Periodic beam-like structures homogenization by transfer matrix eigen-analysis: A direct approach

摘要

Highlights?Closed form solutions are obtained for the Pratt and X-braced unit cell force transmission modes and the related equivalent stiffnesses are determined.?The Timoshenko couple-stress beam is the equivalent continuum medium adopted for the homogenization.?An innovative approach for calculating the principal vectors of the unit cell state transfer matrix is given.?The main advantage of the method is to operate directly on the sub-partitions of the unit cell stiffness matrix of condensed size.?Unit cells with complex geometries must be analysed numerically and ill-conditioning problems, affecting all the transfer methods till now proposed, are do not show up in the method.AbstractThe paper deals with a direct approach to homogenize lattice beam-like structures via eigen- and principal vectors of the state transfer matrix. Since the girders unit cells transmit two bending moments, one given by the axial forces, the other originated by nodal moments, the Timoshenko couple-stress beam is employed as substitute continuum. The main advantage of the method is the possibility of operating directly on the sub-partitions of the unit cell stiffness matrix. Closed form solutions for the Pratt and X-braced girders are achieved and used into the homogenization. Unit cells with more complex geometries are numerically addressed with direct approach, showing that the principal vector problem corresponds to the inversion of a well-conditioned matrix. Finally, a validation of the procedure is carried out comparing the predictions of the homogenized models with the outcomes of f.e. analyses performed on a series of girders.]]>