The elastica with pre-stress due to natural curvatures

The axial buckling behavior is determined for an elastic beam or rod which has a uniform curvature in its natural state, is straightened by pure bending, and clamped at its ends. Buckling can be either identical to the classical two-dimensional behavior determined by Euler, or it can be three-dimensional involving twist and deflection out of the plane of natural curvature depending on the bending and torsional stiffnesses and the natural curvature. While the classical twodimensional buckling behavior of Euler's elastica is stable under applied load, the threedimensional buckling behavior can be stable or unstable. Theoretical and experimental examples are presented illustrating the full range of possibilities.