A triangular hybrid equilibrium plate element of general degree

Abstract

SUMMARYHybrid stress-based finite elements with side displacement fields have been used to generate equilibrium models having the property of equilibrium in a strong form. This paper establishes the static and kinematic characteristics of a flat triangular hybrid equilibrium element with both membrane and plate bending actions of general polynomial degree p. The principal characteristics concern the existence of hyperstatic stress fields and spurious kinematic modes. The former are shown to exist for p > 3, and their significance to finite element analysis is reviewed. Knowledge of the latter is crucial to the determination of the stability of a mesh of triangular elements, and to the choice of procedure adopted for the solution of the system of equations. Both types of characteristic are dependent on p, and are established as regards their numbers and general algebraic forms. Graphical illustrations of these forms are included in the paper.

The stability of stars of triangular equilibrium plate elements

“…Two of these modes involve only displacements of the sides incident with the end nodes 0 and 2 of the link. These modes are the same as for single elements, which are defined and illustrated in [22].…”

“…Tables I and II give these numbers as the dimensions of kinematic spaces derived by Maunder and Almeida [22], where n e rbm denotes the number of rigid body modes. In the case of flexural actions, Reissner-Mindlin plate theory is used, which includes shear deformations, and side displacements include rotations of normal fibres through the thickness as well as transverse translations.…”

“…It should be noted that there are no rigid body modes for the flexural element since transverse displacements are not included in the side freedoms when p = 0. There are thus three independent spurious kinematic modes, and these may be defined to correspond to point O situated at the three midpoints A, B, and C. The spurious kinematic modes for higher degree triangular elements with both membrane and flexural actions are fully described in [22].…”

“…When p = 2, n e skm = 4 and the extra admissibility condition can be expressed as in the following equation [22]: where indices 4, 5, and 6 refer to midside nodes opposite corner nodes 1, 2, and 3, respectively. The components of moment and shear intensities are illustrated in Figure B5.…”

“…Figure 2(b), and using properties of the spurious modes for a pair of elements. The two usual modes of an open star are restricted to displacements of the pairs of sides incident with the end nodes of the link as defined for single elements in [22] and are illustrated in Figure A5(a).…”