On lateral-torsional buckling of discrete elastic systems: A nonlocal approach

Abstract

The lateral-torsional buckling of a discrete repetitive elastic beam-like structure, composed of rigid links connected by bending and torsional elastic springs, is investigated herein using discrete and continuum approaches. It is shown that the governing equations of the microstructured model are equivalent to a finite difference formulation of a continuous lateral-torsional buckling problem. The discrete equations are introduced through variational arguments and solved exactly for the hinged-hinged boundary conditions using a finite difference approach. A nonlocal equivalent continuum is sought via a continualization method. It is shown that the equivalent continuum is an Eringen's based elastic nonlocal continuum, which perfectly fits the exact discrete problem. The length scale effect related to the size of the repetitive cell tends to soften the lateral-torsional buckling limit of the asymptotically local continuum. Prandtl's lateral-torsional buckling solution is a particular case, associated with an infinite number of cells. Warping can also be included into the discrete thin-walled structural model, leading to a microstructured-based nonlocal thin-walled model. Love's equations augmented by the so-called Vlasov effect (warping bimoment) are used to compare the nonlocal approach with the one arising from the continualization of the discrete equation. It is shown that, in this last case, the length scale effect may depend on the warping stiffness.