Keywords: finite element, refined shell model, multilayered structures,.

The aim of this work is to analyze the mechanical behaviour of multilayered structures by shell finite element including transverse shear effects and continuity requirements between layers in order to predict displacements and stresses of such composite structures for design applications. Based on a conform finite element method, a new C¹ triangular six node finite element is presented. This element is based on a refined kinematic model [1], using only five generalized displacements, to assure :

• a cosine distributions for the transverse shear stresses along the thickness, which avoid to calculate the shear correction factors,
• the continuity conditions between layers of the laminate for both displacements and transverse shear stresses,
• the satisfaction of the boundary conditions at the top and bottom surfaces of the shells.

The displacement field for each elastic layer denoted of a laminated shallow shell is given by :

where are the two transverse shear strain components at the middle surface of the shell () and . In this expressions, we denote the in-surface displacements, the transverse displacement and the two rotations. In addition, are trigonometric functions of and are linear functions of determined from the boundary conditions on the top and bottom surfaces of the shell, and from the continuity requirements at the layer interfaces for displacements and stresses, see Reference [1].

Several strain models, called Complete model, Love model and Love-Donnell model are deduced from the displacement field according to appropriate assumptions such as (i.e where denotes curvature tensor) due to geometry or/and neglecting membrane coupling in transverse shear expression, ...

Concerning the geometry of the shell, this one is described with an explicit map between curvilinear co-ordinates associated with the middle surface of the shell and cartesian co-ordinates. The main interest is to calculate analytically all the geometric characteristics of the shell : local covariant basis, metric and curvature tensors, Christoffel symbols, ...are needed.

The generalized displacements and are approximated by higher-order polynomia [2] based on :

• Argyris interpolation for the transverse normal displacement,
• Ganev interpolation for the membrane displacements and for the transverse shear rotations.

This permits to ensure a conform finite element method and to avoid locking phenomenon.

1

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

2

M. Bernadou. Finite Element Methods for Thin Shell Problems. John Wiley et Sons, 1996.

3

N. N. Huang. Influence of shear correction factors in the higher order shear deformation theory. Int. J. Solids Struc., 31(9):1263-1277, 1994. doi:10.1016/0020-7683(94)90120-1

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