Finite element theory for curved and twisted beams based on exact solutions for three-dimensional solids Part 1: Beam concept and geometrically exact nonlinear formulation Original Research Article

A geometrically exact and completely consistent finite element theory for curved and twisted beams is proposed. It is based on the kinematical hypothesis generally formulated for large deformation and accounts for all kinds of deformation of a three-dimensional solid: translational and rotational displacements of the cross-sections, warping of their plane and distortion of their contours. The principle of virtual work is applied in a straightforward manner to all non-zero six components of the strain and stress tensors. Expressions are given for tangent matrices of elastic, inertia and external forces and specific techniques for discretization and updating are developed for the analysis of beams in inertial and non-inertial frames. Finally, the numerical properties of the finite element models are demonstrated through examples.