Hierarchical theories of structures based on Legendre polynomial expansions with finite element applications

This paper discusses classical and refined beam and plate theories based on the Carrera Unified Formulation (CUF). Attention is focussed on (but not limited to) a new refined beam element with enhanced kinematics based on Legendre polynomial expansions of the primary mechanical variables. By employing CUF, the governing equations and the related finite element arrays are written in a hierarchical, compact and general manner. Readily, these characteristics are used to arbitrarily tune the finite element model at the cross-sectional level, by locally enriching the theory kinematics up to the desired accuracy. The uncompromising accuracy of the present beam model is demonstrated by considering various numerical examples, including solid and thin walled beams with open and close cross-sections as well as plate structures. The results are compared with those from classical and already established refined CUF models. Eventually, three-dimensional elasticity solutions by the commercial tool MSC Nastran are also given to underline the high accuracy of the present methodology. The numerical efficiency and the capabilities of the Legendre-based CUF beam models to deal with complex structures with no geometrical approximations result clear from the analyses conducted.