Membrane triangles with corner drilling freedoms—I. The EFF element

Abstract

This paper is the first of a three-part series that studies the formulation of 3-node, 9-dof membrane elements with normal-to-element-plane rotations (the so-called drilling freedoms) within the context of parametrized variational principles. These principles supply a unified basis for several advanced element-construction techniques; in particular: the free formulation (FF), the extended free formulation (EFF) and the assumed natural deviatoric strain (ANDES) formulation. In Part I we construct an element of this type using the EFF. This derivation illustrates the basic steps in the application of that formulation to the construction of high-performance, rank-sufficient, nonconforming elements with corner rotations. The element is initially given the twelve degress of freedom of the linear strain triangle (LST), which allows the displacement expansion to be a complete quadratic in each component. The expansion basis contains the six linear basic functions and six energy-orthogonal quadratic higher-order functions. Three degrees of freedom, defined as the midpoint deviations from linearity along the triangle-median directions, are eliminated by kinematic constraints. The remaining hierarchical midpoint freedoms are transformed to corner rotations. The performance of the resulting element is evaluated in Part III.