Rotation shape functions for a low‐order quadrilateral plate element with mid‐side rotations

Abstract

SUMMARYThis short paper re-visits an earlier paper by Nagtegaal and Slater in which a quadrilateral shell element was derived which used similar kinematics to that in the Morley facet shell. Because the paper seems to be relatively unknown and also is rather long and involved, we here concentrate on the plate-bending aspects of the element and describe the shape functions.

A finite-strain quadrilateral shell element based on discrete Kirchhoff-Love constraints

“…, IV. Symmetry requirements (Nagtegaal and Slater [29] and Crisfield and Tan [35] did not require their shape functions to be symmetric) result in the following relations:…”

“…We also use shape functions with quadratic terms, but they are different from those in References [29,30] (note that Batoz et al favour the so-called 'rational' shape functions [25]). In References [30,35], a central node was considered in the derivation of the shape functions, which did not possess the usual spatial symmetries. In addition, those authors took other special steps in the derivations, including the introduction of two parameters to satisfy the patch test.…”

“…• Director shape functions are newly developed, characterized by the absence of nonsymmetric terms (see previous developments [29,30,35]). …”

“…Reference [30]) such as moderate rotations and co-rotational formulations based on the super-position of a plane element with a plate bending element, valid for triangular elements only. Non-symmetric shape functions are employed in References [29,35]. No complete theory, including the constraints, has been developed for them in the fully non-linear range.…”