Modeling beam-like planar structures by a one-dimensional continuum: an analytical-numerical method

In this paper, beam-like structures, macroscopically behaving as planar Timoshenko beams, are considered. Planar frames, made by periodic assemblies of micro-beams and columns, are taken as examples of these structures and the effectiveness of the equivalent beam model in describing their mechanical behavior, is investigated. The Timoshenko beam (coarse model) is formulated via the direct one-dimensional approach, by considering rigid cross-sections and flexible axis-line, while its constitutive laws is determined through a homogenization procedure. An identification algorithm for evaluation of the constitutive constants is illustrated, based on Finite Element analyses of the cell of the periodic system. The inertial properties of the equivalent model are instead analytically identified under the hypothesis the masses are lumped at the joints. The advantages in using the equivalent model are discussed with reference to the linear static and dynamic responses of some planar frames, taken as case-studies, for which both analytical and numerical tools are used. Numerical results, obtained by the equivalent model, are compared with Finite Element analyses on planar frames (fine models), considering both symmetric and not-symmetric layouts, in order to show to effectiveness of the proposed algorithm. A comparison with analytical results is carried out to validate the limits of applicability of the method.