Keywords: composite shells, multiquadrics, higher-order theory.

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 stresses 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 [2] and Bert [3,4]. These theories are expected to produce accurate results when the side-to thickness ratio 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 stresses.

The introduction of transverse shear and normal stresses represent an improvement to classical theories. However, for (side to radius ratio) the transverse normal stresses are negligible when compared to the transverse shear stresses.

The effect of transverse shear and normal stresses in shells studied by Hildebrand et al [5], Lure [6] and Reissner [7]. The effect of transverse shear deformation was also considered by Vinson [8], Dong et al [9,10] and Whitney and Sun [11,12].

Reddy [1] presented a higher-order theory that is based on five degrees of freedom (same number as in a first-order shear deformation theory by Reddy [13]). 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 distribution of the transverse shear stresses and zero transverse normal strain. Therefore no shear correction factors are used. In Reddy [1] a Navier-type solution is presented. This solution is only feasible for special loading and boundary conditions.

The solution of laminated composite shells by finite elements has been the subject of intense research. 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 method can be successful in the analysis of laminated composite shells.

The multiquadric radial basis function method was introduced by Hardy [14] for the interpolation of scattered geographical data. The use of multiquadrics for the solution of partial differential equations was first proposed by Kansa [15,16]. The authors have used the method for the solution of laminated beams and plates in bending. The solution of laminated elastic shells in bending by multiquadrics is here performed.

1

J. N. Reddy and C. F. Liu, "A higher-order shear deformation theory of laminated elastic shells", Int. J. Engng Sci., 23(3):319-330, 1985. doi:10.1016/0020-7225(85)90051-5

2

P. M. Naghdi, "A survey of recent progress in the theory of elastic shells", Appl. Mech. Reviews, 9(9):365-368, 1956.

3

C. W. Bert, "Dynamics of composite and sandwich panels" part I; doi:10.1177/058310247600801006 and II; doi:10.1177/058310247600801104, Shock Vib. Dig., 8(10,11):37-48, 15-24, 1976.

4

C. W. Bert, "Analysis and Performance of Composites", chapter Analysis of shells, pages 207-258. Wiley,New York, 1980.

5

F. B. Hildebrand, E. Reissner, and G. B. Thomas, "Note on the foundations of the theory of small displacements of orthotropic shells", National Advisory Comm. Aero. Tech. Notes, (1933), 1949.

6

A. I. Lurie, "Statics of Thin Elastic Shells", (in Russian), Moscow, 1947.

7

E. Reissner, "Stress-strain relations in the theory of thin elastic shells", J. Math. Phys, 31:109-119, 1952.

8

J. A. Zukas and J. R. Vinson, "Laminated transversely isotropic cylindrical shells", Journal of Applied Mechanics, 38:400-407, 1971.

9

S. B. Dong, K. S. Pister, and R. L. Taylor, "On the theory of laminated anisotropic shells and plates", Journal of Aerospace Sciences, 29, 1962.

10

S. B. Dong and F. K. W. Tso, "On a laminated orthotropic shell theory including transverse shear deformation", J. Appl. Mech., 39:1091-1096, 1972.

11

J. M. Whitney and C. T. Sun, "A higher order theory for extensional motion of laminated anisotropic shells and plates", J. Sound and Vibration, 30(85), 1973.

12

J. M. Whitney and C. T. Sun, "A refined theory for laminated anisotropic cylindrical shells", Journal of Applied Mechanics, 41(47), 1974.

13

J. N. Reddy, "Bending of laminated anisotropic shells by a shear deformable finite element", Fibre Science and Technology, 17(1):9-24, 1982. doi:10.1016/0015-0568(82)90058-6

14

R. L. Hardy, "Multiquadric equations of topography and other irregular surfaces", Geophys. Res., 176:1905-1915, 1971. doi:10.1029/JB076i008p01905

15

E. J. Kansa, "Multiquadrics- a scattered data approximation scheme with applications to computational fluid dynamics. I: Surface approximations and partial derivative estimates", Comput. Math. Appl., 19(8/9):127-145, 1990. doi:10.1016/0898-1221(90)90270-T

16

E. J. Kansa, "Multiquadrics- a scattered data approximation scheme with applications to computational fluid dynamics. ii: Solutions to parabolic, hyperbolic and elliptic partial differential equations", Comput. Math. Appl., 19(8/9):147-161, 1990. doi:10.1016/0898-1221(90)90271-K

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