Summary
The finite element approximation to the continuum problem is examined from the viewpoint of the principle of virtual work. It is shown that the usual nodal equilibrium equations for triangular elements are a consistent consequence of a piecewise constant strain field, thus guaranteeing that many results of general continuum theory can be directly applied to the finite element model, and also clarifying the relation between the two models.
Übersicht
Das Verfahren, ein Kontinuum durch finite Elemente anzunähern wird vom Standpunkt des Prinzips der virtuellen Arbeiten untersucht. Es wird gezeigt, daß die üblichen Knotenpunktsgleichungen für dreieckförmige Elemente eine Folge des stückweise konstanten Verformungsfeldes sind. Auf diese Weise wird sichergestellt, daß viele Ergebnisse der allgemeinen Kontinuumstheorie unmittelbar auf das aus endlichen Elementen aufgebaute Modell übertragen werden können. Gleichzeitig werden die Beziehungen zwischen beiden Modellen geklärt.
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References
Philip G. Hodge, Jr., Continuum Mechanics, Mc. Graw Hill Book Co. New York (1970), Chap. 8.
W. Prager, The general theory of limit design, Proc. 8th Intern. Congr. Appl. Mech., (Istanbul 1952) 2, 1956, pp. 65–72.
W. Prager, Variational principles of linear elastostatics for discontinous displacements, strains, and stresses, Recent Progress in Applied Mechanics; the Folke Odqvist Volume, Stockholm, 1969, pp. 463–474.
Pedro V. Marcal, Finite-element analysis of combined problems of non-linear material and geometric behavior, Computational Approaches in Applied Mechanics (ASME Computer Conference, Chicago, 1969), The American Society of Mechanical Engineers, New York, 1970, pp. 133–149.
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Dedicated to Professor Dr. H. Ziegler on the occasion of his 60th birthday.
This research was sponsored by the National Science Foundation, Grant GK 10549.
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Hodge, P.G. A consistent finite element model for the two-dimensional continuum. Ing. arch 39, 375–382 (1970). https://doi.org/10.1007/BF00538758
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DOI: https://doi.org/10.1007/BF00538758