Shear and flexural factors for static analysis of homogenized beam models of planar frames

One-dimensional equivalent beam models, embedded in a two-dimensional space, are developed in this work for static analysis of beam-like structures, such as planar frames (fine models). Extensible shear and Timoshenko beam models (coarse models) are formulated in the framework of the direct one-dimensional approach, under the assumption of rigid cross-section, while their constitutive laws are determined through a homogenization procedure. A linear constitutive equation is obtained, also taking into account for the presence of bracing elements in the frame, with axial and shear forces coupled with bending. The warping of the cross-section, caused both by shear and bending, is taken into account, in an approximate way, by resorting to the concept of shear and flexural factors. These factors are introduced to correct the constitutive coefficients of the equivalent beam. They are determined by equating the elastic potential energy of the fine model and that of the coarse models, once the same macroscopic displacement is given. The limits of applicability of the corrected homogenized beam models are discussed with reference to the linear static response of some planar frames, taken as case-studies. Numerical results obtained by the homogenized models are compared with finite element analyses on planar frames, in order to show the effectiveness of the procedure.