Further discussion on four-node isoparametric quadrilateral elements in plane bending

The aim of the study is: "rst*to show that the "nite element method for plane bending is not sensitive to the distorted geometry of isoparametric quadrilateral elements, second*to close out further investigations on irregular elements based on Wilson incompatible functions employing ad hoc methods to force elements to pass the patch test, regardless of the type of the variational formulation used for the problem. A systematic review of papers dealing with "nite four-node plane elements with two degrees of freedom at each node is presented. Selected results of the benchmark example of the mesh distortion test used by all cited authors are compared suggesting that no improvement has been made in more than two decades of intensive development of new elements and methods in the "eld. An attempt has been made to localize the pathological behaviour of irregular elements under consideration by calculating the areas of the elements after deformation for di!erent values of the distortion parameter. It made possible to observe the mass density of the whole deformed system. The tip de#ections at the lower and upper edge at the free end of the cantilever beam were depicted as well, illustrating unexpected physical phenomena of the longitudinal "bers deviating from being parallel. The horizontal displacement components at two common nodes of the adjacent elements have been compared with the analytical solution of the problem for the varying distortion parameter. The resulting horizontal displacement distribution along the common side of the elements has been shown to be the source of unreal behavior of all classes of irregular elements based on Wilson incompatible functions. Increasing the order of the interpolation functions for the displacement "eld, i.e. application of subparametric elements, makes the "nite element method insensitive to element distortion