Finite-element formulation of geometrically exact three-dimensional beam theories based on interpolation of strain measures

Zupan, Dejan; Saje, Miran

doi:10.1016/j.cma.2003.07.008

Cited by 117 publications

(90 citation statements)

References 32 publications

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“…The orientation of the cross section is defined by an angle of axial rotation relative to a certain reference direction, which is introduced in order to perform the rotation in a correct manner and to avoid singularities originating from the usage of the rotation parameter. There exist different interpolation procedures in literature, in which displacements and rotations (see Simo and Vu-Quoc, 1986), displacements and slopes (see Shabana, 1997) or strains (see Zupan and Saje, 2003) are used as basic interpolated variables. In contrast to the above-mentioned formulations, the present approach is based on the interpolation of displacements, slopes and a rotation around the beam axis.…”

Section: Introductionmentioning

confidence: 99%

A novel director-based Bernoulli–Euler beam finite element in absolute nodal coordinate formulation free of geometric singularities

et al. 2013

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Abstract.A three-dimensional nonlinear finite element for thin beams is proposed within the absolute nodal coordinate formulation (ANCF). The deformation of the element is described by means of displacement vector, axial slope and axial rotation parameter per node. The element is based on the Bernoulli-Euler theory and can undergo coupled axial extension, bending and torsion in the large deformation case. Singularities -which are typically caused by such parameterizations -are overcome by a director per element node. Once the directors are properly defined, a cross sectional frame is defined at any point of the beam axis. Since the director is updated during computation, no singularities occur. The proposed element is a three-dimensional ANCF Bernoulli-Euler beam element free of singularities and without transverse slope vectors. Detailed convergence analysis by means of various numerical static and dynamic examples and comparison to analytical solutions shows the performance and accuracy of the element.

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Section: Introductionmentioning

confidence: 99%

A novel director-based Bernoulli–Euler beam finite element in absolute nodal coordinate formulation free of geometric singularities

et al. 2013

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“…The complete set of Reissner's beam equations [14] consists of the constitutive equations (27)-(28), the equilibrium equations (29)-(30) and the kinematic equations (31)-(32) (see [23,24]):…”

Section: Governing Equations Of the Beammentioning

confidence: 99%

The wavelet-based theory of spatial naturally curved and twisted linear beams

Zupan

Saje

2008

Comput Mech

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The paper presents the wavelet-based discretization of the linearized finite-strain beam theory which assumes small displacements, rotations and strains but is capable of considering an arbitrary initial geometry and material behaviour. In the numerical solution algorithm, we base our derivations on the vector of strain measures as the only unknown functions in a finite element. In such a way the determination of the beam quantities does not require the differentiation. This is an important advantage which allows a wider range of shape functions. In the present paper, the classical polynomial interpolation is compared to scaling and wavelet function interpolations. The computational efficiency of the method is demonstrated by analyzing initially curved and twisted beams.

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“…An important exception to this general trend can be found in the works by Zupan and Saje [40,41],Češarek, Saje and Zupan [42] (mainly addressing linearized beam equations), Su and Cesnik [43], and Schröppel and Wackerfuß [44]. There, discretization is performed at the level of Lie algebraic fields, called strains, but nodal values are of the essence and interpolation schemes are again needed to reconstruct the shape of a rod.…”

Section: The Lie Algebra and The Lie Group Associated With The Rod Dementioning

confidence: 99%

Importance and Effectiveness of Representing the Shapes of Cosserat Rods and Framed Curves as Paths in the Special Euclidean Algebra

Giusteri

Fried

2017

J Elast

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We discuss how the shape of a special Cosserat rod can be represented as a path in the special Euclidean algebra. By shape we mean all those geometric features that are invariant under isometries of the three-dimensional ambient space. The representation of the shape as a path in the special Euclidean algebra is intrinsic to the description of the mechanical properties of a rod, since it is given directly in terms of the strain fields that stimulate the elastic response of special Cosserat rods. Moreover, such a representation leads naturally to discretization schemes that avoid the need for the expensive reconstruction of the strains from the discretized placement and for interpolation procedures which introduce some arbitrariness in popular numerical schemes. Given the shape of a rod and the positioning of one of its cross sections, the full placement in the ambient space can be uniquely reconstructed and described by means of a base curve endowed with a material frame. By viewing a geometric curve as a rod with degenerate point-like cross sections, we highlight the essential difference between rods and framed curves, and clarify why the family of relatively parallel adapted frames is not suitable for describing the mechanics of rods but is the appropriate tool for dealing with the geometry of curves.

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Finite-element formulation of geometrically exact three-dimensional beam theories based on interpolation of strain measures

Abstract: DRUGG-Digitalni repozitorij UL FGG http://drugg.fgg.uni-lj.si/ Ta članek je avtorjeva zadnja recenzirana različica, kot je bila sprejeta po opravljeni recenziji. Prosimo, da se pri navajanju sklicujte na bibliografske podatke, kot je navedeno:

Cited by 117 publications

References 32 publications

A novel director-based Bernoulli–Euler beam finite element in absolute nodal coordinate formulation free of geometric singularities

A novel director-based Bernoulli–Euler beam finite element in absolute nodal coordinate formulation free of geometric singularities

The wavelet-based theory of spatial naturally curved and twisted linear beams

Importance and Effectiveness of Representing the Shapes of Cosserat Rods and Framed Curves as Paths in the Special Euclidean Algebra

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