Vierendeel girder homogenization into a polar beam

The homogenization of a Vierendeel girders with elastic joints is treated in the paper. The starting of the proposed method is the Stephen transfer matrix analysis, and the polar Timoshenko beam as equivalent continuum of the Vierendeel girder. The polar character of the homogenized beam raises by the analysis of the pure bending eigenvector components of the girder force transfer matrix. Two bending moments are transmitted by the unit cell of the girder: one is caused by the couple of axial forces applied on each nodal section of the beam, the other is due to the moments present at the nodes of each ones by the adjacent cells and is modelled as the resultant of the micro-polar moments. The Transfer force eigenvector analysis bring out that the unit cell bends maintaining the webs undeformed. This property allows evaluating both the bending stiffnesses and the equivalent material micro-polar scale parameter with straightforward equations. The results of the homogenization procedure proposed are tested by comparison with literature data and with the outcomes of the analysis of a series of girders carried out by the finite element method.