Vol. 7, No. 8-9, 2012

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ISSN: 1559-3959
On the choice of functions spaces in the limit analysis for masonry bodies

Massimiliano Lucchesi, Miroslav Šilhavý and Nicola Zani

Vol. 7 (2012), No. 8-9, 795–836
Abstract
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The kinematic and static problems of limit analysis of no-tension bodies are formulated. The kinematic problem involves the infimum of kinematically admissible multipliers, and the static problem the supremum of statically admissible multipliers. The central question of the paper is under which conditions these two numbers coincide. This involves choices of function spaces for the competitor displacements and competitor stresses. A whole ordered scale of these spaces is presented. These problems are formulated as convex variational problems considered by Ekeland and Témam. The static problem is unconditionally shown to be the dual problem (in the sense of the mentioned reference) of the kinematic problem. A necessary and sufficient condition, the normality, guarantees that the kinematic and static problems give the same result. The normality is not always satisfied, as examples show (one of which is presented here). The qualification hypothesis of Ekeland and Témam, sufficient for the equality of the static and kinematic problems, is never satisfied in the spaces of admissible displacements of bounded deformation or of functions integrable together with the gradient in the power p, 1 p < . In the cases of lipschitzian displacements and of smooth displacements, the qualification hypothesis is equivalent to simple conditions that can be satisfied in the case of the pure traction problem. However, it is shown that then the space of admissible stresses must be enlarged to contain stress fields represented by finitely or countably additive tensor-valued measures.

Keywords
limit analysis, static and kinematic theorems, duality of displacements and forces
Milestones
Received: 1 June 2012
Revised: 27 September 2012
Accepted: 4 October 2012
Published: 23 January 2013
Authors
Massimiliano Lucchesi
Dipartimento di Ingegneria Civile e Ambientale
Università di Firenze
Piazza Brunelleschi 6
I-50121 Firenze
Italy
Miroslav Šilhavý
Institute of Mathematics
Academy of Sciences of the Czech Republic
Žitná 25
115 67 Praha 1
Czech Republic
Nicola Zani
Dipartimento di Ingegneria Civile e Ambientale
Università di Firenze
Piazza Brunelleschi 6
I-50121 Firenze
Italy