On the Bending and Buckling of Very Thin Beams

Abstract

Bending and buckling deformations of very thin beams with curved cross-sections are discussed. Nonconvex strain-energy density functions are considered, and the critical loads and corresponding critical conditions are defined. The Euler-Lagrange equation with Weierstrass-Erdmann corner conditions is acceptable for globally stable deformations. The deformations are defined in the context of globally stable equilibrium deformations. Regions of low and high strain are revealed, separated by a discontinuity in the curvature of the elastic curve. Since the curvature is almost zero in one section, when combined with the other section, the beam will be treated as two separate sections. The one remaining is undeformed, and the other is deformed with constant curvature. Those phenomena are easily observed in the metallic measuring bands.