A new quadrilateral shell element using 16 degrees of freedom

Abstract

Purpose

The purpose of this paper is to present a quadrilateral shell element using 16 degrees of freedom (dof) (12 translations and four rotations) which makes a pair with Morley's triangle at 12 dof. This latter has been updated by Batoz who later proposed an extension to a quadrilateral (“DKQ16”) but only with special interpolation functions for an elastic behaviour of the material. Precisely, it is in order to release from this strong limitation that a completely different formulation is proposed here.

Design/methodology/approach

The development of this new quadrilateral called “DKS16” involves three stages. The first one starts from Morley's triangle updated by Batoz (“DKT12”) to derive a rotation‐free (RF) triangular element (“S3”). The second stage consists in generalising this triangle to a RF quadrilateral (“S4”). During the final leg, the S4 and DKT12 main features are combined to give the quadrilateral “DKS16”.

Findings

Other parameters being equal, the type of finite element chosen for the forming stage simulation has a great influence on further springback result even in software with automatic remeshing. Particularly, it is pointed out that the RF shell elements S3 and S4 as well as the triangle DKT12 are less sensitive to the mesh size than classical shell elements with six dof per node. But, even if some improvements of in‐plane shear have been proposed, stamping codes users are reluctant to use triangles. That is why this paper presents an attempt to extrapolate a quadrilateral (DKS16) from the triangle DKT12 via S3 and S4 elements formulation. Numerous examples showing convergence and accuracy are presented: irregular meshes, large displacement analyses and deep‐drawing simulations.

Practical implications

The triangular “S3” element is already implemented in RADIOSS^® software and its implementation – as well as the one of “DKT12” – is in progress in Pam‐Stamp, both as “user elements”. The next step will be the implementation of the quadrilateral “S4” (RF) and, maybe, the element “DKS16” since both are cheaper in terms of computation time and are found interesting for sheet forming.

Originality/value

It seems obvious that curvatures are more exactly captured in RF elements (when nodes slide on die radius) since they are imposed in terms of translations instead of traditional nodal rotations not managed by contact conditions. As the neighbours are involved, a drawback of these RF elements is their complex formulation in case of branching surfaces and/or abrupt variations in material behaviour and/or thickness. This is not the case for elements such as DKT12 or DKS16, good candidates to add to the (long) list of cheap shell elements for large scale computations typical of sheet metal forming.

Citation

Sabourin, F., Carbonniere, J. and Brunet, M. (2009), "A new quadrilateral shell element using 16 degrees of freedom", Engineering Computations, Vol. 26 No. 5, pp. 500-540. https://doi.org/10.1108/02644400910970176