Noël Challamel: An introduction into lateral-torsional buckling of elastic beams

Lateral-torsional buckling of elastic structures under combined loading will be considered in this seminar. This problem has been first reported  in the habilitation thesis of Prandtl dated 1899. Closed-form solutions based on Bessel's functions are available for some specific types of loading. However, numerical methods such as the Finite Element Methods (FEM) or other approximate methods are needed in the general case. More generally, approximation of the buckling curve (limit of the stable domain in the loading parameters space) is investigated from the stationary property of the Rayleigh’s quotient. The approximation is then compared to a numerical approach, namely the iterative method of Vianello and Stodola. Closed-form solutions give upper bounds with relative error less than 0.2%. It is shown that the stable domain of the loading parameter space is convex. The Papkovitch–Schaefer theorem proven in 1934 is extended for this specific problem, despite the nonlinear dependence of the equilibrium equations on the loading parameters for the one-dimensional system. The boundary of the stable domain is clearly nonlinear, but this nonlinearity is weak. It is shown that Dunkerley’s lower bound is relevant for the two structural cases considered, and the maximum relative error induced by such a lower bound is lower than 2%. Prandtl's linear approximation is then validated approximately one century later the pioneer works of Prandtl devoted to elastic instability.

Noël Challamel is Professor at the Department of Civil Engineering (LIMATB), University of South-Brittany, Lorient, France, and Marie Curie fellow at the Department of Mathematics, University of Oslo, Oslo, Norway.