The first locking-free plane-elastic finite element: historia mathematica Original Research Article

We consider a two-parameter family of conforming bilinear quadrilateral finite element formulations for plane-elastic problems with homogeneous isotropic material. The approach, taken earlier by Belytschko et al. is based on numerical strain reductions and stabilization and may be viewed as an extension & reformulation of the classical Turner rectangle of 1956. We present the full theory of this approach in the energy norm framework. The analysis is based on splitting the error in two orthogonal components, the approximation error and the equilibrium or consistency error. The theory covers the full parametric dependence of the plane-elastic problem, emphasizing in particular the case of beam-like bodies with arbitrarily small aspect ratios. The theory shows that, depending on the mesh, the stabilized reduced-strain formulation is capable of avoiding some, or even all, the parametric locking effects characteristic to 2D elastic problems. We analyze first these locking effects in the context of standard bilinear FEM.