A method for the closed-form evaluation of critical buckling loads for planar strongly curved bars

The calculation of critical buckling loads of planar curved bars, subjected to a general co-planar continuous external load (or a general co-planar terminal loading), leads to the solution of transcendental (nonlinear) equations. In this investigation a new method for the closed-form solution of such types of equations is presented. In particular, the transcendental equation u tan γ cot uγ = 1, corresponding to the buckling problem of a cantilever circular bar of high curvature loaded by two co-planar forces acting along its chord, is solved in a closed-form. Finally, several numerical results are presented, based on the Gauss integration rule.