Keywords: multilayered shells, shear-membrane refinements, triangular C¹-C^1/2 finite element, piezoelectric laminates, bending-torsion warping coupling, stress computations.

Despite an enormous literature on the subject, structural mechanics modeling including plates and shells is still open research both at fundamental and applied levels. In fact, structural plates and shells are three-dimensional bodies, one dimension (the thickness) of which happens to be much smaller than the two others. Thus the quality of a plate or shell theory must be judged on the basis of how well its exact solution approximates the exact three-dimensional elasticity problem. In addition, the exact solution depends not only on the choice of the model, but also on the geometry, material properties, loading and boundary conditions. Finally, finite element approximations of these basic models or theories are themselves dependent on the interpolation with its particular numerical phenomena, especially in structures (numerical locking).

Because of difficulties involved in deriving two-dimensional theories of shells from three-dimensional equations of elasticity, assumptions of one kind or another must be introduced in the derivation. Approximate bidimensional linear theories for shells have therefore been developed by making use of an assumed displacement field in powers of the thickness coordinate and a variational theorem. An integration with respect to the thickness coordinate supplies the governing differential equations and consistent boundary conditions in terms of unknown generalized coordinates which are independent of the thickness coordinate. An asymptotic integration of the elasticity equations has been employed for nonhomogeneous shells, Widera and Logan [1].

For laminated shells, recent papers have been published for the purpose of removing the inaccuracies of the first-order shear deformation theory which accounts only for constant transverse shear stresses through thickness, see Doxsee [2], Touratier [3], Fraternali and Reddy [4], Shu [5], He [6], and Carrera [7] accounting for layerwise mixed description. Note that references [4,5,6,7], and Ossadzow et al [8] show various approaches to take into account of interface conditions and top and bottom boundary conditions for laminated shells.

In this paper we present an approach for developing a simple and refined theory for deep, doubly curved laminated shells, which allows verifying exactly the continuity of transverse shear stresses and displacements at the layer interfaces and boundary conditions at the top and bottom surfaces of the shell, while, at the same time, the membrane and shear terms are refined. Finite element approximations of the shell/plate theory have been proposed in Béakou and Touratier [9], Polit and Touratier [10]. Finally, this work has been extended to sandwich beam finite element under bending-torsion coupling for laminates in Ganapathi et al. [11], as well as piezoelectric laminated beams and shells.

1

Widera, G.E.O., Logan, D.L., "Refined theories for nonhomogeneous anisotropic cylindrical shells: Part I-derivation", J. Engng Mech. Div., 106, 1053- 1074, 1980.

2

Doxsee, L.E., "A higher-order theory of hygrothermal behavior of laminated composite shells", Int. J. Solids Structures, 25(4), 339-355, 1989. doi:10.1016/0020-7683(89)90052-8

3

Touratier, M., "A refined theory of laminated shallow shells", Int. J. Solids Structures, 29(11), 1401-1415, 1992. doi:10.1016/0020-7683(92)90086-9

4

Fraternali, F., Reddy, J.N., "A penalty model for the analysis of laminated composite shells", Int. J. Solids Structures, 30(24), 3337-3355, 1993. doi:10.1016/0020-7683(93)90088-O

5

Shu, X.P., "An improved simple higher-order theory for laminated composite shells", Computers & Structures, 60(3), 343-350, 1996. doi:10.1016/0045-7949(95)00406-8

6

He, L.H., "A linear theory of laminated shells accounting for continuity of displacements and transverse shear stresses at layer interfaces", Int. J. Solids Structures, 31(5), 613-627, 1994. doi:10.1016/0020-7683(94)90141-4

7

Carrera, E., "Multilayered shell theories accounting for layerwise mixed description, Part 1: Governing equations", AIAA Journal, 37(9), 1107-1116, 1999. doi:10.2514/2.821

8

Ossadzow, C., Touratier, M., Muller, P., "Deep doubly curved multilayered shell theory", AIAA Journal, 37(1), 100-109, 1999. doi:10.2514/2.670

9

Béakou, A., Touratier, M., "A rectangular finite element for analysing composite multilayered shallow shells in statics, vibration, and buckling", Int. J. Numer. Meth. Engng, 36, 627-653, 1993. doi:10.1002/nme.1620360406

10

Polit, O., Touratier, M., "High-order triangular sandwich plate finite element for linear and non-linear analyses", Comput. Meth. Appl. Mech. Engng, 185, 305-324, 2000. doi:10.1016/S0045-7825(99)00264-9

11

Ganapathi, M., Patel, B.P., Polit, O., Touratier, M., "A C¹ finite element including transverse shear and torsion warping for rectangular sandwich beams", Int. J. Numer. Meth. Engng, 45, 47-75, 1999. doi:10.1002/(SICI)1097-0207(19990510)45:1<47::AID-NME578>3.0.CO;2-B

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