Variational formulations, instabilities and critical loadings of space curved beams. (1st June 2016) - Liu, Liping; Lu, Nanshu
Abstract: 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.