Efficient Finite Element Formulation for Geometrically Exact Beams

This paper proposes a new approach to the modeling of geometrically exact beams based on motion interpolation schemes. Motion interpolation schemes yield simple expressions for the sectional strains and linearized strain-motion relationships at the mesh nodes. The classical formulation of the finite element method starts from the weak form of the continuous governing equations obtained from a variational principle. Approximations, typically of a polynomial nature, are introduced to express the continuous displacement field in term of its nodal values. Introducing these approximations into the weak form of the governing equations then yields nonlinear discrete that can be solved with the help of a linearization process. In the proposed approach, the order of the first two steps of the procedure is reversed: approximations are introduced in the variational principle directly and the discrete equations of the problem then follow. This paper has shown that for geometrically exact beams, the discrete equations obtained from the two procedure differ significantly: far simpler discrete equations are obtained from the proposed approach.