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On Drilling Degrees of Freedom

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Computational Modelling and Advanced Simulations

Part of the book series: Computational Methods in Applied Sciences ((COMPUTMETHODS,volume 24))

Abstract

In this paper the development of a new quadrilateral membrane finite element with drilling degrees of freedom is discussed. A variational principle employing an independent rotation field around the normal of a plane continuum element is derived. This potential is based on the Cosserat continuum theory where skew symmetric stress and strain tensors are introduced in connection with the rotation of a point. From this higher continuum theory a formulation that incorporates rotational degrees of freedom is extracted, while the stress tensor is symmetric in a weak form. The resulting potential is found to be similar to that obtained by the procedure of Hughes and Brezzi. However, Hughes and Brezzi derived their potential in terms of pure mathematical investigations of Reissner’s potential, while the present procedure is based on physical considerations. This framework can be enhanced in terms of assumed stress and strain interpolations, if the numerical model is based on a modified Hu-Washizu functional with symmetric and asymmetric terms. The resulting variational statement enables the development of a new finite element that is very efficient since all parts of the stiffness matrix can be obtained analytically even in terms of arbitrary element distortions. Without the addition of any internal degrees of freedom the element shows excellent performance in bending dominated problems for rectangular element configurations.

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Notes

  1. 1.

    One spurious mode is elastically restrained for non rectangular element shapes.

  2. 2.

    Invariant with respect to rotations.

  3. 3.

    This represents the key step of our procedure. We are not involved with a continuum theory where boundary moments are acting since we focus on classical elasticity formulations.

  4. 4.

    The choice \(\alpha = \mu\) is derived straightforward in terms of a convergence proof in [5].

  5. 5.

    This procedure also leads to the derivation of elements without drilling degrees of freedom, i.e. classical plane continuum elements where all parts of the stiffness matrix are evaluated analytically.

  6. 6.

    Admirably, even the hourglass stiffness matrices avoid any numerical quadrature which considerably furnishes the element’s effectiveness.

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Authors and Affiliations

  1. Department of Applied and Numerical Mechanics, University of Applied Sciences Wiener Neustadt, Wiener Neustadt, Austria

    Stephan Kugler & Peter A. Fotiu

  2. Department of Mechanics, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology, Bratislava, Slovakia

    Justín Murín

Authors
  1. Stephan Kugler
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  2. Peter A. Fotiu
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  3. Justín Murín
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Correspondence to Stephan Kugler .

Editor information

Editors and Affiliations

  1. in Bratislava, Department of Mechanics, Slovak University of Technology, Ilkovičova 3, Bratislava, 812 19, Slovakia

    Justín Murín

  2. AFA General Milan Rastislav Štefánik, Demänová 393, Liptovsky Mikulas, 031 19, Slovakia

    Vladimír Kompiš

  3. Department of Mechanics, Slovak Univ. of Technology in Bratislava, Ilkovičova 3, Bratislava, 812 19, Slovakia

    Vladimír Kutiš

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Kugler, S., Fotiu, P.A., Murín, J. (2011). On Drilling Degrees of Freedom. In: Murín, J., Kompiš, V., Kutiš, V. (eds) Computational Modelling and Advanced Simulations. Computational Methods in Applied Sciences, vol 24. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0317-9_15

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  • DOI: https://doi.org/10.1007/978-94-007-0317-9_15

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