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Keywords: structural analysis, operational research, NRT model, duality, non-linear programming, limit analysis.

Summary

Masonry behaviour is often modelled by the Not Resisting Tension (NRT) assumption. As well known, basic NRT models exhibit a simple linear elastic behaviour under compression stress states and no resistance in tension, thus resulting in an overall fragile non-linear behaviour.

It has been shown (see Heyman [1,2]) that the loading capacity of NRT structures can be investigated by means of the tools of the Limit Analysis (LA) theory, according to some extensions to NRT solids of the static and kinematic LA theorems (see e.g. Baratta [3,4], Como and Grimaldi [5]).

On the basis of the static theorem, the collapse multiplier definitely limiting the loading capacity of the structure is recognized as the upper bound of the class of statically admissible multipliers, i.e. its maximum value, while, by means of the kinematical approach, it is recognized as the lower bound of the class of kinematically sufficient multipliers, i.e. its minimum value.

Anyway both approaches, particularized to some specific cases (such as the evaluation of the loading capacity of a masonry wall loaded by in-plane forces and modelled by finite elements with constant stress/constant strain), lead to constrained extremum problems, governed by linear objective functions under linear and non- linear constraint conditions, thus resulting in non-linear programming problems, whose solution can be numerically pursued by means of Operational Research methods.

On the other side, since one deals with non-linear programming problems, they may be approached by means of the tools of duality theory (see Mangasarian, 1969 [6]); one should emphasize that application of duality to non-linear programming is related to the reciprocal principles of the calculus of variations, which have been known since as far back as 1927 (see Mangasarian [6]); therefore it yields interesting results when applied to the solution of problems, such as the evaluation of the loading capacity of masonry panels, which obey NRT LA theorems, basically deriving from the fundamental energy principles.

In the paper one demonstrates that, starting from the application of the duality theory to the non-linear program defined by the static theorem approach for the above mentioned discrete masonry model, this procedure results in the definition of a dual problem that has a significant physical meaning: the expression of the kinematic theorem.

References

1: J. Heyman, "The stone skeleton" Journal of Solids and Structures, 2, 269-279, 1966. doi:10.1016/0020-7683(66)90018-7
2: J. Heyman, "The Safety of Masonry Arches", Journal of Mechanic Sciences, 2, 363-384, 1969. doi:10.1016/0020-7403(69)90070-8
3: A. Baratta, "Structural Analysis of Masonry Buildings", in "Seismic Risk of Historic Centers. A Preliminary Approach to the Naples Case", A. Baratta and T. Colletta Eds., La Città del Sole B.C., Napoli, 76-122, 1996.
4: A. Baratta, "Il Materiale Non Reagente a Trazione Come Modello per il Calcolo delle Tensioni nelle Pareti Murarie", Journal of "Restauro", 75, 53-77, 1984.
5: M. Como, A. Grimaldi, "A Unilateral model for Limit Analysis of Masonry Walls", in "Unilateral Problems in Structural Analysis", Ravello, 25-46, 1983.
6: O.L. Mangasarian, "Nonlinear Programming", McGraw-Hill, USA, 1969.

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Front Page Browse CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Proceedings ISSN 1759-3433 CCP: 77 PROCEEDINGS OF THE NINTH INTERNATIONAL CONFERENCE ON CIVIL AND STRUCTURAL ENGINEERING COMPUTING Edited by: B.H.V. Topping Paper 103 Limit Analysis of No Tension Bodies and Non Linear Programming A. Baratta and O. Corbi Department of "Scienza delle Costruzioni", University of Naples "Federico II", Italy doi:10.4203/ccp.77.103 purchase the full-text of this paper Full Bibliographic Reference for this paper A. Baratta, O. Corbi, "Limit Analysis of No Tension Bodies and Non Linear Programming", in B.H.V. Topping, (Editor), "Proceedings of the Ninth International Conference on Civil and Structural Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 103, 2003. doi:10.4203/ccp.77.103 Keywords: structural analysis, operational research, NRT model, duality, non-linear programming, limit analysis. Summary Masonry behaviour is often modelled by the Not Resisting Tension (NRT) assumption. As well known, basic NRT models exhibit a simple linear elastic behaviour under compression stress states and no resistance in tension, thus resulting in an overall fragile non-linear behaviour. It has been shown (see Heyman [1,2]) that the loading capacity of NRT structures can be investigated by means of the tools of the Limit Analysis (LA) theory, according to some extensions to NRT solids of the static and kinematic LA theorems (see e.g. Baratta [3,4], Como and Grimaldi [5]). On the basis of the static theorem, the collapse multiplier definitely limiting the loading capacity of the structure is recognized as the upper bound of the class of statically admissible multipliers, i.e. its maximum value, while, by means of the kinematical approach, it is recognized as the lower bound of the class of kinematically sufficient multipliers, i.e. its minimum value. Anyway both approaches, particularized to some specific cases (such as the evaluation of the loading capacity of a masonry wall loaded by in-plane forces and modelled by finite elements with constant stress/constant strain), lead to constrained extremum problems, governed by linear objective functions under linear and non- linear constraint conditions, thus resulting in non-linear programming problems, whose solution can be numerically pursued by means of Operational Research methods. On the other side, since one deals with non-linear programming problems, they may be approached by means of the tools of duality theory (see Mangasarian, 1969 [6]); one should emphasize that application of duality to non-linear programming is related to the reciprocal principles of the calculus of variations, which have been known since as far back as 1927 (see Mangasarian [6]); therefore it yields interesting results when applied to the solution of problems, such as the evaluation of the loading capacity of masonry panels, which obey NRT LA theorems, basically deriving from the fundamental energy principles. In the paper one demonstrates that, starting from the application of the duality theory to the non-linear program defined by the static theorem approach for the above mentioned discrete masonry model, this procedure results in the definition of a dual problem that has a significant physical meaning: the expression of the kinematic theorem. References 1 J. Heyman, "The stone skeleton" Journal of Solids and Structures, 2, 269-279, 1966. doi:10.1016/0020-7683(66)90018-7 2 J. Heyman, "The Safety of Masonry Arches", Journal of Mechanic Sciences, 2, 363-384, 1969. doi:10.1016/0020-7403(69)90070-8 3 A. Baratta, "Structural Analysis of Masonry Buildings", in "Seismic Risk of Historic Centers. A Preliminary Approach to the Naples Case", A. Baratta and T. Colletta Eds., La Città del Sole B.C., Napoli, 76-122, 1996. 4 A. Baratta, "Il Materiale Non Reagente a Trazione Come Modello per il Calcolo delle Tensioni nelle Pareti Murarie", Journal of "Restauro", 75, 53-77, 1984. 5 M. Como, A. Grimaldi, "A Unilateral model for Limit Analysis of Masonry Walls", in "Unilateral Problems in Structural Analysis", Ravello, 25-46, 1983. 6 O.L. Mangasarian, "Nonlinear Programming", McGraw-Hill, USA, 1969. purchase the full-text of this paper (price £20) go to the previous paper go to the next paper return to the table of contents return to the book description purchase this book (price £123 +P&P)
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