Research Papers

Structural and Continuum Mechanics Approaches for a 3D Shear Deformable ANCF Beam Finite Element: Application to Static and Linearized Dynamic Examples

[+] Author and Article Information
Karin Nachbagauer

Institute of Technical Mechanics,
Johannes Kepler University Linz,
Altenbergerstraße 69, Linz, 4040, Austria
e-mail: karin.nachbagauer@jku.at

Peter Gruber

e-mail: peter.gruber@lcm.at

Johannes Gerstmayr

e-mail: johannes.gerstmayr@lcm.at
Linz Center of Mechatronics GmbH,
Altenbergerstraße 69, Linz, 4040, Austria

Gerstmayr et al. [18] show the exact thickness mode frequencies in Eq. (70) for a beam with rectangular cross section and in Eq. (72) the thickness mode frequencies following from mass and stiffness of a linear oscillator. This obtained eigenfrequency in Eq. (72), which is about 1.1 times higher as the given exact value, was taken as a reference value for the first thickness eigenfrequency in Tab. 5.

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August 29, 2011; final manuscript received April 23, 2012; published online July 23, 2012. Assoc. Editor: Arend L. Schwab.

J. Comput. Nonlinear Dynam 8(2), 021004 (Jul 23, 2012) (7 pages) Paper No: CND-11-1141; doi: 10.1115/1.4006787 History: Received August 29, 2011; Revised April 23, 2012

In the present paper, a three-dimensional shear deformable beam finite element is presented, which is based on the absolute nodal coordinate formulation (ANCF). The orientation of the beam’s cross section is parameterized by means of slope vectors. Both a structural mechanics based formulation of the elastic forces based on Reissner’s nonlinear rod theory, as well as a continuum mechanics based formulation for a St. Venant Kirchhoff material are presented in this paper. The performance of the proposed finite beam element is investigated by the analysis of several static and linearized dynamic problems. A comparison to results provided in the literature, to analytical solutions, and to the solution found by commercial finite element software shows high accuracy and high order of convergence, and therefore the present element has high potential for geometrically nonlinear problems.

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Fig. 1

Different configurations of the finite beam element: (a) scaled straight reference element and (b) the reference element depicted in the global coordinate system

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Fig. 2

Instead of position and slope vectors, here the respective displacement-based quantities are chosen as degrees of freedom

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Fig. 3

A cantilever beam under a vertical tip load F is tested to show the performance for in-plane bending problems. Therefore, the displacement in the third direction uz is zero.

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Fig. 4

Comparison of convergence of the different beam formulations: With respect to the converged reference solution of the proposed quadratic CMF (locking free) and considering Poisson ratio ν = 0.3, the standard cubic continuum mechanics based ANCF beam element [6] shows convergence of order two, while the proposed quadratic CMF (locking free) shows convergence of order four. The standard quadratic CMF, which suffers from Poisson locking, does not converge to the correct solution. With respect to the converged solution of Simo-Vu-Quoc (Maple) [10] and considering Poisson ratio ν = 0.3, the proposed quadratic SMF converges with order four as well. In addition, the plot shows that the standard cubic continuum mechanics based ANCF beam element [6] does not converge to the solution of the proposed CMF (locking free), while considering Poisson ratio ν = 0.

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Fig. 5

A cantilever beam under a bending and a torsional moment is tested to analyze a general bending problem

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Fig. 6

Simply supported beam in Sec. 4.5 investigated for eigenfrequency analysis




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