On the homogenization of periodic beam-like structures

Abstract

A homogenization method for periodic beam-like structures that is based on the unit cell force transmission modes is presented. Its main advantage is that to identify the principal vectors of the state transfer matrix corresponding to the transmission modes it operates directly on the sub-partitions of the unit cell stiffness matrix and allows to overcome the problems due to ill-conditioning of the transfer matrix. As case study, the Pratt girder is considered. Closed form solutions for the transmission modes of this girder are achieved and used into homogenization. Since the pure bending mode shows that the Pratt unit cell transmits two kinds of bending moments, one given by the axial forces and the other originated by nodal moments, the Timoshenko couple-stress beam is employed as substitute continuum. Finally, a validation of the proposed procedure is carried out comparing the predictions of the homogenized models with the results of a series of girder f.e. analyses.