Lateral-torsional buckling of beams under combined loading: A reappraisal of Papkovitch - Schaefer theorem

Lateral-torsional buckling of elastic structures under combined loading is considered in this paper. Closed-form solutions are available for a specific range of loading. 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 in the loading parameters space is convex. The Papkovitch-Schaefer theorem is extended from previous theoretical works of Plaut, (12) 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%.