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As the need to model flexibility arose in multibody dynamics, the floating frame of reference formulation was developed, but this approach can yield inaccurate results when elastic displacements becomes large. While the use of three-dimensional finite element formulations overcomes this problem, the associated computational cost is overwhelming. Consequently, beam models, which are one-dimensional approximations of three-dimensional elasticity, have become the workhorse of many flexible multibody dynamics codes. Numerous beam formulations have been proposed, such as the geometrically exact beam formulation or the absolute nodal coordinate formulation, to name just two. New solution strategies have been investigated as well, including the intrinsic beam formulation or the DAE approach. This paper provides a systematic comparison of these various approaches, which will be assessed by comparing their predictions for four benchmark problems. The first problem is the Princeton beam experiment, a study of the static large displacement and rotation behavior of a simple cantilevered beam under a gravity tip load. The second problem, the four-bar mechanism, focuses on a flexible mechanism involving beams and revolute joints. The third problem investigates the behavior of a beam bent in its plane of greatest flexural rigidity, resulting in lateral buckling when a critical value of the transverse load is reached. The last problem investigates the dynamic stability of a rotating shaft. The predictions of eight independent codes are compared for these four benchmark problems and are found to be in close agreement with each other and with experimental measurements, when available

Recently, the authors proposed a geometrically exact beam finite element formulation on the Lie group SE(3). Some important numerical and theoretical aspects leading to a computationally efficient strategy were obtained. For instance, the formulation leads to invariant equilibrium equations under rigid body motions and a locking free element. In this paper we discuss some important aspects of this formulation. The invariance property of the equilibrium equations under rigid body motions is discussed and brought out in simple analytical examples. The discretization method based on the exponential map is recalled and a geometric interpretation is given. Special attention is also dedicated to the consistent interpolation of the velocities.

This paper focuses on the interpolation of the kinematic fields describing the configuration of geometrically exact beams, namely, the position and rotation fields. Two kinematic representations are investigated: the classical approach that treats the displacement and rotation fields separately and the motion approach that treats those two fields as a unit. The latter approach is found to be more consistent with the kinematic description of beams. Then, two finite element interpolation strategies are presented and contrasted. The first interpolates the displacement and rotation fields separately, whereas the second interpolates both fields as a unit, in a manner consistent with the motion approach. The performance of both strategies is evaluated in light of the fundamental requirements for the convergence of the finite element method: the ability to represent rigid-body motions and constant strain states. It is shown that the traditional uncoupled interpolation scheme for the position field approximates that based on the motion approach and that the coupling induced by the interpolation of motion yields superior convergence rates for the representation of constant strain states. This property is known to lead to finite elements that are less prone to the locking phenomenon.Several issues with these early techniques have been documented in the literature. Crisfield and Jelenić [7,8] were the first to point out that the strain measures used in the formulation should be objective, that is, should be invariant under the addition of a rigid-body motion to the configuration of the system. They showed that several commonly used discretization schemes for rotation fields do not satisfy this requirement: interpolated strain measures do not remain invariant under a superimposed rigid-body motion. Typically, the use of mesh and load step size refinements alleviate these problems. Romero and Armero [9] developed interpolation schemes that guarantee objectivity and the same goal was achieved by Betsch and Steinmann [10] who used a redundant set of generalized coordinates to represent the configuration of the system. Another recurring issue of these early schemes is the locking phenomenon. The requirement for the convergence of the finite element method is that the interpolated field be able to capture constant strain states accurately; see Zienkiewicz [11]. Unfortunately, when using the polynomial interpolation functions found in finite element textbooks for interpolating beam problems, it is not possible to represent constant or vanishing shear strain distributions, for instance, resulting in the well-known shear locking phenomenon. Typically, reduced integration, higher-order interpolation schemes, or both, are implemented to mitigate this problem.The deficiencies of these early schemes limit their performance and indicate that fundamental underlying concepts are not treated properly. The goal of this paper is to present a fresh look at beam formulation. The choice of an interpolation strategy is rooted in the kinematic de...

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