Postbuckling and vibration of linearly elastic and softening columns under self-weight

The critical height for buckling of a linearly elastic cantilevered column due to its self-weight was determined by Greenhill in 1881. Postbuckling behavior also has been studied, often assuming that the column is an elastica (inextensible, with its bending moment proportional to its curvature). The bifurcation point at the critical height is supercritical, so that the postbuckling path is stable as the height increases past its critical value. Subcritical bifurcation may occur if the column is nonlinearly elastic with a softening behavior. This results in a sudden jump from the straight vertical configuration to a severely-drooped shape. The governing equation is solved numerically with the use of a shooting method to obtain the equilibrium paths. Also, small vibrations about the straight and postbuckled equilibrium states are examined, and vibration frequencies (and hence stability properties) are obtained. An initial curvature of the column is included in the analysis. Experiments are conducted to verify the results qualitatively for linearly elastic and softening materials. © 2004 Elsevier Ltd. All rights reserved.