Keywords: FEA, elastic, laminate, fibre reinforced composite, transverse shear.

A reliability-based finite element formulation, capable of analysing laminated plate and shell structures made from fibre-reinforced composite materials is presented in this paper. The problem is one of non-linear optimisation in which we wish to calculate the equilibrium geometry of a given linear-elastic plate which is subjected to some system of forces and/or moments. A total Lagrangian frame of reference, rather than the classical displacement vector, is used to derive a suitable form for the model equations that can then be solved to determine the equilibrium state.

There are a number of formulations that have used the displacement vector approach to derive finite element methods suitable for laminated plate deformation analyses[1,2,3,4]. Some of them take into account transverse shear strains. For problems involving large displacements, rotations and strains, the total Lagrangian description used here has been shown to be more accurate at quantifying non-linear plate geometry and has a computational advantage because local element co-ordinate transformations are not required at run-time. The laminated plate theory of Beakou and Touratier[1] is used to incorporate transverse shear strains into our model and also to enforce inter-layer continuity of displacements and transverse shear stresses.

For simplicity the initial model presented here is restricted to the analysis of plates that are shallow, i.e. the ratio of the plate thickness to either of the principle curvatures is small, as was the case in Beakou and Touratier[1]. The degrees of freedom in our finite element algorithm are the absolute co-ordinates (and their derivatives) of a point on the mid-surface (derived from the total Lagrange description) along with two rotational degrees of freedom (which follow from the Touratier kinematic model[1]). We show how the related system of non-linear equations is constructed along with the elemental stiffness matrix.

The next steps are coding, the formulation of suitable benchmark cases and testing. The longer term goal of the project, of which this work is part of, is to integrate such a deterministic finite element code into a larger statistical software tool that takes into account the level of uncertainty in the experimentally measured values of the elastic constants and material strengths of a given lamina.

1

A. Beakou, M. Touratier, "A Rectangular Finite Element for Analysing Composite Multilayered Shallow Shells in Statics, Vibration and Buckling", International Journal for Numerical Methods in Engineering, 36, 627-653, 1993. doi:10.1002/nme.1620360406

2

M. Ganapathi, B.P. Patel, O. Polit, M. Touratier, "A C¹ Finite Element Including Transverse Shear and Torsion Warping for Rectangular Sandwich Beams", International Journal of Numerical Methods in Engineering, 45, 47-75, 1999. doi:10.1002/(SICI)1097-0207(19990510)45:1<47::AID-NME578>3.0.CO;2-B

3

O. Polit, M. Touratier, "A multilayered/sandwich triangular finite element applied to linear and non-linear analyses", Composite Structures, 58, 121-128, 2002. doi:10.1016/S0263-8223(02)00033-8

4

M. Ganapathi, B.P. Patel, M. Touratier, "A C¹ finite element for flexural and torsional analysis of rectangular piezoelectric laminated/sandwich composite beams", International Journal of Numerical Methods in Engineering, 61, 584-610, 2004. doi:10.1002/nme.1082

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