Elastic buckling of end-loaded, tapered, cantilevered beams with initial curvature

The elastic deflections and Euler buckling loads are investigated for a class of tapered and initially curved cantilevered beams subjected to loading at the tip. The beam's width increases linearly and its depth decreases linearly with the distance from the fixed end to the tip. Unloaded, the beam forms a circular are perpendicular to the axis of bending. The beam's deflection responses, obtained by solving the differential equations in closed form, are presented in terms of four nondimensional system parameters: taper ratio ${\kappa}$, initial shape ratio ${\Delta}_0$, end load ratio f, and load angle ${\theta}$. Laboratory measurements of the Euler buckling loads for scale models of tapered initially straight, corrugated beams compared favorably with those computed from the present analysis. The results are applicable to future designs of the end structures of highway guardrails, which can be designed to give the appropriate balance between the capacity to deflect a nearly head-on vehicle back to its right-of-way and the capacity to buckle sufficiently that penetration of the vehicle may be averted.