Evaluation of the accuracy of classical beam FE models via locking-free hierarchically refined elements

It is well known that the classical 6-DOF (Degrees of Freedom) beam theories that are incorporated in commercial finite element (FE) tools are not able to foresee higher-order phenomena, such as elastic bending/shear coupling, restrained torsional warping and three-dimensional strain effects. In this work, the accuracy of one-dimensional (1D) finite elements based on the classical theories (Euler-Bernoulli and Timoshenko theories as well as a 6-DOF model including torsion) is evaluated for a number of problems of practical interest and modelling guidelines are given. The investigation is carried out by exploiting a novel hierarchical, locking-free, finite beam element based on the well-known Carrera Unified Formulation (CUF). Thanks to CUF, the FE arrays of the novel beam element are written in terms of fundamental nuclei, which are invariant with respect to the theory approximation order. Thus, results from classical as well as arbitrarily refined beam models can be formally obtained by the same CUF beam element. Linear Lagrange shape functions are used in this paper to interpolate the generalized unknowns and shear locking phenomena are avoided by adopting an MITC (Mixed Interpolation of Tensorial Components) scheme. Different sample problems are addressed, including rectangular and warping-free circular cross-sections as well as thin-walled beams. The results from classical theories and the 6-DOF model are compared to those from higher-order refined beam models, both in terms of displacement and stress fields for various loading conditions. The discussion focuses on the limitations of the commonly used 1D FEs and the need for refined kinematics beams for most of the problems of common interest. The research clearly depicts CUF as a valuable framework to assess FE formulations such as the 6-DOF model herein considered, which is one of the most known and used finite element for the analysis of structures.