A Legendre Spectral Finite Element Implementation of Geometrically Exact Beam Theory

This paper considers Legendre spectral finite elements (LSFEs) for linear and nonlinear elastic deformation of composite beams. LSFEs are high-order Lagrangian-interpolant finite elements with nodes located at the Gauss-Lobatto-Legendre quadrature points. Geometrically exact beam theory (GEBT) is adopted as the theoretical framework, where coupling effects (which usually exist in composite structures) and geometric nonlinearity are taken into consideration. Preliminary results are shown for two example problems. In the first example, the planar linear deflection and natural frequencies of a tapered beam are calculated with LSFEs and with first-order finite elements found in a commercial code. In the second example, the planar nonlinear deflection of a beam subjected to a tip moment is examined with LSFEs and first-order finite elements. For both cases, the LSFEs exhibit exponential convergence rates and are dramatically more accurate than low-order finite elements for a given model size.