E-mail address: [email protected] (A.J.M. Ferreira). 0955-7997/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2006.05.002
stress are negligible when compared to the transverse shear stress. The effect of transverse shear and normal stress in shells was studied by Reissner [4]. The effect of transverse shear deformation was also considered by Vinson [5], Dong et al. [6,7] and Whitney and Sun [8,9]. The higher-order theory of Reddy [10] for composite shells is based on five degrees of freedom (same number as in a first-order shear deformation theory by Reddy [11]). This theory assumes a constant transverse deflection through the thickness and the displacements of the middle surface are expanded as cubic functions of the thickness coordinate. The displacement field leads to parabolic distribution of the transverse shear stress and zero transverse normal strain. Therefore, no shear correction factors are used. The analysis of laminated composite shells by meshless methods is still starting. The purpose of the present research is to demonstrate that a truly meshless method such as the multiquadric radial basis function (RBF) method can be successful in the static and free vibration analysis of laminated composite shells. This paper addresses the use of third-order theory in combination with a
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recent meshless technique based on RBFs in a strong form. This approach was not yet tried as far as the author’s knowledge. The multiquadric RBF method was introduced by Hardy [12] for the interpolation of scattered geographical data. The use of multiquadrics for the solution of partial differential equations was first proposed by Kansa [13,14], followed by others in the analysis of boundary-value problems. Here we show for the first time that the RBFs method can be very successful in the static and free vibration analysis of doubly curved laminated elastic shells. We show that the static displacements and stress and the natural frequencies obtained from present method are in excellent agreement with analytical solutions. We prove that this technique is as good as the exact solutions by [10], with a simple yet accurate formulation.
ð0Þ 2 ð2Þ 1 ¼ ð0Þ 1 þ zðk 1 þ z k1 Þ; 2 ð1Þ 4 ¼ ð0Þ 4 þ z k4 ;
The third-order theory of Reddy for the analysis of laminated composite elastic shells is presented. The shell is composed of a finite number of orthotropic layers of uniform thickness (h). Let ðx1 ; x2 ; zÞ denote the orthogonal curvilinear coordinates (or shell coordinates) such that the x1 - and x2 -curves are lines of curvature on the middle surface z ¼ 0, and z-curves are straight lines perpendicular to the middle surface z ¼ 0. For cylindrical and spherical shells the lines of principal curvature coincide with the coordinate lines. The values of the principal radii of curvature are denoted by R1 and R2 . The displacement field is obtained as [10] z 4 1 qw0 u¼ 1þ u0 þ zf1 2 z3 f1 þ , R1 a1 qx1 3h z 4 1 qw0 v¼ 1þ v0 þ zf2 2 z3 f2 þ , R2 a2 qx2 3h (1)
where ai are the surface metrics, u0 , v0 and w0 are the displacements of the middle surface of the shell and f1 and f2 the rotations of the normals to the x1 - and x2 -axes, respectively. This displacement field is used to compute strains and stress and the equations of motion are then obtained by the dynamic version of the principle of virtual work. The first-order displacements can be readily obtained by deleting the z3 terms in (1). If we assume the term z=R in the definition of stress resultants and assume constant radii of curvatures, we can simplify the shell theory. If we consider thin shallow shells we can therefore have z z 1þ 1; 1þ 1 (2) R1 R2 and we can simplify the calculation of stress resultants as mentioned before. As will be seen further in numerical examples these assumptions still produce quite good results for a large range of shell cases.
ð0Þ 2 ð2Þ 2 ¼ ð0Þ 2 þ zðk 2 þ z k2 Þ,
2 ð1Þ 5 ¼ ð0Þ 5 þ z k5 ,
ð0Þ 2 ð2Þ 6 ¼ ð0Þ 6 þ zðk 6 þ z k6 Þ,
ð3Þ
where ð0Þ 1 ¼
qu0 w þ ; qx1 R1
ð0Þ 2 ¼
qv0 w þ ; qx2 R2
ð0Þ 4 ¼
qw0 þ f2 ; qx2
ð0Þ 5 ¼
qw0 þ f1 , qx1
kð1Þ 4
qf1 ; qx1
ð0Þ 6 ¼
qu0 qv0 þ , qx2 qx1
qf2 qf2 qf1 ; kð0Þ þ , 6 ¼ qx2 qx1 qx2 4 qw0 4 qw0 ¼ 2 f2 þ ¼ f þ ; kð1Þ , 1 5 qx2 qx1 h h2
kð0Þ 1 ¼
2. Third-order shear theory
w ¼ w0 ,
The strain–displacement relations referred to an orthogonal curvilinear coordinate system lead to the following deformation field, where xi denote the cartesian coordinates ðdx1 ¼ ai dxi Þ; i ¼ 1; 2 [10].
kð0Þ 2 ¼
4 qf1 q2 w0 þ , qx21 3h2 qx1 4 qf2 q2 w0 ¼ 2 þ , qx22 3h qx2
kð2Þ 1 ¼ kð2Þ 2 kð2Þ 6
4 qf2 qf1 q2 w0 ¼ 2 þ þ2 . qx1 qx2 3h qx1 qx2
(4)
The stress–strain relation for the kth layer are given by rk ¼ Dk ek ,
(5)
where ðkÞ ðkÞ ðkÞ ðkÞ T rk ¼ ½sðkÞ 1 s2 s6 s4 s5 , ðkÞ ðkÞ ðkÞ ðkÞ T ek ¼ ½ðkÞ 1 2 6 4 5
and
2
QðkÞ 11
6 6 6 6 Dk ¼ 6 6 symm: 6 6 4
ð6Þ
QðkÞ 12
QðkÞ 16
0
0
QðkÞ 22
QðkÞ 26
0
0
QðkÞ 66
0
0
QðkÞ 44
0 QðkÞ 55
3 7 7 7 7 7, 7 7 7 5
(7)
where QðkÞ ij are the material constants of the kth layer in the laminate coordinate system [15]. In the following we substitute x1 ; x2 by x; y. Using Hamilton’s principle, the equations of motion of the third-order shell theory are obtained as [15] qN xx qN xy € c1 I 3 qw€ 0 , þ ¼ I 0 u€ 0 þ J 1 f x qx qy qx
(8)
qN xy qN yy qw€ 0 þ ¼ I 0 v€0 þ J 1 f€ y c1 I 3 , qx qy qy
(9)
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qQx qQy 4 qK x qK y þ 2 þ qx qy qx qy h
ð0Þ ð2Þ M xy ¼ B6j ð0Þ j þ D6j kj þ F 6j kj
! q2 Pxy q2 Pxx q2 Pyy N xx N yy þ þ2 þq qx2 qy2 qx qy R1 R2 2 q w€ 0 q2 w€ 0 ¼ I 0 w€ 0 c21 I 6 þ qx2 qy2 " !# qu€ 0 q€v0 qf€ x qf€ y þ c1 I 3 þ þ þ J4 , ð10Þ qx qy qx qy
4 þ 2 3h
qM xx qM xy 4 4 qPxx qPxy þ Qx þ 2 K x 2 þ qx qy qx qy h 3h qw€ 0 , ¼ J 1 u€ 0 þ K 2 f€ x c1 J 4 qx qM xy qM yy 4 4 qPxy qPyy þ Qy þ 2 K y 2 þ qx qy qx qy h 3h qw€ 0 , ¼ J 1 v€0 þ K 2 f€ y c1 J 4 qy
ðQx ; K x Þ ¼
NL Z X k¼1
ðQy ; K y Þ ¼
zk1
NL Z X k¼1
zk
zk
zk1
sðkÞ xz ð1; z
(22)
ð0Þ ð2Þ Pyy ¼ E 2j ð0Þ j þ F 2j kj þ H 2j kj ,
(23)
ð0Þ ð2Þ Pxy ¼ E 6j ð0Þ j þ F 6j kj þ H 6j kj
ðj ¼ 1; 2; 6Þ,
(24)
(25)
ð11Þ ð1Þ Qx ¼ A5j ð0Þ j þ D5j kj
ð12Þ
ðj ¼ 4; 5Þ,
Þ dz,
(26)
ð1Þ K y ¼ D4j ð0Þ j þ F 4j kj ,
ð1Þ K x ¼ D5j ð0Þ j þ F 5j kj
(27)
ðj ¼ 4; 5Þ,
(28)
where Aij , Bij , etc. are the laminate stiffness components obtained by ðAij ; Bij ; Dij ; E ij ; F ij ; H ij Þ ¼
NL Z X k¼1
zk
zk1
2 3 4 6 QðkÞ ij ð1; z; z ; z ; z ; z Þ dz.
(29) 2 sðkÞ yz ð1; z Þ dz,
ð13Þ
where NL denotes the number of the orthotropic layers of the shell and zk , zk1 the top and bottom z-coordinates of the kth lamina. The inertia terms are given by NL Z zk X rðkÞ zi dz ði ¼ 0; 1; 2; . . . ; 6Þ, (14) Ii ¼ k¼1
(21)
ð0Þ ð2Þ Pxx ¼ E 1j ð0Þ j þ F 1j kj þ H 1j kj ,
zk1
2
ðj ¼ 1; 2; 6Þ,
ð1Þ Qy ¼ A4j ð0Þ j þ D4j kj ,
where q is the distributed transverse load, N i , M i , etc. are the stress resultants, given by NL Z zk X 3 sðkÞ ðN i ; M i ; Pi Þ ¼ i ð1; z; z Þ dz ði ¼ xx; yy; xyÞ, k¼1
721
zk1
J i ¼ I i c1 I iþ2 ; K 2 ¼ I 2 2c1 I 4 þ c21 I 6 , 4 ð15Þ c1 ¼ 2 ; c2 ¼ 3c1 3h being rðkÞ the density of the material of layer k. The stress resultants can also be obtained in terms of the strain components by (13) ð0Þ ð2Þ N xx ¼ A1j ð0Þ j þ B1j kj þ E 1j kj ,
(16)
ð0Þ ð2Þ N yy ¼ A2j ð0Þ j þ B2j kj þ E 2j kj ,
(17)
ð0Þ ð2Þ N xy ¼ A6j ð0Þ j þ B6j kj þ E 6j kj
ðj ¼ 1; 2; 6Þ,
(18)
ð0Þ ð2Þ M xx ¼ B1j ð0Þ j þ D1j kj þ F 1j kj ,
(19)
ð0Þ ð2Þ M yy ¼ B2j ð0Þ j þ D2j kj þ F 2j kj ,
(20)
The equations of motion [15] can now be expressed in terms of the displacements u0 , v0 , w0 and the rotations fx , fy , as
2 q2 u0 1 qw0 q v0 1 qw0 þ þ þ A 12 qx2 R1 qx qx qy R2 qx 2 2 q v0 q u0 qw€ 0 þ , ¼ I 0 u€ 0 þ J 1 f€ x c1 I 3 þ A66 qx qy qy2 qx
A11
2 q2 v 0 1 qw0 q u0 A12 qw0 þ þ þ A 12 qy2 R2 qy qx qy R1 qy 2 2 q v 0 q u0 qw€ 0 , þ þ A66 ¼ I 0 v€0 þ J 1 f€ y c1 I 3 qx2 qx qy qy
ð30Þ
A22
q3 fy 32 q3 f q4 w0 2 x 2 2 2 H 66 4 2 qx qy qy qx qy qx 9h 4 3 16 q w0 q fx þ 4 H 11 4 qx qx3 9h
!
! q3 fy q4 w0 q3 fy 4 q3 fx 16 4 þ 4 H 22 3 þ 2 F 22 þ 2 F 11 3 4 qx qy qy qy3 3h 9h 3h ! q3 fy qfy q2 w0 4 q3 fx 16 þ þ þ þ 2 F 12 F 44 qy2 qx qx2 qy qy qy2 3h h4
ð31Þ
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722
! q3 fy 16 q3 fx q4 w0 þ 4 H 12 2 2 2 2 2 qx qy qy qx qy qx 9h 2 8 q w0 qf þ 2 D55 2 x qx qx h 2 2 8 q w0 qfy 16 q w0 qfx þ 4 F 55 þ þ 2 D44 2 qy qy qx2 qx h h ! 3 3 q fy 8 q fx qfx q2 w0 þ þ A þ þ 2 F 66 55 qx2 qy qy2 qx qx qx2 3h qfy q2 w0 2 1 qv0 1 qu0 þ þ A w þ A44 12 0 R1 R2 R1 qy R2 qx qy qy2
þ
A22 qv0 1 A11 qu 1 w0 þ w0 þ q qx R1 R2 qy R2 R1 " 2 2 € € q q qu€ 0 q€v0 w w 0 0 2 þ þ þ c1 I 3 ¼ I 0 w€ 0 c1 I 6 qx2 qy2 qx qy !# qf€ x qf€ y þ þJ 4 , ð32Þ qx qy 2
3
2
!
3
2
q fy q fy 16 q w0 q fx q w0 4 þ 2 þ þ 2 H 12 2 F 22 4 qx qy qx qy qx qy qy qx qy2 9h 3h ! 3 2 4 q w0 q2 f x 16 q3 w0 q fy þ 2 F 11 3 2 2 þ 4 H 22 þ qx qx qy3 qy2 3h 9h 3 16 q w 0 q2 f x 4 þ þ 4 H 11 þ 2 3 2 qx qx 9h 3h ! 2 3 2 q fy q w0 q fx 2 F 12 2 qy qx qx qy qx qy ! 2 q2 fy qw0 q2 fx q fy fx þ D66 þ A55 þ þ D 12 qx qy2 qx qy qx qy ! 2 q fy q2 fx 4 q3 w0 þ 2 F 66 2 2 2 qx qy qy qy qx 3h 8 qw0 þ 2 D55 fx þ qx h 2 q fx 16 qw0 f þ F þ D11 55 x qx2 qx h4 € c1 J 4 qw€ 0 , ð33Þ ¼ J 1 u€ 0 þ K 2 f x qx q2 fy 8 qw0 qw0 þ f f D þ A þ D 44 44 22 y y qy qy qy2 h2 ! q2 fy 4 q3 w 0 4 þ 2 F 22 3 2 2 þ 2 qy qy 3h 3h 3 2 q w0 q fx F 12 2 2 qx qy qx qy 2 8 q3 w 0 q2 fx q fy þ 2 F 66 2 qx qy qx qy qx2 3h
!
þ D66
2 q2 fx q fy þ qxqy qx2
!
q2 fx 16 qw0 þ F 44 fy þ D12 qx qy h4 qy 2 3 16 q fx q w0 þ þ 4 H 12 qx qy qx2 qy 9h q2 f y q2 f x 16 q3 w 0 þ 2 þ 4 H 66 þ qx2 qx qy qx2 qy 9h ! q2 fy q3 w0 16 þ 4 H 22 þ qy2 qy3 9h
!
€ c1 J 4 qw€ 0 . ¼ J 1 v€0 þ K 2 f y qy
ð34Þ
3. Free vibration analysis For free vibration problems we assume harmonic solution in terms of displacements and rotations u0 ; v0 ; w0 ; fx ; fy in the form u0 ðx; y; tÞ ¼ u0 ðw; yÞeiot ,
(35)
v0 ðx; y; tÞ ¼ v0 ðw; yÞeiot ,
(36)
w0 ðx; y; tÞ ¼ w0 ðw; yÞeiot ,
(37)
yx ðx; y; tÞ ¼ yx ðw; yÞeiot ,
(38)
yy ðx; y; tÞ ¼ yy ðw; yÞeiot ,
(39)
where o is the frequency of natural vibration. Removing the external force q and substituting the harmonic expansions into equations of motion we obtain equations of motion in terms of the amplitudes as
2 q2 u0 1 qw0 q v0 1 qw0 þ A11 þ þ A12 qx2 R1 qx qx qy R2 qx 2 q v0 q2 u 0 þ 2 þ A66 qx qy qy qw0 ¼ I 0 o2 u0 J 1 o2 yx þ c1 I 3 o2 , qx 2 q2 v 0 1 qw0 q u0 A12 qw0 þ A22 þ þ A12 qy2 R2 qy qx qy R1 qy 2 q v 0 q2 u0 þ þ A66 qx2 qx qy qw0 , ¼ I 0 o2 v0 J 1 o2 yy þ c1 I 3 o2 qy
ð40Þ
ð41Þ
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q3 fy 32 q3 fx q4 w0 2 H 66 qx2 qy qy2 qx qy2 qx2 9h4 4 16 q w0 q 3 fx þ 4 H 11 4 qx qx3 9h
!
! q3 fy q4 w0 q3 fy 4 q3 fx 16 4 þ 4 H 22 3 þ 2 F 22 þ 2 F 11 3 4 qx qy qy qy3 3h 9h 3h ! q3 fy qfy q2 w0 4 q 3 fx 16 þ þ þ þ 2 F 12 F 44 qy2 qx qx2 qy qy qy2 3h h4 ! q3 fy 16 q3 f q4 w0 þ 4 H 12 2 2 x 2 2 2 qx qy qy qx qy qx 9h 2 8 q w0 qf þ 2 D55 2 x qx qx h 2 2 8 q w0 qfy 16 q w0 qfx þ 4 F 55 þ 2 D44 2 þ qy qy qx2 qx h h ! 3 3 2 q fy 8 q fx qfx q w0 þ þ A þ þ 2 F 66 55 qx2 qy qy2 qx qx qx2 3h qfy q2 w0 2 1 qv0 1 qu0 þ w þ A þ A44 12 0 R1 R2 R1 qy R2 qx qy qy2
A22 qv0 1 A11 qu 1 þ w0 þ w0 qx R1 R2 qy R2 R1 2 q w0 q2 w0 þ ¼ I 0 o2 w0 þ c21 I 6 o2 qx2 qy2 qu0 qv0 qyx qyy c1 o 2 I 3 þ þ þ J4 , qx qy qx qy
ð42Þ
! q2 fy q2 fy 16 q3 w 0 q2 f x q3 w0 4 þ þ þ H F 12 22 qx qy qx2 qy qx qy qy2 qx qy2 9h4 3h2 ! 3 2 4 q w0 q2 f x 16 q3 w0 q fy þ 2 F 11 3 2 2 þ 4 H 22 þ qx qx qy3 qy2 3h 9h 3 16 q w 0 q2 f x þ þ 4 H 11 qx3 qx2 9h ! q2 fy 4 q3 w 0 q2 f x þ 2 F 12 2 2 qy qx qx qy qx qy 3h ! 2 q2 f y qw0 q2 fx q fy fx þ D66 þ A55 þ þ D 12 qx qy2 qx qy qx qy ! 2 q fy q2 fx 4 q3 w0 þ 2 F 66 2 2 2 qx qy qy qy qx 3h 8 qw0 þ 2 D55 fx þ qx h 2 q fx 16 qw0 f þ F þ D11 55 x qx2 qx h4 qw0 , ð43Þ ¼ J 1 o2 u0 K 2 o2 yx þ c1 J 4 o2 qx q2 fy 8 qw0 qw0 þ fy þ A44 fy þ D22 D 2 44 qy qy qy2 h
! q2 fy 4 q3 w0 þ 2 F 22 3 2 2 qy qy 3h 3 4 q w0 q2 fx þ 2 F 12 2 2 qx qy qx qy 3h 2 8 q3 w0 q2 f x q f y þ 2 F 66 2 qx qy qx qy qx2 3h
723
! þ D66
q2 fx 16 qw0 þ 4 F 44 fy qx qy h qy 2 3 16 q fx q w0 þ þ 4 H 12 qx qy qx2 qy 9h
2 q2 fx q fy þ qxqy qx2
!
þ D12
q2 fy q2 fx 16 q3 w 0 þ 2 þ 4 H 66 þ qx2 qx qy qx2 qy 9h ! q2 f y q3 w 0 16 þ 4 H 22 þ qy2 qy3 9h ¼ J 1 o2 v0 K 2 o2 yy þ c1 J 4 o2
!
qw0 . qy
ð44Þ
This set of equations of motion can now be interpolated by multiquadrics RBFs by a global unsymmetric collocation scheme [13,14]. The resulting linear system of equations is later modified by the introduction of boundary conditions, by introducing new lines with the new interpolation equations. The multiquadric RBF method is explained in the next section. 4. The multiquadric radial basis method 4.1. Kansa’s unsymmetric collocation method The discretization is based on the multiquadrics radial basis method [13,14] which is here resumed. Consider a set of nodes x1 ; x2 ; . . . ; xN 2 O Rn . The RBFs centered at xj are defined as gj ðxÞ gðkx xj kÞ 2 Rn ; j ¼ 1; . . . ; N where kx xj k is the Euclidian norm. In this work we use multiquadrics, in the form gj ðxÞ ¼ ðkx xj k2 þ c2 Þ1=2 where c is a shape (user-defined) parameter. In this paper we used c ¼ 2=N 0:5 , N being the total number of nodes, as proposed by Fasshauer [16]. One of the main advantages of RBFs is the insensitivity to spatial dimension, making the implementation of this method much easier than, e.g., finite elements [13,14]. The method does not require a grid or mesh, the pairwise distances between the points being the only geometric properties required by the method. In this paper it is proposed to use Kansa’s unsymmetric collocation method [13,14]. Consider a boundary-valued problem with a domain O Rn and a linear elliptic partial differential equation of the form LuðxÞ ¼ xðxÞ 2 O Rn , BuðxÞjqO ¼ f ðxÞ;
qO Rn ,
(45) (46)
ARTICLE IN PRESS A.J.M. Ferreira et al. / Engineering Analysis with Boundary Elements 30 (2006) 719–733
724
where qO represents the boundary of the problem. We use points along the boundary ðxj ; j ¼ 1; . . . ; N B Þ and in the interior ðxj ; j ¼ N B þ 1; . . . ; NÞ. Let the RBF interpolant to the solution uðxÞ be xðx; cÞ ¼
N X
4.3. Solution of the eigenproblem aj gðkx xj kÞ.
(47)
j¼1
Collocation with the boundary data at the boundary points and with PDE at the interior points leads to equations xB ðx; cÞ
eigenvectors of this matrix are then evaluated by standard techniques.
N X
aj Bgðkxi xj kÞ ¼ lðxi Þ;
i ¼ 1; . . . ; N B ,
j¼1
We follow a simple scheme, as proposed also by Platte and Driscoll [24] for the solution of the eigenproblem (51) and (52). We consider N I nodes in the interior of the domain and N B nodes on the boundary, with N ¼ N I þ N B. We denote interpolation points by xi 2 O; i ¼ 1; . . . ; N I and xi 2 qO; i ¼ N I þ 1; . . . ; N. For the interior points we have that
(48) xL ðx; cÞ
N X
N X
aj Lgðkxi xj kÞ ¼ Fðxi Þ;
i ¼ N B þ 1; . . . ; N,
j¼1
(49) where lðxi Þ, Fðxi Þ are the prescribed values at the boundary nodes and the function values at the interior nodes, respectively. This corresponds to a system of equations with an unsymmetric coefficient matrix, structured in matrix form as " # Bg l ½a ¼ . (50) Lg U The use of globally supported RBFs for large problems can bring problems due to the full populated matrices. To solve this drawback a localization scheme is advisable. Domain decomposition methods [17,18], localization of the basis functions [19,18] claim to be able to deal with tens of thousands of nodes. These techniques are not used in this paper. The discretization method applied in the present work follows closely the work of the authors for composite plates in [20–23] for first-order and third-order shear deformation theories.
Consider a linear elliptic partial differential operator L and a bounded region O in Rn with some boundary qO. The eigenproblem seeks eigenvalues (l) and eigenvectors ðuÞ that satisfy
LB u ¼ 0
in O,
on qO,
j ¼ 1; 2; . . . ; N I
(53)
or LI a ¼ le uI ,
(54)
where LI ¼ ½Lgðkx xj kÞN I N .
(55)
For the boundary conditions we have N X
ai LBg ðkx xj kÞ ¼ 0;
j ¼ N I þ 1; . . . ; N
(56)
i¼1
or B a ¼ 0.
(57)
Therefore we can write a finite-dimensional problem as a generalized eigenvalue problem " # " # LI AI a¼l a, (58) B 0 where AI ¼ g½ðkx xj kÞN I N ;
4.2. The eigenproblem
Lu lu ¼ 0
ai Lgðkx xj kÞ ¼ le uðxj Þ;
i¼1
BI ¼ LBg ½ðkx xj kÞN B N .
We seek the generalized eigenvalues and eigenvectors of these matrices.
5. Multiquadrics interpolation of equations of motion
(51) (52)
where LB is a linear boundary operator. The eigenproblem of (51) and (52) is replaced by a finite-dimensional eigenvalue problem, based on RBF approximations. The operator L is then approximated by a matrix that incorporates the boundary conditions. The eigenvalues and
Applying the multiquadrics method previously explained, the equations of motion are now interpolated, for each node. In the following equations, a is equivalent to a. " # " # 2 N N X X q2 gj q2 gj u0 v 0 q gj ðA12 þ A66 Þ aj A11 2 þ A66 2 þ aj qx qy qx qy j¼1 j¼1
ARTICLE IN PRESS A.J.M. Ferreira et al. / Engineering Analysis with Boundary Elements 30 (2006) 719–733
þ
N X
awj 0
j¼1
"
qgj A11 A12 þ qx R1 R2
¼ I 0 o2
N X
auj 0 gj J 1 o2
j¼1
þ c1 I 3 o2
N qgj X qgj þ avj 0 c1 o I 3 qx j¼1 qy j¼1 !# N N X qgj X y qgj þ ayj x aj y þJ 4 , qx j¼1 qy j¼1
ayj x gj
j¼1
N X
awj 0
j¼1
qgj , qx
ð59Þ
N X
awj 0
j¼1 N X
"
#
"
N X q2 gj q2 gj q2 gj ðA66 þ A12 Þ þ avj 0 A66 2 þ A22 2 qx qy qx qy j¼1 j¼1 N X qgj A12 A22 þ awj 0 þ qy R1 R2 j¼1
#
auj 0
¼ I 0 o2
N X
avj 0 gj J 1 o2
j¼1
þ c1 I 3 o2
N X
y
aj y gj
j¼1
N X
awj 0
j¼1
qgj , qy
ð60Þ
X N qgj A12 A11 qgj A22 A12 v0 þ aj þ þ qx R2 R1 qy R2 R1 j¼1 j¼1 " N X A12 A22 A11 awj 0 gj 2 2 2 þ R1 R2 R2 R1 j¼1 q2 gj 16 8 þ 2 þ 4 F 55 2 D55 þ A55 qx h h 2 q gj 16 q4 gj 8 16 þ 2 F D þ A H 44 44 44 11 qy h4 qx4 h2 9h4 # q4 gj q4 gj 16 64 32 4 H 22 4 2 2 H 66 þ 4 H 12 qy qy qx 9h4 9h 9h " N X 16 8 fx qgj þ A55 þ 4 F 55 2 D55 aj qx h h j¼1 3 q gj 4 16 F 11 4 H 11 þ 3 2 qx 3h 9h # 3 q gj 4 32 8 16 þ 2 F 12 4 H 66 þ 2 F 66 4 H 12 qy qx 3h2 9h 3h 9h
N X
auj 0
"
qgj 16 8 F D þ A 44 44 44 qy h4 h2 j¼1 q3 gj 4 16 F 22 4 H 22 þ 3 qy 3h2 9h # 3 q gj 8 4 32 16 þ 2 F 66 þ 2 F 12 4 H 66 4 H 12 qx qy 3h2 3h 9h 9h þ
N X
f aj y
2
¼ I 0 o
N X j¼1
awj 0 gj
þ
N X
2
N X
c21 I 6 o2
N X j¼1
awj 0
725
q2 gj q2 gj þ 2 qx2 qy
!
!
auj 0
ð61Þ
qgj 8 16 D F A 55 55 55 qx h2 h4
q3 gj q3 gj 16 16 H þ H 22 12 qy3 9h4 qx2 qy 9h4 3 q gj 16 q3 gj 4 H F þ 3 þ 11 11 qx 9h4 qy2 qx 3h2 16 4 8 H F F 12 12 66 9h4 3h2 3h2 N X 8 16 fx aj gj 2 D55 A55 4 F 55 þ h h j¼1 2 q gj 8 16 þ 2 D11 2 F 11 þ 4 H 11 qx 3h 9h # 2 q gj q2 gj 16 4 4 þ 2 D66 2 F 66 þ H 12 2 F 12 qy qx qy 9h4 3h 3h " 2 N X f q gj 16 4 þ aj y H 22 2 F 22 4 2 qy 9h 3h j¼1 # q2 gj 8 4 16 D12 2 F 12 2 F 66 þ D66 þ 4 H 12 þ qx qy 3h 3h 9h þ
¼ J 1 o2
N X
auj 0 gj K 2 o2
j¼1
þ c1 J 4 o 2
N X j¼1
"
N X
ayj x gj
j¼1
awj 0
qgj , qx
ð62Þ
qgj 8 16 D A F 44 44 44 qy h2 h4 j¼1 q3 gj 16 4 H 22 2 F 22 þ 3 qy 9h4 3h # 3 q gj 16 32 8 4 þ 2 H 12 þ 4 H 66 2 F 66 2 F 12 qx qy 9h4 9h 3h 3h " N X q2 gj 8 8 f þ D66 D12 2 F 66 2 F 12 aj x qxqy 3h 3h j¼1 # 16 16 þ 4 H 66 þ 4 H 12 9h 9h " N X 8 16 f þ aj y gj 2 D44 A44 4 F 44 h h j¼1 q2 gj 8 16 þ 2 D66 2 F 66 þ 4 H 66 qx 3h 9h N X
awj 0
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726
# q 2 gj 8 16 þ 2 D22 2 F 22 þ 4 H 22 qy 3h 9h ¼ J 1 o2
N X
avj 0 gj K 2 o2
j¼1
þ c1 J 4 o 2
N X
N X
the model is validated for a composite plate. The second example consists of a spherical shell in bending by distributed loading. All examples consider laminated composites and are subjected to bending loads. The material properties for each layer are the following:
y
aj y gj
j¼1
awj 0
j¼1
E 1 ¼ 25:0E 2 ; G 12 ¼ G 13 ¼ 0:5E 2 ; n12 ¼ n13 ¼ 0:25,
qgj . qy
wðx ¼ aÞ ¼ 0,
(63)
uðx ¼ aÞ ¼ 0,
(64)
fx ðx ¼ aÞ ¼ 0,
(65)
M x ðx ¼ aÞ ¼ 0,
(66)
N x ðx ¼ aÞ ¼ 0,
(67)
can be interpolated by RBF, respectively, by awj 0 gj ¼ 0,
(68)
auj 0 gj ¼ 0,
(69)
j¼1 N X j¼1 N X
f
aj x gj ¼ 0,
(70)
j¼1
"
q2 gj 16 4 H F 11 11 qx2 9h4 3h2 j¼1 # q2 gj 16 4 H 12 2 F 12 þ 2 qy 9h4 3h N X 8 16 f qgj D11 2 F 11 þ 4 H 11 þ aj x qx 3h 9h j¼1 N X 8 16 f qgj D12 2 F 12 þ 4 H 12 þ aj y ¼ 0, qy 3h 9h j¼1
N X
N X j¼1
ð71Þ
ð72Þ
6. Numerical problems 6.1. Static problems The accuracy of the present model for static analysis is examined in two numerical examples. In the first example
6.1.1. Composite plate in bending A square simply supported laminated plate under sinusoidal pressure is considered. The sinusoidal force is given by px py sin (74) P ¼ P0 sin a a acting on a square plate of side a, where P0 is the uniform pressure. In order to model the plate (infinite radii), the shell radii are set equal to R1 =a ¼ R2 =a ¼ 109 . In this example the present model is compared in Tables 1 and 2 with other models. The finite strip formulation of Akhras [25], the finite element formulation of Reddy [26], both considering a third-order theory, the finite strip model of Akhras et al. [27] considering a first-order theory and the work of Ferreira et al. [22,28] using a third-order shear deformation theory and also a layerwise theory and multiquadrics for plates are considered for comparison. The exact (elasticity) solution of Pagano [29] is also presented for comparison. Results are presented in Tables 1 and 2, in normalized form as
txz
N N qgj X qgj X A11 A12 þ þ avj 0 A12 awj 0 gj þ qx j¼1 qy R1 R2 j¼1
¼ 0.
Two cross-ply laminates are considered: ½0 =90 =90 =0 and ½0 =90 =0 , with the origin of the coordinate system being located at the lower left corner on the midplane.
102 E 2 h3 ; P0 a4 h . ¼ txz P0 a
w¼w
awj 0
auj 0 A11
ð73Þ
The interpolation of boundary conditions follows a similar procedure as in the interpolation equilibrium equations. A typical simply supported boundary condition in the form
N X
G 23 ¼ 0:2E 2 ,
si ¼ si
h2 ; i ¼ x; y; P0 a2
txy ¼ txy
h2 , P0 a2 ð75Þ
Here we consider a uniform grid with N being the number of points in x-direction and y-direction. Therefore 11 11, 15 15, and 21 21 uniform grids are used. Thick (a=h ¼ 4) to thin (a=h ¼ 100) plates are considered. In all cases our shell model can predict with good accuracy the transverse deflections, the normal stress and the transverse shear stress. Even in thicker plates our results are in good agreement with other higher-order formulations. 6.1.2. Spherical shell in bending A laminated composite spherical shell is here considered. The shell is loaded with the same sinusoidal load as in the plate example and is simply supported in all edges. In Table 3 we compare the present shell model with results of Reddy shell formulation using first-order and third-order shear-deformation theories. We consider various values of R=a and two values of a=h. Results are in good agreement for various a=h ratios, in particular with
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727
Table 1 Non-dimensional deflection and stress results for different methods and shear deformation theories, for a ½0 =90 =90 =0 shell, with R=a ¼ 109 a=h
Method
w
sx ða=2; a=2; h=2Þ
sy ða=2; a=2; h=4Þ
tzx ð0; a=2; 0Þ
txy ð0; 0; h=2Þ
4
3 strip [25] HSDT [26] FSDT [27] Elasticity [29] Ferreira et al. [22] ðN ¼ 21Þ Ferreira (layerwise) [28] ðN ¼ 21Þ Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ
1.8939 1.8937 1.7100 1.954 1.8864 1.9075 1.8848 1.8885 1.8900
0.6806 0.6651 0.4059 0.720 0.6659 0.6432 0.6616 0.6604 0.6599
0.6463 0.6322 0.5765 0.666 0.6313 0.6228 0.6293 0.6307 0.6312
0.2109 0.2064 0.1398 0.270 0.1352 0.2166 0.2185 0.2148 0.2128
0.0450 0.0440 0.0308 0.0467 0.0433 0.0441 0.0458 0.0463 0.0466
10
3 strip [25] HSDT [26] FSDT [27] Elasticity [29] Ferreira et al. [22] ðN ¼ 21Þ Ferreira (layerwise) [28] ðN ¼ 21Þ Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ
0.7149 0.7147 0.6628 0.743 0.7153 0.7309 0.7115 0.7120 0.7121
0.5589 0.5456 0.4989 0.559 0.5466 0.5496 0.5413 0.5411 0.5411
0.3974 0.3888 0.3615 0.403 0.4383 0.3956 0.3873 0.3874 0.3874
0.2697 0.2640 0.1667 0.301 0.3347 0.2888 0.2981 0.2911 0.2921
0.0273 0.0268 0.0241 0.0276 0.0267 0.0273 0.0269 0.0271 0.0272
Table 2 Non-dimensional deflection and stress results for different methods and shear deformation theories, for a ½0 =90 =90 =0 shell, with R=a ¼ 109 a=h
Method
w
sx ða=2; a=2; h=2Þ
sy ða=2; a=2; h=4Þ
tzx ð0; a=2; 0Þ
txy ð0; 0; h=2Þ
20
3 strip [25] HSDT [26] FSDT [27] Elasticity [29] Ferreira et al. [22] ðN ¼ 21Þ Ferreira (layerwise) [28] ðN ¼ 21Þ Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ
0.5061 0.5060 0.4912 0.517 0.5070 0.5121 0.5039 0.5035 0.5032
0.5523 0.5393 0.5273 0.543 0.5405 0.5417 0.5365 0.5358 0.5354
0.3110 0.3043 0.2957 0.309 0.3648 0.3056 0.3028 0.3027 0.3026
0.2883 0.2825 0.1749 0.328 0.3818 0.3248 0.3238 0.3259 0.3262
0.0233 0.0228 0.0221 0.0230 0.0228 0.0230 0.0226 0.0227 0.0228
100
3 strip [25] HSDT [26] FSDT [27] Elasticity [29] Ferreira et al. [22] ðN ¼ 21Þ Ferreira (layerwise) [28] ðN ¼ 21Þ Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ
0.4343 0.4343 0.4337 0.4347 0.4365 0.4374 0.4496 0.4367 0.4327
0.5507 0.5387 0.5382 0.539 0.5413 0.5420 0.5546 0.5408 0.5364
0.2769 0.2708 0.2705 0.271 0.3359 0.2697 0.2739 0.2706 0.2694
0.2948 0.2897 0.1780 0.339 0.4106 0.3232 0.3737 0.3466 0.3454
0.0217 0.0213 0.0213 0.0214 0.0215 0.0216 0.0226 0.0216 0.0213
the higher-order results of Reddy [27]. It is important to notice that our present approach agrees quite well with Reddy third-order theory for shells, but can present small differences when compared with Reddy first-order shell theory for thicker shells. This shows that the formulation is consistent. This remark is relevant to understand the results of Tables 4 and 5. In Tables 4 and 5 we compare our solution with an exact solution and various finite element solutions [11] for a ½0 =90 =90 =0 simply supported spherical shell, under sinusoidal pressure. Here we set h ¼ 0:1 and we consider
various R=h ratios. The transverse deflection and stress are normalized as 102 E 2 h3 h2 ; s ¼ s ; i ¼ x; y, i i P0 a4 P0 a2 h2 tij ¼ tij ð1 dij Þ; i; j ¼ x; y; z. P 0 a2 w¼w
ð76Þ
Our results are in reasonable agreement with CFS and finite element results of Reddy [28] using a first-order shear deformation theory for shells. The agreement is better for
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Table 3 Non-dimensional central deflection, w ¼ wð102 E 2 h3 =P0 a4 Þ variation with various number of grid points per unit length, N for different R=a ratios, for R1 ¼ R2 a=h
½0 =90 =0
½0 =90 =90 =0
Method
R=a 5
10
20
50
100
109
10 10 10 10 10
Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ HSDT [10] FSDT [10]
6.7047 6.7308 6.7396 6.7688 6.4253
6.9900 6.9994 7.0028 7.0325 6.6247
7.0652 7.0700 7.0718 7.1016 6.6756
7.0865 7.0900 7.0914 7.1212 6.6902
7.0896 7.0928 7.0942 7.1240 6.6923
7.0906 7.0938 7.0951 7.125 6.6939
100 100 100 100 100
Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ HSDT [10] FSDT [10]
0.9608 1.0084 1.0253 1.0321 1.0337
2.3356 2.3810 2.3964 2.4099 2.4109
3.6423 3.6104 3.6003 3.617 3.6150
4.3199 4.2209 4.1897 4.2071 4.2027
4.4379 4.3254 4.2900 4.3074 4.3026
4.4786 4.3614 4.3244 4.3420 4.3370
10 10 10 10 10
Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ HSDT [10] FSDT [10]
6.7237 6.7515 6.7608 6.7865 6.3623
7.0127 7.0239 7.0277 7.0536 6.5595
7.0889 7.0955 7.0978 7.1237 6.6099
7.1105 7.1158 7.1176 7.1436 6.6244
7.1136 7.1187 7.1205 7.1464 6.6264
7.1146 7.1196 7.1214 7.1474 6.6280
100 100 100 100 100
Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ HSDT [10] FSDT [10]
0.9572 1.0034 1.0198 1.0264 1.0279
2.3335 2.3752 2.3894 2.4024 2.4030
3.6495 3.6101 3.5974 3.6133 3.6104
4.3349 4.2254 4.1908 4.2071 4.2015
4.4544 4.3309 4.2919 4.3082 4.3021
4.4957 4.3672 4.3284 4.3430 4.3368
Table 4 Non-dimensional deflection and stress results for different methods, for a ½0 =90 =90 =0 shell, with R1 ¼ R2 , a=h ¼ 10 R=h
Method
w
sx ða=2; a=2; h=2Þ
sy ða=2; a=2; h=4Þ
txy ð0; 0; h=2Þ
txz ð0; a=2; 0Þ
10
CFS FEM Q4-R [11] FEM Q2-F9 [11] FEM L2-F [11] Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ
0.3229 0.3229 0.3241 0.2337 0.2912 0.3015 0.3052
0.1678 0.1678 0.1632 0.1459 0.1663 0.1746 0.1775
0.1032 0.1032 0.1000 0.1165 0.1034 0.1095 0.1117
0.04055 0.04055 0.03947 0.02184 0.03753 0.03934 0.04001
0.01824 0.01824 0.01772 0.01153 0.00883 0.01112 0.01235
20
CFS FEM Q4-R [11] FEM Q2-F9 [11] FEM L2-F [11] Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ
0.5254 0.5254
0.3426 0.3426
0.2331 0.2330
0.04295 0.04294
0.03201 0.03201
0.4276 0.5221 0.5310 0.5341
0.2597 0.3475 0.3555 0.3582
0.1994 0.2343 0.2408 0.2430
0.02615 0.04367 0.04480 0.04522
0.02245 0.01883 0.02097 0.02207
lower R=h ratios which proves the flexibility of the theory for analyzing both plates and shells of higher curvature. In Tables 6 and 7 the present approach is compared with third-order theory of Reddy and first-order theory results by Reddy [10], for various values of R=a and various laminates for spherical shells, under uniform load. The results clearly show an excellent agreement with Reddy third-order theory and some expected deviation from firstorder theory as mentioned before. For all laminates and for all R=a ratios results for central deflection are very good.
6.2. Free vibration problems In free vibration problems we consider nodal regular grids with 7 7 up to 17 17 points. Tables 8–10 contain nondimensionalized natural frequencies obtained using the first- and higher-order theories for various cross-ply spherical shells. Present results are compared with analytical solutions by Reddy and Liu [10] who considered both the first-order (FSDT) and the third-order (HSDT) theories. Analogous to cylindrical shells, the first-order
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Table 5 Non-dimensional deflection and stress results for different methods, for a ½0 =90 =90 =0 shell, with R1 ¼ R2 , a=h ¼ 10 Method
w
sx
sy
txy
txz
CFS FEM Q4-R [11] FEM Q2-F9 [11] FEM L2-F [11] Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ
0.6362 0.6361 0.6357
0.4541 0.4540 0.4427
0.3216 0.3215 0.3128
0.03474 0.03473 0.03391
0.03955 0.03955 0.03865
0.6724 0.6752 0.6761
0.4859 0.4887 0.4896
0.3402 0.3428 0.3437
0.03777 0.03822 0.03840
0.02661 0.02771 0.02829
100
CFS FEM Q4-R [11] FEM Q2-F9 [11] FEM L2-F [11] Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ
0.6559 0.6558 0.6357 0.5813 0.7013 0.7024 0.7028
0.4797 0.4796 0.4678 0.3349 0.5202 0.5211 0.5214
0.3437 0.3437 0.3344 0.2484 0.3682 0.3694 0.3698
0.02979 0.02978 0.02909 0.02000 0.03295 0.03326 0.03339
0.04090 0.04089 0.03997 0.03111 0.02857 0.02918 0.02953
109 (Plate)
CFS FEM Q4-R [11] FEM Q2-F9 [11] FEM L2-F [11] Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ
0.6627 0.6627 0.6623 0.5901 0.7115 0.7120 0.7121
0.4995 0.4954 0.4832 0.3339 0.5413 0.5211 0.5214
0.3589 0.3589 0.3492 0.2454 0.3873 0.3694 0.3698
0.02397 0.02396 0.02340 0.01629 0.02688 0.03326 0.03339
0.04136 0.04136 0.04043 0.03161 0.02981 0.02994 0.03005
R=h 50
Table 6 Non-dimensional central deflection, w ¼ wð102 E 2 h3 =P0 a4 Þ variation with various number of grid points per unit length, N for different R=a ratios, for R1 ¼ R2 , with uniform load (½0 =90 , ½0 =90 =0 ) a=h
½0 =90
½0 =90 =0
Method
R=a 5
10
20
50
100
109
10 10 10 10 10
Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ HSDT [10] FSDT [10]
17.1112 17.4314 17.5408 17.566 19.944
18.4090 18.6475 18.7296 18.744 19.065
18.7638 18.9780 19.0518 19.064 19.365
18.8659 19.0730 19.1442 19.155 19.452
18.8807 19.0867 19.1578 19.168 19.464
18.8858 19.0914 19.1623 19.172 19.469
100 100 100 100 100
Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ HSDT [10] FSDT [10]
1.5323 1.6849 1.7370 1.7519 1.7535
5.0401 5.3711 5.5008 5.5388 5.5428
10.9689 11.1385 11.2346 11.268 11.273
16.1840 15.7804 15.7221 15.711 15.714
17.3556 16.7727 16.6678 16.642 16.645
17.7844 17.1314 17.008 16.977 16.980
10 10 10 10 10
Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ HSDT [10] FSDT [10]
10.132 10.285 10.339 10.332 9.7937
10.741 10.841 10.876 10.862 10.191
10.777 10.874 10.908 10.893 10.214
10.783 10.878 10.913 10.898 10.218
10.784 10.88 10.914 10.899 10.220
100 100 100 100 100
Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ HSDT [10] FSDT [10]
1.3436 1.4589 1.499 1.5092 1.5118
theory underpredicts fundamental natural frequencies of antisymmetric cross-ply shells; for symmetric thick shells and symmetric shallow thin shells the trend reverses. The present radial basis function method is compared with analytical results by Reddy [10] and shows excellent
10.614 10.725 10.765 10.752 10.110 3.3984 3.562 3.6235 3.6426 3.6445
5.3948 5.4899 5.5358 5.5503 5.5473
6.4392 6.4562 6.4791 6.4895 6.4827
6.6216 6.6221 6.6421 6.6496 6.6421
6.6846 6.6793 6.6975 6.7047 6.6970
agreement. In fact for both first- and third-order approaches differences are lower than 1% for all ratios and laminates. Tables 11 and 12 contain nondimensionalized natural frequencies obtained using the higher-order theory for
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Table 7 Non-dimensional central deflection, w ¼ wð102 E 2 h3 =P0 a4 Þ variation with various number of grid points per unit length, N for different R=a ratios, for R1 ¼ R2 , with uniform load (½0 =90 =90 =0 ) a=h
½0 =90 =90 =0
R=a
Method
5
10
20
50
100
109
10.761 10.876 10.917 10.904 10.141
10.891 10.994 11.031 11.017 10.222
10.928 11.028 11.063 11.049 10.245
10.933 11.033 11.068 11.053 10.249
10.935 11.034 11.069 11.055 10.251
10 10 10 10 10
Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ HSDT [10] FSDT [10]
10.269 10.427 10.482 10.476 9.8249
100 100 100 100 100
Present ðN ¼ 11Þ Present ðN ¼ 15Þ Present ðN ¼ 21Þ HSDT [10] FSDT [10]
1.3666 1.482 1.5225 1.5332 1.5358
3.4735 3.6379 3.6997 3.7195 3.7208
5.5172 5.608 5.6511 5.666 5.6618
6.5875 6.5953 6.6153 6.6234 6.6148
6.7745 6.7649 6.7785 6.7866 6.7772
6.8392 6.8233 6.8366 6.8427 6.8331
Table 8 pffiffiffiffiffiffiffiffiffiffiffi Nondimensionalized fundamental frequencies of cross-ply laminated spherical shells, o ¼ oða2 =hÞ r=E 2 , laminate (½0 =90 =90 =0 ) a=h
½0 =90 =90 =0
R=a
Method
5
10
20
50
100
109
10
Present ðN ¼ 7Þ Present ðN ¼ 9Þ Present ðN ¼ 11Þ Present ðN ¼ 13Þ Present ðN ¼ 15Þ Present ðN ¼ 17Þ HSDT [10]
12.239 12.127 12.085 12.066 12.056 12.051 12.040
11.933 11.881 11.862 11.854 11.850 11.848 11.840
11.854 11.818 11.806 11.800 11.798 11.796 11.790
11.832 11.800 11.790 11.785 11.783 11.782 11.780
11.829 11.798 11.787 11.783 11.781 11.780 11.780
11.828 11.797 11.787 11.782 11.780 11.779 11.780
100
Present ðN ¼ 7Þ Present ðN ¼ 9Þ Present ðN ¼ 11Þ Present ðN ¼ 13Þ Present ðN ¼ 15Þ Present ðN ¼ 17Þ HSDT [10]
35.544 33.213 32.176 31.681 31.430 31.296 31.100
21.118 20.816 20.612 20.506 20.457 20.422 20.380
15.436 16.260 16.467 16.545 16.579 16.595 16.630
13.412 14.730 15.102 15.250 15.318 15.359 15.420
13.097 14.498 14.896 15.055 15.130 15.170 15.230
12.990 14.420 14.827 14.990 15.067 15.109 15.170
Table 9 pffiffiffiffiffiffiffiffiffiffiffi Nondimensionalized fundamental frequencies of cross-ply laminated spherical shells, o ¼ oða2 =hÞ r=E 2 , laminate (½0 =90 =0 ) a=h
Method
R=a 5
10
20
50
100
109
10
Present ðN ¼ 7Þ Present ðN ¼ 9Þ Present ðN ¼ 11Þ Present ðN ¼ 13Þ Present ðN ¼ 15Þ Present ðN ¼ 17Þ HSDT [10]
12.242 12.134 12.094 12.076 12.068 12.063 12.060
11.935 11.889 11.873 11.867 11.863 11.861 11.860
11.856 11.826 11.817 11.813 11.811 11.810 11.810
11.834 11.809 11.801 11.798 11.797 11.796 11.790
11.831 11.806 11.799 11.796 11.794 11.794 11.790
11.830 11.805 11.798 11.795 11.794 11.793 11.790
100
Present ðN ¼ 7Þ Present ðN ¼ 9Þ Present ðN ¼ 11Þ Present ðN ¼ 13Þ Present ðN ¼ 15Þ Present ðN ¼ 17Þ HSDT [10]
35.590 33.177 32.114 31.609 31.353 31.216 31.020
21.221 20.828 20.600 20.484 20.425 20.394 20.350
15.587 16.297 16.480 16.547 16.578 16.595 16.620
13.588 14.778 15.124 15.261 15.326 15.361 15.420
13.277 14.548 14.920 15.068 15.139 15.176 15.240
13.172 14.471 14.851 15.004 15.077 15.115 15.170
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Table 10 pffiffiffiffiffiffiffiffiffiffiffi Nondimensionalized fundamental frequencies of cross-ply laminated spherical shells, o ¼ oða2 =hÞ r=E 2 , laminate (½0 =90 ) a=h
Method
R=a 5
10
Present ðN ¼ 7Þ Present ðN ¼ 9Þ Present ðN ¼ 11Þ Present ðN ¼ 13Þ Present ðN ¼ 15Þ Present ðN ¼ 17Þ HSDT [10]
100
Present ðN ¼ 7Þ Present ðN ¼ 9Þ Present ðN ¼ 11Þ Present ðN ¼ 13Þ Present ðN ¼ 15Þ Present ðN ¼ 17Þ HSDT [10]
9.5185 9.4085 9.3697 9.3536 9.3460 9.3420 9.3370 33.721 31.149 30.011 29.470 29.195 29.049 28.840
10 9.1233 9.0859 9.0754 9.0717 9.0702 9.0694 9.0680 17.698 17.287 17.014 16.874 16.805 16.762 16.710
20 9.0213 9.0031 8.9999 8.9995 8.9995 8.9995 8.9990 10.169 11.379 11.647 11.740 11.785 11.806 11.840
100
109
8.9926 8.9798 8.9787 8.9792 8.9796 8.9798 8.9800
8.9886 8.9765 8.9757 8.9763 8.9767 8.9770 8.9770
8.9873 8.9755 8.9747 8.9753 8.9758 8.9761 8.9760
6.6587 9.0525 9.6176 9.8312 9.9302 9.9822 10.060
5.9895 8.6690 9.2917 9.5167 9.6389 9.6929 9.7840
5.7476 8.5372 9.1806 9.4196 9.5375 9.5905 9.6880
50
Table 11 pffiffiffiffiffiffiffiffiffiffiffi Nondimensionalized fundamental frequencies of cross-ply cylindrical shells, o ¼ oða2 =hÞ r=E 2 R=a
Method
[0/90]
[0/90/0]
[0/90/90/0]
a=h ¼ 100
a=h ¼ 10
a=h ¼ 100
a=h ¼ 10
a=h ¼ 100
a=h ¼ 10
5
Present ðN ¼ 7Þ Present ðN ¼ 9Þ Present ðN ¼ 11Þ Present ðN ¼ 13Þ Present ðN ¼ 15Þ Present ðN ¼ 17Þ FSDT [10] HSDT [10]
20.2936 18.4646 17.5967 17.1762 16.9639 16.8523 16.668 16.690
9.1302 9.0993 9.0943 9.0941 9.0945 9.0949 8.9082 9.0230
24.7600 22.5379 21.4708 20.9488 20.6803 20.5357 20.332 20.330
11.9952 11.9068 11.8754 11.8617 11.8550 11.8513 12.207 11.850
23.3888 21.8946 21.1533 20.7891 20.6023 20.5033 20.361 20.360
11.9661 11.8873 11.8592 11.8469 11.8408 11.8376 12.267 11.830
10
Present ðN ¼ 7Þ Present ðN ¼ 9Þ Present ðN ¼ 11Þ Present ðN ¼ 13Þ Present ðN ¼ 15Þ Present ðN ¼ 17Þ FSDT [10] HSDT [10]
11.2950 11.8050 11.8397 11.8349 11.8251 11.8250 11.831 11.840
9.0109 9.0053 9.0084 9.0112 9.0129 9.0139 8.8879 8.9790
16.8901 16.8697 16.7605 16.6947 16.6500 16.6417 16.625 16.620
11.8717 11.8310 11.8177 11.8120 11.8092 11.8076 12.173 11.800
16.2680 16.6195 16.6423 16.6353 16.6293 16.6289 16.634 16.630
11.8626 11.8196 11.8050 11.7986 11.7954 11.7937 12.236 11.790
20
Present ðN ¼ 7Þ Present ðN ¼ 9Þ Present ðN ¼ 11Þ Present ðN ¼ 13Þ Present ðN ¼ 15Þ Present ðN ¼ 17Þ FSDT [10] HSDT [10]
7.4668 9.4336 9.8965 10.0695 10.1436 10.1955 10.265 10.27
8.9872 8.9824 8.9850 8.9874 8.9888 8.9895 8.8900 8.972
14.1975 15.1074 15.3513 15.4441 15.4915 15.5106 15.556 15.55
11.8402 11.8118 11.8032 11.7995 11.7976 11.7966 12.166 11.79
13.8853 15.0009 15.3017 15.4182 15.4727 15.5022 15.559 15.55
11.8364 11.8025 11.7913 11.7865 11.7840 11.7827 12.230 11.78
cross-ply cylindrical shells with lamination schemes [0/90], [0/90/0], [0/90/90/0]. Present results are compared with analytical solutions by Reddy and Liu [10] who considered both the first-order (FSDT) and the third-order (HSDT) theories. The first-order theory underpredicts fundamental natural frequencies of antisymmetric cross-ply shells; for
symmetric thick shells and symmetric shallow thin shells the trend reverses. The present radial basis function method is compared with analytical results by Reddy [10] and shows excellent agreement. In fact for both first- and third-order approaches differences are lower than 1% for all ratios and laminates.
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Table 12 pffiffiffiffiffiffiffiffiffiffiffi Nondimensionalized fundamental frequencies of cross-ply cylindrical shells, o ¼ oða2 =hÞ r=E 2 R=a
Method
[0/90]
[0/90/0]
[0/90/90/0]
a=h ¼ 100
a=h ¼ 10
a=h ¼ 100
a=h ¼ 10
a=h ¼ 100
a=h ¼ 10
50
Present ðN ¼ 7Þ Present ðN ¼ 9Þ Present ðN ¼ 11Þ Present ðN ¼ 13Þ Present ðN ¼ 15Þ Present ðN ¼ 17Þ FSDT [10] HSDT [10]
6.0147 8.6719 9.2903 9.5239 9.6417 9.7022 9.7816 9.783
8.9847 8.9763 8.9773 8.9787 8.9796 8.9802 8.8951 8.973
13.3418 14.5747 14.9324 15.0764 15.1422 15.1789 15.244 15.24
11.8313 11.8065 11.7991 11.7960 11.7944 11.7935 12.163 11.79
13.1374 14.5144 14.9043 15.0616 15.1326 15.1726 15.245 15.23
11.8290 11.7977 11.7875 11.7831 11.7808 11.7796 12.228 11.78
100
Present ðN ¼ 7Þ Present ðN ¼ 9Þ Present ðN ¼ 11Þ Present ðN ¼ 13Þ Present ðN ¼ 15Þ Present ðN ¼ 17Þ FSDT [10] HSDT [10]
5.7976 8.5648 9.2045 9.4466 9.5542 9.6201 9.7108 9.712
8.9854 8.9756 8.9757 8.9768 8.9775 8.9602 8.8974 8.975
13.2148 14.4969 14.8716 15.0216 15.0932 15.1306 15.198 15.19
11.8301 11.8057 11.7985 11.7955 11.7940 11.7930 12.163 11.79
13.0270 14.4435 14.8467 15.0076 15.0840 15.1261 15.199 15.19
11.8279 11.7970 11.7869 11.7826 11.7804 11.7792 12.227 11.78
Plate
Present ðN ¼ 7Þ Present ðN ¼ 9Þ Present ðN ¼ 11Þ Present ðN ¼ 13Þ Present ðN ¼ 15Þ Present ðN ¼ 17Þ FSDT [10] HSDT [10]
5.7476 8.5372 9.1806 9.4230 9.5364 9.5985 9.6873 9.6880
8.9873 8.9755 8.9747 8.9753 8.9758 8.9755 8.8998 8.9760
13.1722 14.4709 14.8513 15.0039 15.0737 15.1157 15.183 15.170
11.8296 11.8054 11.7983 11.7953 11.7938 11.7929 12.162 11.790
12.9899 14.4198 14.8274 14.9902 15.0672 15.1087 15.184 15.170
11.8276 11.7968 11.7867 11.7824 11.7802 11.7790 12.226 11.780
7. Concluding remarks
References
In this paper the Reddy higher-order shear deformation theory of laminated orthotropic elastic shells was implemented for the first time through a multiquadrics discretization of equations of motion and boundary conditions. The theory accounts for parabolic distribution of the transverse shear strains through the thickness of the shell and tangential stress-free boundary conditions on the boundary surfaces of the shell. The multiquadric RBF method for the solution of shell bending and free vibration problems was presented. Results for static deformations and natural frequencies were obtained and compared with other sources. This meshless approach demonstrated that is very successful in the static deformations and free vibration analysis of laminated composite shells or plates. Advantages of RBFs are absence of mesh, ease of discretization of boundary conditions and equations of equilibrium or motion and very easy coding. We show that the static displacements and stress and the natural frequencies obtained from present method are in excellent agreement with analytical solutions. We illustrate that this combination of a thirdorder theory for shells and the discretization by multiquadrics produces results for static and free vibrations as good as the exact solutions from [10], with a simple yet accurate formulation.
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