A review on buckling and postbuckling of thin elastic beams

This paper provides a review of models and solutions for the buckling and postbuckling of beams available from the 50s of the last century to date. Beams with axially unrestrained (movable) ends and restrained (immovable) ends are covered. In each class, the formulation of the nonlinear buckling problem for the buckling loads and the postbuckling states is discussed and the underlying assumptions are highlighted. For relatively large-amplitude buckling of beams with movable ends, approximate analytical solutions up to the third order are presented and compared with the exact solutions expressed in terms of elliptic integrals. For beams with immovable ends, buckling involves midplane stretching that makes the nonlinear buckling problem takes the same form of the standard eigenvalue problem and, hence, exact solutions are affordable. This review combines the research outcomes on buckled beams from two scientific viewpoints: the structural dynamics and the nonlinear vibration viewpoints, respectively. Moreover, it presents in one place the formulation and the exact solutions for the buckling of beams to serve as an on-demand resource for researchers concerned with the buckling and post buckling of beams.

Automated summary

This paper reviews models and solutions for the buckling and postbuckling of beams from the 1950s to present day, discussing the formulation of nonlinear buckling problems for beams with movable and immovable ends, providing both approximate and exact solutions. It aims to serve as a comprehensive resource for researchers interested in the buckling and postbuckling of beams, combining outcomes from structural dynamics and nonlinear vibration perspectives.