Locking, rank and singularity of penalty-linked stiffness matrix and consistency of strain-field

Abstract The poor behaviour of conventionally formulated displacement type C 0 finite elements (i.e., with exactly integrated stiffness matrices) is often attributed to the high rank and non-singularity of the penalty-linked stiffness matrix. In this paper, we show that the correct rank and non-singularity required emerges directly from the consistency of the discretized strain field approximations. A simple study using Ritz-type approximations of the Timoshenko beam problem shows how these aspects are all linked. This investigation therefore unifies many of the statements made about such problems and provides a single, consistent viewpoint.