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Computational Science, Engineering & Technology Series
ISSN 17593158 CSETS: 7
COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping, Z. Bittnar
Chapter 9
Refined Computational Models for Laminated Shells M. Touratier
LMSP UMR CNRS  ENSAM  ESEM, Paris, France M. Touratier, "Refined Computational Models for Laminated Shells", in B.H.V. Topping, Z. Bittnar, (Editors), "Computational Structures Technology", SaxeCoburg Publications, Stirlingshire, UK, Chapter 9, pp 221238, 2002. doi:10.4203/csets.7.9
Keywords: multilayered shells, shearmembrane refinements, triangular C^{1}C^{1/2} finite element, piezoelectric laminates, bendingtorsion warping coupling, stress computations.
Summary
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 threedimensional 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 threedimensional 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 twodimensional theories of shells from threedimensional 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 firstorder 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 bendingtorsion coupling for laminates in Ganapathi et al. [11], as well as piezoelectric laminated beams and shells. References
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