Finite element formulation of a homogenized beam for reticulated structure dynamics

This paper presents a finite element formulation of an existing homogenized beam-like model for the description of reticulated structures transverse dynamics. This work deals with elastic periodic lattice structures such as foams, crystals, honeycombs, or multi-story buildings, whose unit cell is made of interconnected beams or plates and repeats itself in one direction. The studied model is a one-dimensional enriched form of the fourth-order Timoshenko beam equation, and the motion is described by a sixth order differential equation. The higher differential equation order is attributed to an additional kinematic mechanism that may appear under large stiffness contrasts between the cell elements. In this context, the construction of an original finite element approximate solution is proposed. Weak formulations lead to generalized elementary stiffness and mass matrices. Through the analysis of a realistic building in the linear elasticity framework, it is shown that this homogenized beam finite element solution recovers the analytical results and is close to the full detailed finite element structural model. This new formulation simplifies the implementation of the beam-like model in complex configurations, enables parametric studies, and could be easily used for a wide range of applications in structural dynamics, other than free vibration analysis, at reduced computational costs. (authors)