Energy equations for elastic flexural–torsional buckling analysis of plane structures

Lateral–torsional buckling is a critical mode of failure of metal structures. When the values of the loadings on a member of a structure reach a limiting state, the member will experience out-of-plane bending and twisting. This type of failure occurs suddenly in members with a much greater in-plane bending stiffness than torsional or lateral bending stiffness. Slender members of a structural system may buckle laterally and twist before their in-plane capabilities can be reached. Energy equations are derived by considering the total potential energy of a beam-column element. The second variation of the total potential energy equal to zero indicates the transition from a stable state to an unstable state, which is the critical condition for buckling. Several energy equations are derived analytically by calculating the second variation of the total potential energy of a double symmetric thin wall beam-column element. In this article, in-plane deformations of the beam-column element are disregarded. Then energy equations are derived expressing in dimensional and non-dimensional forms. These energy equations will be implemented in a future article to derive elastic and geometric stiffness matrices for the beam-column element and calculate the lateral–torsional buckling of plane structures. Examples are provided to show the accuracy of the equations and applications.