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TECHNICAL PAPERS

A Linear Beam Finite Element Based on the Absolute Nodal Coordinate Formulation

[+] Author and Article Information
Kimmo S. Kerkkänen

Department of Mechanical Engineering, Institute of Mechatronics and Virtual Engineering, Lappeenranta University of Technology, P.O. Box 20, FIN-53851 Lappeenranta, Finlandkkerkkan@lut.fi

Jussi T. Sopanen

Department of Mechanical Engineering, Institute of Mechatronics and Virtual Engineering, Lappeenranta University of Technology, P.O. Box 20, FIN-53851 Lappeenranta, Finlandjsopanen@lut.fi

Aki M. Mikkola

Department of Mechanical Engineering, Institute of Mechatronics and Virtual Engineering, Lappeenranta University of Technology, P.O. Box 20, FIN-53851 Lappeenranta, Finlandmikkola@lut.fi

J. Mech. Des 127(4), 621-630 (Aug 29, 2004) (10 pages) doi:10.1115/1.1897406 History: Received January 27, 2004; Revised August 29, 2004

In this paper, a new two-dimensional shear deformable beam element based on the absolute nodal coordinate formulation is proposed. The nonlinear elastic forces of the beam element are obtained using a continuum mechanics approach, without employing a local element coordinate system. In this study, linear polynomials are used to interpolate both the transverse and longitudinal components of the displacement. This is different from other absolute nodal-coordinate-based beam elements where cubic polynomials are used in the longitudinal direction. The use of linear interpolation polynomials leads to the phenomenon known as shear locking. This defect is avoided through the adoption of selective integration within the numerical integration method. The proposed element is verified using several numerical examples. The results of the proposed element are compared to analytical solutions and the results for an existing shear deformable beam element. It is shown that by using the proposed element, accurate linear and nonlinear static deformations, as well as realistic dynamic behavior including the capturing of the centrifugal stiffening effect, can be achieved with a smaller computational effort than by using existing shear deformable two-dimensional beam elements.

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Copyright © 2005 by American Society of Mechanical Engineers
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Figures

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Figure 1

(a) The correct deformation mode of a rectangular block in pure bending. (b) The shear locking of the element results in the incorrect deformation mode of a rectangular block in pure bending.

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Figure 2

A simply supported beam for linear deformations

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Figure 3

The cantilever beam model for nonlinear deformations

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Figure 4

A free-falling flexible pendulum for dynamic verification in the initial position

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Figure 5

Vertical displacement of the falling flexible beam tip point using 2, 4, 8, and 16 elements

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Figure 6

The energy components and energy balance of the falling flexible beam modeled using four elements

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Figure 7

A comparison of the vertical displacement between the proposed element and that presented by Omar and Shabana using eight elements

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Figure 8

A rotating flexible cantilever beam

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Figure 9

The difference of the end-point vertical displacements between the modeled beam and the straight shadow beam

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Figure 10

The difference of the end-point horizontal displacements between the modeled beam and the straight shadow beam and the steady-state extension of the rotating beam

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