Keywords: shear deformation, anisotropic, laminated shells, cross-ply, stacking.

A shell may be said to be shallow whenever the smallest radius of curvature at every point is larger than the greatest length measured along the middle surface of the shell, i.e, every point of the middle surface is sufficiently close to a plane. An alternative definition for shallow shells can be given that the Gaussian curvature at any point of the middle surface is small compared with , where and are principal radii of curvature, and is the smallest length in planform projections.

The purpose of this paper is to derive a more general theory of laminated shells with shear deformation in cartesian coordinates, based on the improved Love- Kirchhoff hypotheses and the consideration of Love's higher approximation, through the variational principle[1] and to give the fundamental equations of motion and corresponding types of boundary conditions and energy functionals for laminated shallow shells. Extensive numerical study for laminated shells made to investigate the various stacking sequence effects in the static deflection results by using closed-form solutions for shallow shells having simple supported boundary. Also, static analysis is carried out for cross-ply laminated shells considering the effects of various geometrical parameters, e.g., the ratios of length-to-radius and length-to-thickness, etc. The results are compared with existing one and show good agreement.

In this paper, bellows are included. (1)The theories for the anisotropic laminated shallow shell are obtained by employing the generalized Hooke's law [2,3]. Careful considerations such as examining the validity of the assumptions for a single-layer shell theory when applied to a nonhomogeneous, multi-layer shell, are made. The basic equations of motion, boundary conditions and energy functionals for laminated shells [4] are presented. (2)Closed-form solutions of Navier's type for cross-ply laminated shells having simple supported boundary are shown. (3)To observe the effects of different parameters such as shallowness ratio, curvature ratio and thickness ratio, illustrative numerical analyses for anisotropic laminated shells with transverse shear deformation, in static deflection are presented. Meanwhile, comparisons are made among the proposed theory and thin shell theory. (4)For six different laminates having two symmetrical and four asymmetrical stacking sequences, the effects of parameters of geometric and material properties are examined for shells of rectangular planform, having simple supported boundary. Comparisons among the results obtained from the most accurate of the present theory and the thin shell theory are also given. (5)Finally, this paper contains the conclusions and major findings of this study.

The conclusions of this paper are as follows. (1)There occurs some differences between the symmetric stacking sequences and asymmetric stacking sequence. The symmetrical models [0/90/90/0] and [90/0/0/90] show the similar behaviors and the behaviors of anisotropic laminated shells consisted by asymmetrically as [0/90] and [90/0], [0/90/0/90] and [90/0/90/0] are nearly identical. (2)Generally, the central deflections of [0/90/90/0] and [90/0/0/90] are less than those of [0/90/0/90] and [90/0/90/0], i.e., it means difference of stiffness as stacking sequence. These results can verify the fact that the central deflections of symmetric laminates are less than those of asymmetric laminates because of the stretching-bending coupling which exists for asymmetrical laminates. (3)In case of shells with four layers, the differences between the results obtained from this paper and TST are less than 1% for three typical shells with . However, the differences between the results obtained from this paper and TST are higher than 40% for three typical shells having . These facts imply that it is necessary to consider the shear deformation effect for the thick shell structures. (4)The central deflections of two layers laminated shells as [0/90] and [90/0] are more than those of four layers laminated shells as [0/90/90/0], [90/0/0/90], [0/90/0/90] and [90/0/90/0]. It means that the stiffness of laminated shell is related to the number of laminates. (5)The central deflections in cylindrical shells and spherical shells are decreased according to increasing of the shallowness ratio. On the other hand, the central deflections of hyperbolic paraboloidal shells are increased according to increasing of the shallowness. (6)As the results, it is shown that the shear effects and the stacking sequences become more important factors in structural behaviors of laminated shells, and presented theory and analysis procedure are useful for the cross-ply anisotropic laminated shells.

1

J.N. Reddy, "Energy and Variational Methods in Applied Mechanics", John Wiley & Sons, 1984.

2

D.H. Kim, "Composite Structures for Civil and Architectural Engineering", 1st ed., E & FN Spon, 1995.

3

J.M. Whitney, "Structural Analysis of laminated Anisotropic Plates", Technomic Publishing Company, 1987.

4

Ding, Kewei and Tang, Limin, "Exact Solution for Axisymmetric Thick Laminated Shells", Composite Structures, Vol.46, 125-129, 1999. doi:10.1016/S0263-8223(99)00047-1

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