Problems in geometrically exact modeling of highly flexible beams: Fid-Bau Portal

Problems in geometrically exact modeling of highly flexible beams

Pai, P. Frank

Abstract Because the deformed beam geometry often is the most important information for applications of highly flexible beams, a geometrically exact beam theory needs to be displacement-based in order to directly and exactly describe any greatly deformed geometry. Main challenges in geometrically exact beam modeling are how to describe a beam's large reference-line deformation and cross-sectional rotations without singularity and how to derive objective directional strains in terms of global displacements and rotations that contain elastic deformation and rigid-body movement. By comparing with a geometrically exact displacement-based beam theory this paper shows that theoretical and numerical problems of other geometrically nonlinear beam theories are mainly caused by: (1) use of independent variables to account for bending-shear rotations, (2) use of problematic energy-based Green–Lagrange strains in order to have objective strain measures, and/or (3) use of strain-based formulations in order not to use problematic Green–Lagrange strains. The theoretical problems include inconsistent governing equations from energy- and momentum-based formulations, inexistence of material property matrices for the chosen strain and stress measures, and non-directional stresses. The numerical problems include shear locking in finite-element analysis, the need of internal nodes and hence more degrees of freedom in finite-element modeling, singularity of mathematics-based rotational variables, deformed geometry being obtained by approximate post-processing numerical integration, and difficult to match secondary (force) variables with deformed conditions.

Highlights • It summarizes a truly geometrically exact displacement-based beam theory. • It reveals problems of other geometrically nonlinear beam theories. • Rotation variables for beam modeling are prone to singularity and shear locking. • Modeling flexible beams using Green–Lagrange strains is problematic. • Strain-based beam theories are difficult to obtain geometrically exact solutions.