Keywords: finite element, multilayered plates, multilayered shell, refined model, geometrically non linear analysis.

The aim of this work is to analyse the geometrically non linear mechanical behaviour of multilayered structures by a high order plate/shell finite element in order to predict displacements and stresses of such composite structures for design applications. Based on a conforming finite element method, a C¹ triangular six node finite element was previously presented for linear static and dynamic evaluations, see [1,2]. This element is based on a refined kinematic model [3], and only five generalized displacements are used to ensure :

• a cosine distribution for the transverse shear stresses with respect to the thickness co-ordinate, avoiding 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 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 [3].

An explicit map between curvilinear co-ordinates associated with the middle surface of the shell and cartesian co-ordinates is used to described the shell geometry. Thus, all the geometric characteristics of the shell are analytically computed : local covariant basis, metric and curvature tensors, Christoffel symbols, ... Therefore, this is a "pure" shell model.

The generalized displacements and are approximated by higher-order polynomia [4] based on :

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

The geometrically non-linear formulation is based on Von-Karmann assumptions where deflection is moderately large, while rotations and strains are small. The element performances are evaluated on some tests from the literature [5,6,7] for multilayered plates and shells in non-linear (moderately large deflection) statics.

All results indicate that the present element has very fast convergence properties and also gives very accurate results for displacements and stresses.

1

O. Polit and M. Touratier. "A new laminated triangular finite element assuring interface continuity for displacements and stresses." J. Composite Structures, 38(1-4 37, 1997. doi:10.1016/S0263-8223(97)00039-1

2

F. Dau, O. Polit, and M. Touratier. "A c triangular finite element for analysis multilayered shell structures." Comp. and Struc., 2004. to appear.

3

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

4

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

5

S.A. Zaghloul and J.B. Kennedy. "Nonlinear behaviour of symmetrically laminated plates." J. Applied Mech. ASME, 42(1):234-236, 1975.

6

T.Y. Chang and K. Sawamiphakdi. "Large deformation analysis of laminated shells by finite element method", Comp. and Struc., 13:331-340, 1981. doi:10.1016/0045-7949(81)90141-3

7

I. Kreja, R. Schimdt, and J.N. Reddy. "Finite element based on a first order shear deformation moderate rotation shell theory with applications to the analysys of composite structures", Int. J. Non-Linear Mech., 32(6):1123-1142, 1997. doi:10.1016/S0020-7462(96)00124-2

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