Flexural-torsional buckling with enforced axis of twist and related problems

The classical lateral-torsional instability of a cantilever beam with continuous elastic lateral restraint and a transverse point-load applied at the free end is discussed here through the lens of asymptotic simplifications. One of our main goals is to provide analytical approximations for the critical buckling load, as well as its dependence on various key non-dimensional groups. The first part of this study is concerned with a scenario in which a doubly symmetric beam is constrained to rotate about a fixed axis situated at an arbitrary height above the shear centroidal axis. The second part examines a beam that features a continuous elastic lateral restraint spanning its entire length. Assuming that the stiffness of the constraint, κ (say), is finite, the buckling equations in the second case are described by a system of two coupled fourth-order differential equations in the lateral displacement and the angle of twist. We show that as κ → ∞, the problem discussed in the first part provides an outer solution for the aforementioned system; the relevant boundary conditions for the next-to-leading-order outer approximation are also derived by using matched asymptotics. Our theoretical findings are confirmed by comparisons with direct numerical simulations of the full buckling problem.