The elastica with pre-stress due to natural curvature

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 two-dimensional buckling behavior of Euler's elastica is stable under applied load, the three-dimensional buckling behavior can be stable or unstable. Theoretical and experimental examples are presented illustrating the full range of possibilities.