Euler-Bernoulli's and Timoshenko's theories are the classical models for beams made of isotropic materials. The former does not account for the transverse shear effects on the cross-section deformations. The latter provides a model that, at best, foresees a constant shear deformation distribution on the cross-section. Both theories yield better results for slender than for short beams. The static analysis requires refined beam elements for the proper detection of non-classical effects, such as the out-of-plane warping. As far as the free-vibration analysis is concerned, higher-order models are necessary for the detailed evaluation of high number modes. These issues are specially relevant for aircraft wings and rotor blades that require a detailed evaluation of the deformation field and natural modes for the proper investigation of the aeroelastic phenomena (e.g. flutter, divergence, etc.).

The finite element analysis has been conducted in the framework of the Carrera unified formulation (CUF). CUF was introduced during the last decade and is focused on higher-order shell and beam theories. The main feature of CUF is represented by its hierarchical capabilities, that is, in CUF the order of the formulation is considered as a free parameter of the analysis. Taylor-type polynomials have been used to model the beam cross-section kinematic field. The finite element formulation has been introduced to deal with arbitrary geometries, loading and boundary conditions. Four-node elements have been used along the longitudinal axis of the beam. The principle of virtual displacements (PVD) has been exploited to compute the stiffness and mass matrices, and the loading vectors.

Two different beam structures have been addressed: an aircraft wing and a wind turbine blade. Isotropic materials have been adopted. Homogenous cross-sections have been considered. Static and free-vibration analyses have been conducted. The results have been evaluated in terms of displacements, natural frequencies and vibration modes. They have been compared with those furnished by commercial finite element codes and experimental data (in the case of the rotor blade model).

It has mainly been concluded that the enhanced refined beam element, which has been formulated via CUF, is able to detect the so-called shell-like mechanical behavior, that is, shell-like results can be obtained using higher-order beam elements. The shell-like capabilities include the detection of the local displacement field induced by a concentrated load, and natural modes characterized by the presence of waves along the cross-section contour. Moreover, the computational cost of the higher-order beam models is strongly smaller than those requested by shell and solid elements.

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