A Micro-Polar Model for Buckling Analysis of Vierendeel Periodic Beams

A method is presented for evaluating the axial critical loads and deformed shapes of a Vierendeel periodic girder. They are obtained by solving the large-deflections equilibrium problem of a micro-polar equivalent model. The elastic properties of this model have been derived from those of an ideal girder whose cells may deform only according to the inner forces transferring modes of the unit cell. In particular, the strain energy density of the equivalent medium is obtained by evaluating the limit of the girder elastic energy for the cell to girder size ratio tending to zero. To solve the Engesser/Haringx discord, the large-deflections equilibrium equations are deduced by the virtual work principle, without any a priori assumption on the shear force. For this aim, also the external work of the substitute medium is evaluated by the same procedure as the strain energy density. Actually, it is first written for the ideal girder under the assumption of negligibility of the axial effects at the cell scale and then specialized for the continuous medium by examining the limit value for cell to girder size ratio going to zero. Closed form solutions for the critical deformed shapes are given, together with an accurate formula for the girder buckling load.