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Rotation-based finite elements: reference-configuration geometry and motion description

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Abstract

Infinitesimal-rotation finite elements allow creating a linear problem that can be exploited to systematically reduce the number of coordinates and obtain efficient solutions for a wide range of applications, including those governed by nonlinear equations. This paper discusses the limitations of conventional infinitesimal-rotation finite elements (FE) in capturing correctly the initial stress-free reference-configuration geometry, and explains the effect of these limitations on the definition of the inertia used in the motion description. An alternative to conventional infinitesimal-rotation finite elements is a new class of elements that allow developing inertia expressions written explicitly in terms of constant coefficients that define accurately the reference-configuration geometry. It is shown that using a geometrically inconsistent (GI) approach that introduces the infinitesimal-rotation coordinates from the outset to replace the interpolation-polynomial coefficients is the main source of the failure to capture correctly the reference-configuration geometry. On the other hand, by using a geometrically consistent (GC) approach that employs the position gradients of the absolute nodal coordinate formulation (ANCF) to define the infinitesimal-rotation coordinates, the reference-configuration geometry can be preserved. Two simple examples of straight and tapered beams are used to demonstrate the basic differences between the two fundamentally different approaches used to introduce the infinitesimal-rotation coordinates. The analysis presented in this study sheds light on the differences between the incremental co-rotational solution procedure, widely used in computational structural mechanics, and the non-incremental floating frame of reference formulation (FFR), widely used in multibody system (MBS) dynamics.

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Acknowledgements

This research was supported, in part, by the National Science Foundation (Grant 1852510). The authors would like also to thank Ph.D. graduate student Mr. Ahmed Eldeeb of University of Illinois at Chicago for providing some of the figures and data presented in this paper.

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Authors and Affiliations

  1. Department of Mechanical and Industrial Engineering, University of Illinois At Chicago, Chicago, IL, 60607, USA

    Ahmed A. Shabana

  2. Faculty of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, Hangzhou, 310018, China

    Lingmin Xu

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  1. Ahmed A. Shabana
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Correspondence to Ahmed A. Shabana.

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Executive Editor: Gui-Lin Wen

Appendix

Appendix

The ANCF mass matrix of straight beam can be written as:

$$ \begin{aligned} {\mathbf{M}}_{As} & = \rho t\iint {{\mathbf{S}}_{A}^{{\text{T}}} {\mathbf{S}}_{A} {\text{d}}x{\text{d}}y} \\ & = m\left[ {\begin{array}{*{20}c} {13/35} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ 0 & {13/35} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ {11l/210} & 0 & {l^{2} /105} & {} & {} & {} & {} & {} & {{\text{symmetric}}} & {} & {} & {} \\ 0 & {11l/210} & 0 & {l^{2} /105} & {} & {} & {} & {} & {} & {} & {} & {} \\ 0 & 0 & 0 & 0 & {h^{2} /36} & {} & {} & {} & {} & {} & {} & {} \\ 0 & 0 & 0 & 0 & 0 & {h^{2} /36} & {} & {} & {} & {} & {} & {} \\ {9/70} & 0 & {13l/420} & 0 & 0 & 0 & {13/35} & {} & {} & {} & {} & {} \\ 0 & {9/70} & 0 & {13l/420} & 0 & 0 & 0 & {13/35} & {} & {} & {} & {} \\ { - 13l/420} & 0 & { - l^{2} /140} & 0 & 0 & 0 & { - 11l/210} & 0 & {l^{2} /105} & {} & {} & {} \\ 0 & { - 13l/420} & 0 & { - l^{2} /140} & 0 & 0 & 0 & { - 11l/210} & 0 & {l^{2} /105} & {} & {} \\ 0 & 0 & 0 & 0 & {h^{2} /72} & 0 & 0 & 0 & 0 & 0 & {h^{2} /36} & {} \\ 0 & 0 & 0 & 0 & 0 & {h^{2} /72} & 0 & 0 & 0 & 0 & 0 & {h^{2} /36} \\ \end{array} } \right] \\ \end{aligned} $$
(46)

For the planar straight beam with the coordinate system located at the end point, the mass moment of inertia is

$$ J = {\mathbf{e}}_{Aso}^{{\text{T}}} {\mathbf{M}}_{As} {\mathbf{e}}_{Aso} = m\left( {4l^{2} { + }h^{2} } \right)/12, $$
(47)

where \({\mathbf{e}}_{Aso}\) is the vector of element nodal coordinates defined with respect to the coordinate system located at the end point. If the coordinate system is located at the center for the straight beam, the mass moment of inertia is

$$ J = {\mathbf{e}}_{Asc}^{{\text{T}}} {\mathbf{M}}_{As} {\mathbf{e}}_{Asc} = m\left( {l^{2} + h^{2} } \right)/12, $$
(48)

where \({\mathbf{e}}_{Asc}\) is the vector of element nodal coordinates defined with respect to the coordinate system located at the center of the element. When the ANCF element is used, the tapered geometry is accounted for by stretching the transvers gradient vector [43].

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Shabana, A.A., Xu, L. Rotation-based finite elements: reference-configuration geometry and motion description. Acta Mech. Sin. 37, 105–126 (2021). https://doi.org/10.1007/s10409-020-01030-6

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