Variational formulations, instabilities and critical loadings of space curved beams

Beam theories have been extensively studied for applications in structural engineering. Space curved beams with large displacements, however, have been explored to a much less extent, not to mention explicit solutions concerning instabilities and critical loadings. In this paper, by carefully accounting for geometric nonlinearity and different scalings of kinematic variables, we present a variational framework for large-displacement space curved beams. We show that the variational formulation is consistent with the classic field equations, derive the appropriate boundary value problems for a variety of loading conditions and kinematic constraints, and generalize the Kirchhoff’s helical solutions. Explicit planar solutions for semi-circular arches are obtained upon linearization. Further, two nonlinear asymptotic theories are proposed to address ribbon-like and moderately deformed curved beams, respectively. Based on the method of trial solutions, we obtain explicit approximate solutions to critical loadings for semi-circular arches losing stabilities due to twisting and out-of-plane displacement. The variational framework, nonlinear asymptotic theories, stability analysis and explicit solutions are anticipated to have novel applications in stretchable electronics and biological macromolecules.