An efficient and locking-free flat anistropic plate and shell triangular element

Abstract

The present study discusses, in a unified manner, the application of generalised decomposition principles and physical lumping techniques in finite element analysis of isotropic and laminated plates and shells. Using physical ideas and basic relations of mechanics, as well as geometrical arguments, it attempts to reduce the apparent complexity of the kinematical behaviour of such structures by means of assigning rigid-body and straining modes of deformation to an assembly of elements which comprise a compatible kinematical idealization. Subsequently, it applies these assertions to study a flat three-node 18 degrees-of-freedom triangular shell element applicable to the analysis of isotropic and fibre-reinforced laminated composite structures. Following the presentation of the general thermoelastic lamination theory, the Natural Mode Method is employed to form the elemental natural and Cartesian stiffness matrices. Although the elemental Cartesian stiffness matrix is of the order 18 × 18 and the initial Cartesian thermal load vector of the order 18 × 1, the triangular element essentially incorporates 12 generalized freedoms (the natural straining modes), and therefore neccesitates the computation of a 12 × 12 (not fully populated) natural stiffness matrix, and a 12 × 1 thermal (initial) vector, which to our knowledge make the present element the most inexpensive flat shell element available. The features of economy, simplicity and accuracy, facilitate the application of the present element in the non-linear range, in CPU intensive applications such as optimisation and failure — in general in the analysis of large and complex shell structures. The triangular finite element is supported by significant postprocessing features including the computation of all through-the-thickness stresses being equally important for structures composed from composite materials. Numerical examples are presented for both isotropic and laminated composite structures, and comparisons with analytic solutions are attempted where available. All computational experiments illustrate the accuracy of the conceived simple triangular element and substantiate its physical and theoretical bases.