Geometrically Exact Finite Element Formulations for Slender Beams: Kirchhoff–Love Theory Versus Simo–Reissner Theory

Abstract

The present work focuses on geometrically exact finite elements for highly slender beams. It aims at the proposal of novel formulations of Kirchhoff-Love type, a detailed review of existing formulations of Kirchhoff-Love and Simo-Reissner type as well as a careful evaluation and comparison of the proposed and existing formulations. Two different rotation interpolation schemes with strong or weak Kirchhoff constraint enforcement, respectively, as well as two different choices of nodal triad parametrizations in terms of rotation or tangent vectors are proposed. The combination of these schemes leads to four novel finite element variants, all of them based on a C 1 -continuous Hermite interpolation of the beam centerline. Essential requirements such as representability of general 3D, large-deformation, dynamic problems involving slender beams with arbitrary initial curvatures and anisotropic cross-section shapes, preservation of objectivity and path-independence, consistent convergence orders, avoidance of locking effects as well as conservation of energy and momentum by the employed spatial discretization schemes, but also a range of practically relevant secondary aspects will be investigated analytically and verified numerically for the different formulations. It will be shown that the geometrically exact Kirchhoff-Love beam elements proposed in this work are the first ones of this type that fulfill all these essential requirements. On the contrary, Simo-Reissner type formulations fulfilling these requirements can be found in the literature very well. However, it will be argued that the shear-free Kirchhoff-Love formulations can provide considerable numerical advantages such as lower spatial discretization error level, improved performance of time integration schemes as well as linear and nonlinear solvers or smooth geometry representation as compared to shear-deformable Simo-Reissner formulations when applied to highly slender beams. Concretely, several representative numerical test cases confirm that the proposed Kirchhoff-Love formulations exhibit a lower discretization error level as well as a considerably improved nonlinear solver performance in the range of high beam slenderness ratios as compared to two representative Simo-Reissner element formulations from the literature.

Investigation of the peeling and pull-off behavior of adhesive elastic fibers via a novel computational beam interaction model

“…At this point, we would like to point out that the numerical methods for beam-beam interactions to be presented in the following are independent of the specific beam formulation and have been applied to both Simo-Reissner and Kirchhoff-Love type formulations. Note that the latter are known to be advantageous in the regime of high slenderness ratios where the underlying assumption of negligible shear deformation is met [23,24].…”

A unified approach for beam-to-beam contact

“…for a maximal admissible ratio µ max := Rmax(κ) 1 (see e.g. Geradin and Cardona [1], Linn et al [4] or Meier et al [7]; if beams with different cross-section radii R 1 and R 2 shall be considered, these radii R i should at least be of the same order of magnitude in order to still fulfill the requirement µ max = max(R i )max(κ) 1). Consequently, also this second assumption does typically not represent an additional limitation concerning the range of potential / suitable applications.…”

A Review on Stress and Deformation Analysis of Curved Beams under Large Deflection

“…On the other hand, induced beam theories are derived from three dimensional continuum mechanics with one characteristic direction. Such one dimensional continuum theories describe three dimensional deformation behaviors on the basis of proper kinematic, kinetic and constitutive relations [18].…”

“…In the mid nineteenth century, Kirchhoff [18] developed a generalized beam model based on Euler-Bernoulli hypothesis to predict three dimensional deformation characteristics of arbitrarily initially curved beam. Almost after a century, Love extended the Kirchhoff model for extensible beam [18]. Later on Reissner [6,7] introduced shear deformation in Kirchhoff-Love beam model but unfortunately decreased its exactness.…”