Co-rotational System Definitions for Large Displacement Triangular and Quadrilateral Shell Elements

The co-rotational approach offers exceptional benefits for large-displacement structural analysis problems with deformations of the bending type, particularly when accounting for arbitrarily large rigid body rotations. A principal issue in any co-rotational approach is associated with the specific choice of the local reference system in relation to the current deformed element configuration. Whilst an arbitrary choice that closely follows the current element configuration, for example using the current positions of any two of the element sides, does not significantly affect the large displacement response predictions for small strain problems, this often leads to local system definitions which are not invariant to the specified order of the element nodes. It has been previously argued that this invariance characteristic would be desirable for extending the co-rotational approach to large strain problems [

1

] and, more recently, for identifying the bifurcation points of perfectly symmetric structures [

2

]. Two approaches were employed to achieve the invariance of the local system to nodal ordering [1,2], but these suffered from complexity associated with application to material points within the element domain, and resulted in an asymmetric consistent tangent stiffness matrix which leads to further computational disadvantages.

In this paper, new definitions of the local co-rotational system are proposed for quadrilateral and triangular shell elements, which achieve the invariance characteristic to nodal ordering in a relatively simple manner, and importantly result in a symmetric tangent stiffness matrix. The proposed definitions utilise only the nodal coordinates in the deformed configuration, where two alternative definitions are outlined for each of the quadrilateral and triangular element shapes. The first is a bisector definition utilising alignment along the bisectors of one or more internal element angles, while the second is a zero-‘macro spin’ definition considering the minimisation of the spin for the triangular/quadrilateral shape as described by the element nodes. The paper presents the co-rotational transformations linking the local and global element freedoms for both definitions, and provides a numerical example to demonstrate their relative accuracy in large displacement analysis of plates and shells. It is shown that both definitions are equally accurate for small strain problems, but that the zero-‘macro spin’ definition has more general potential application in large strain problems.