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Adding local rotational degrees of freedom to ANC beams
Abstract
This work shows a simple finite element formulation that enables to impose concentrated moments and rotations to ANC beams which are finite elements that lack rotational degrees of freedom. The idea is based on an specific constraint that expresses in a simple form the relation between the deformation of the beam and the rotation of any of its sections. By controlling this sectional rotation, moments and angles can be easily imposed on any model.
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The absolute nodal coordinate formulation has been developed in order to simu-
The objective of this paper is to investigate the accuracy of the elastic force models that can be used in the absolute nodal coordinate finite element formulation. This study focuses on the description of the elastic forces in three-dimensional beams. The elastic forces of the absolute nodal coordinate formulation can be derived using a continuum mechanics approach. This study investigates the accuracy and usability of such an approach for a three-dimensional absolute nodal coordinate beam element. This study also presents an improvement proposal for the use of a continuum mechanics approach in deriving the expression of the elastic forces in the beam element. The improvement proposal is verified using several numerical examples that show that the proposed elastic force model of the beam element agrees with the analytical results as well as with the solutions obtained using existing finite element formulation. In the beam element under investigation, global displacements and slopes are used as the nodal coordinates, which resulted in a large number of nodal degrees of freedom. This study provides a physical interpretation of the nodal coordinates used in the absolute nodal coordinate beam element. It is shown that a beam element based on the absolute nodal coordinate formulation relaxes the assumption of a rigid cross-section and is capable of representing a distortional deformation of the cross-section. The numerical results also imply that the beam element does not suffer from the phenomenon called shear locking.
The description of a beam element by only the displacement of its centerline leads to some difficulties in the representation of the torsion and shear effects. For instance such a representation does not capture the rotation of the beam as a rigid body about its own axis. This problem was circumvented in the literature by using a local coordinate system in the incremental finite element method or by using the multibody floating frame of reference formulation. The use of such a local element coordinate system leads to a highly nonlinear expression for the inertia forces as the result of the large element rotation. In this investigation, an absolute nodal coordinate formulation is presented for the large rotation and deformation analysis of three dimensional beam elements. This formulation leads to a constant mass matrix, and as a result, the vectors of the centrifugal and Coriolis forces are identically equal to zero. The formulation presented in this paper takes into account the effect of rotary inertia, torsion and shear, and ensures continuity of the slopes as well as the rotation of the beam cross section at the nodal points. Using the proposed formulation curved beams can be systematically modeled.
This part of these two companion papers demonstrates the computer implementation of the absolute nodal coordinate formulation for three-dimensional beam elements. Two beam elements that relax the assumptions of Elder-Bernoulli and Timoshenko beam theories are developed. These two elements take into account the effect of rotary inertia, shear deformation and torsion, and yet they lead to a constant mass matrix. As a consequence, the Coriolis and centrifugal forces are identically equal to zero. Both beam elements use the same interpolating polynomials and have the same number of nodal coordinates. However, one of the elements has two nodes, while the other has four nodes. The results obtained using the two elements are compared with the results obtained using existing incremental methods. Unlike existing large rotation vector formulations, the results of this paper show that no special numerical integration methods need to be used in order to satisfy the principle of work and energy when the absolute nodal coordinate formulation is used. These results show that this formulation can be used in manufacturing applications such as high speed forming and extrusion problems in which the element cross section dimensions significantly change.
Two of the most popular finite element formulations for solving nonlinear beams are the absolute nodal coordinate and the
geometrically exact approaches. Both can be applied to problems with very large deformations and strains, but they differ
substantially at the continuous and the discrete levels. In addition, implementation and run-time computational costs also
vary significantly. In the current work, we summarize the main features of the two formulations, highlighting their differences
and similarities, and perform numerical benchmarks to assess their accuracy and robustness. The article concludes with recommendations
for the choice of one formulation over the other.
Three dimensional absolute nodal coordinate formulation for beam elements: implementation and applications
- R Y Yacoub
- A A Shabana
R.Y. Yacoub and A. A. Shabana. Three dimensional absolute nodal coordinate formulation for beam elements:
implementation and applications. ASME Journal of Mechanical Engineering, 123(4):614-621, 2001.