Keywords: elastic plastic trusses, elastic shakedown, plastic shakedown, Bree diagram, buckling, limited ductility.

In the greater part of engineering applications, modern structures are constituted by materials exhibiting elastic plastic behaviour possessing suitably wide ductility properties. They are always subjected to the slight variable load serviceability conditions, while, sometimes, they can be subjected to very high intensity loads characterized by great variability.

Under such conditions, and for load intensities not exceeding appropriate limits, the elastic shakedown theory provides useful tools in the study of the behaviour of relevant structures [1]. Furthermore, the so-called bounding techniques provide sufficiently approximate limits on suitably chosen measures of the plastic deformation related to the transient phase of the structural response [2]. On the contrary, if the load multipliers exceed the elastic shakedown limit, then the structure is approaching a collapse condition, either due to a plastic shakedown behaviour or to a ratchetting behaviour. Finally, for further increasing values of the load multipliers, the structure is eventually approaching an instantaneous collapse condition.

Therefore, depending on the load multiplier values a structure can alternatively exhibit: a purely elastic behaviour, an elastic shakedown behaviour, a plastic shakedown behaviour, a ratchetting behaviour, or, eventually, it can be exposed to instantaneous collapse.

As it is well known, it would be preferable that structures, even exhibiting an elastic plastic behaviour, behave in a condition of elastic shakedown when subjected to serviceability load intensities and behave in a condition of plastic shakedown when subjected to exceptionally high intensity loads. As a consequence, it is clear the importance of knowing the borderline between the elastic-plastic shakedown domain and the incremental-instantaneous collapse zone for the given structure subjected to the assigned load condition.

Depending on the structure topology and on the special ductility features of the material, the above defined borderline may be also influenced by other constraints which identify ductility limits and, or functionality limits for the structure. Unfortunately, in the case of elastic shakedown behaviour, and at the same way, in the case of plastic shakedown behaviour, the plastic strains accumulated during the initial transient phase remains unknown and, as a consequence, no check can be effected in order to verify the suitable imposed ductility and, or the functionality limits.

So, in order to obtain even approximate information about the plastic deformations occurring at the end of the transient phase, it is necessary to make reference to other analytical and, or numerical procedures and, in the relevant case, to suitable bounding techniques [3]. These techniques allows us to evaluate bounds on some prefixed measures of the plastic deformations, whatever the real loading history is during the unknown transient phase.

In addition, depending on the special given structure, and in particular making reference to truss structures, another effective limit which it is necessary to respect in order to preserve the full functionality of the structure is related to the buckling of the bars. Such a limit can have further influence on the above described borderline between the elastic-plastic shakedown domain and the incremental-instantaneous collapse zone.

Therefore, the aim of the present paper is to propose a new formulation of the elastic-plastic shakedown boundary determination for elastic perfectly plastic truss structures subjected to a combination of fixed and cyclic loads, adopting the restrictive hypothesis of mechanical fixed loads and perfect cyclic mechanical loads, preventing the undesired phenomenon of buckling and simultaneously imposing the respect of assigned limits on the plastic deformations occurring in the initial transient phase of the structural response.

A numerical application related to steel trusses concludes the paper.

1

J. A. König, "Shakedown of elastic plastic structures", PWN-Polish Scientific Publishers, Warsaw and Elsevier, Amsterdam, 1987.

2

C. Polizzotto, "A unified treatment of shakedown theory and related bounding techniques", S. M. Archives, 7, 1, 19-75, 1982.

3

F. Giambanco, L. Palizzolo, A. Caffarelli, "Bounds on Plastic Deformations for Structures in Plastic Shakedown", Sixth World Congress on Computational Mechanics, WCCM VI, Beijing, Cina, Sept. 2004.

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