On the role of mass distribution in free vibration of Hencky beam models: discrete and nonlocal continuous approaches

In this paper, we investigate the role of mass distribution on the vibration frequencies of various discrete Hencky-type beam models. These models are composed of rigid beam elements connected by concentrated rotational springs, and they all asymptotically converge towards the continuous Euler–Bernoulli beam models, for an infinite number of elements. However, the order of convergence and the upper bound or lower bound status of each discrete beam model as compared to the continuous counterpart depends on how the mass is assumed to be distributed along the discrete elements. Three mass distributions assumptions are considered herein: Hencky model with lumped masses at each joint, Hencky model with lumped masses at the middle of the rigid segment, and Hencky model with distributed masses along each rigid element. Exact eigenfrequency formulas are presented for the various Hencky beam models with simply supported ends. A nonlocal continuous approach for each model is also developed. A strong dependence with respect to the length scale effect is observed for each lattice beam model. Stiffening or softening length scale phenomena are shown to be controlled by the mass microstructure, namely concentrated or distributed mass inertia. The wave dispersive behaviour of each Hencky model is also discussed for each mass distribution (concentrated and distributed mass properties). The length scale of the associated nonlocal beam model is also fitted with respect to the exact dispersive curves of Hencky beams.