Buckling Reversals of Axially Restrained Imperfect Beam-Column

Abstract

The stability and postbuckling analysis of an axially restrained prismatic beam-column with single symmetrical cross section and an initial imperfection (camber) is presented. The proposed model is that by Timoshenko but including the effects of small camber of any form and any transverse loading. This model can be used to (1) determine the prebuckling elastic response and initial buckling load; (2) explain the postbuckling elastic behavior including the phenomena of snap-through, snap-back, and reversals of deflections; and (3) determine the effects of high modes of buckling on the stability behavior of beam-columns with small camber. In addition, closed-form equations corresponding to the transverse and axial deflections caused by any transverse loads on a partially restrained beam-column are developed as well as the bending stress along its span. It is shown that the prebuckling, stability, and postbuckling behavior of a beam-column depends on (1) the cross section and material properties (area, inertia, and elastic modulus); (2) the magnitude of the end restraints; and (3) the type and lack of symmetry about the beam-column midspan of the applied transverse loads and initial camber or imperfection. For transverse loads that are not symmetrical with respect to the beam-column midspan, the pre- and postbuckling criterion given by Timoshenko might yield significant errors in both the critical load and deflections. Three examples are presented that show the effectiveness and validity of the proposed equations and the limitations of Timoshenko's criteria.