Higher-order shear beam theories and enriched continuum

The buckling of higher-order shear beam-columns is studied in the light of enriched continuum. We show the equivalence between the enriched kinematics of usual higher-order shear beam theories with the nonlocal and gradient nature of the associated constitutive law. These equivalences are useful for a hierarchical classification of usual beam theories comprising Euler-Bernoulli beam theory, Timoshenko and third-order shear beam theories. A consistent variationnally presentation is derived for all generic theories, leading to meaningful buckling solutions. It is shown that Timoshenko or some other higher-order shear theories can be considered as nonlocal or gradient Euler-Bernoulli theories. The buckling problem of a third-order shear beam-column is analytically studied and treated in the framework of gradient elasticity Timoshenko theory. Some different gradient elasticity Timoshenko models are presented at the end of the paper with available buckling solutions for repetitive structures and microstructured beams.