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Discretization of rotational equilibrium in the finite element method

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Part of the Lecture Notes in Mathematics book series (LNM,volume 606)

Abstract

The theory of equilibrium elements1,2 shows that their stiffness matrices may present a singular behavior due to the presence of mechanisms (deformation modes without strain energy). The origin of such difficulties is easily traced to the rigorous requirement of rotational equilibrium (symmetry of the stress tensor) and equivalently, if the discretization is performed on the basis of stress functions, to the C1 continuity requirement involved. Moreover loss of diffusivity (reciprocity of surface traction distributions at interfaces) is incurred in an isoparametric coordinate transformation to curved boundaries, whenever preservation of C1 continuity is at stake.

Both difficulties are resolved by enforcing rotational equilibrium only in weak form. First order stress functions are used to preserve rigorous translational equilibrium and diffusivity. They need only be Co continuous, a property that remains invariant under isoparametric coordinate transformations.

The theory of discretized rotational equilibrium has been investigated in detail for membrane elements3. The paper is devoted to the more difficult case of axisymmetric elements.

Keywords

  • Stress Function
  • Hoop Stress
  • Surface Traction
  • Complementary Energy
  • Rotation Field

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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12. References

  1. B.Fraeijs de Veubeke: Upper and lowers bounds in Matrix Structural Analysis. AGARDograph 72, Pergamon Press 1964, pp. 165–201.

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  2. Displacement and Equilibrium models in the Finite Element Method. Chap. 9 in "Stress Analysis" Ed. Zienkiewicz and Molister, John Wiley & Sons, 1965.

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  3. Stress Function Approach. World Congress on Finite Element Methods in Structural Mechanics, Bournemouth, 1975, pp. J1–J48.

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© 1977 Springer-Verlag

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Fraeijs de Veubeke, B.M. (1977). Discretization of rotational equilibrium in the finite element method. In: Galligani, I., Magenes, E. (eds) Mathematical Aspects of Finite Element Methods. Lecture Notes in Mathematics, vol 606. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0064458

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  • DOI: https://doi.org/10.1007/BFb0064458

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08432-7

  • Online ISBN: 978-3-540-37158-8

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