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. Depending on the bending and torsional stiffnesses and the natural curvature, buckling can be either identical to the classical two-dimensional behavior determined by Euler, or it can be threedimensional involving twist and deflection out of the plane of 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.