Keywords: bounds, Bree diagram, structural optimization, ductility, plastic strains, shakedown, trusses.

Solving a structural optimal design problem requires finding, among all the feasible designs, the one which minimizes a suitably chosen quantity according to some required (limit) behaviour for the structure. In the last decades many researchers addressed their scientific efforts to the structural optimization problems providing many refined and original formulations of the optimal design as well as several interesting contributions related to the computational procedures.

In the hypothesis of elastic plastic structures subjected to fixed and cyclic loads, usually the minimum weight design has been treated. Actually, the elastic optimal design (see, e.g., [1]), the elastic shakedown optimal design (see, e.g., [2,3]), and the standard limit design (see, e.g., [4]) have been developed. Each one of these criteria takes into account just the corresponding structural limit state disregarding the observance of suitable safety factors for the other possible limit states. In order to avoid this unsafe behaviour of the obtained optimal structure, some multicriteria optimal design formulations (see, e.g., [5]) have been developed, in which the optimal structure must satisfy contemporaneously the three above referenced resistance criteria with appropriate safety factors. Unfortunately, this last formulation, even if it ensures the satisfaction of the imposed limit conditions under the prescribed amplified loads, does not provide any useful information about the behaviour of the structure above the elastic shakedown limit. Actually, for load conditions above the elastic shakedown limit, a plastic shakedown behaviour is preferable with respect to a ratchetting one because in the first case the structure has the ability to suffer a much greater number of load cycles than in the last case, and the second order geometrical effects can be usually considered as negligible.

Recently (see, e.g., [6]), some formulations of the optimal design of elastic plastic structures subjected to a combination of fixed and perfect cyclic loads have been proposed. In particular, the optimal structure is obtained in such a way that, above the elastic shakedown limit, the incremental collapse is prevented as far as the load multipliers do not exceed some given limits. Whatever the special formulation be utilized, it can be very useful to know if the optimal structure at the prescribed limit state fulfils special limits on its ductility behaviour. Actually, in the above referred formulations possible limits on the ductility behaviour of the structure either with regard to the plastic shakedown limit or to the elastic shakedown limit for the structure in serviceability conditions have been disregarded. Some contributions on this topic have been proposed in the case of elastic shakedown design (see, e.g., [7]) as well as in the case of standard limit design (see, e.g., [8]).

The present paper is devoted to an improved formulation of the optimal design of elastic plastic structures subjected to a combination of fixed and perfect cyclic loads contemporaneously according to a plastic shakedown criterion and to an elastic shakedown one, each related to suitably chosen load multiplier values, and with constraints on the plastic strains related to the limit state of plastic shakedown as well as on some suitably chosen measures of the plastic deformation related to the elastic shakedown. The relevant formulation is developed making reference to truss structures. The described minimum problem is a strongly non linear mathematical programming one and, as a consequence, special effort has been devoted to the proposing of a suitable numerical procedure. In the framework of the numerical applications, reference has been made to steel trusses.

1

C. Cinquini, G. Guerlement, D. Lamblin, "Finite element iterative methods for optimal elastic design of circular plates", Computers & Structures, 12(1), 85-92, 1980. doi:10.1016/0045-7949(80)90096-6

2

J.A. König, "On optimum shakedown design", Optimization in structure design, Edited by Sawczuk A. and Mroz Z., 405-414, Springer-Verlag, Berlin, Germany, 1975.

3

F. Giambanco, L. Palizzolo, L. Cirone, "Computational methods for optimal shakedown design of FE structures", Structural Optimization 15(3/4), 284-295, 1998. doi:10.1007/BF01203544

4

G.I.N. Rozvany, Optimal design of flexural systems, Pergamon Press, Oxford, England, 1976.

5

L. Palizzolo, "Optimization of continuous elastic perfectly plastic beams", Computers & Structures, 82(4-5), 397-411, 2004. doi:10.1016/j.compstruc.2003.10.015

6

F. Giambanco, L. Palizzolo, A. Caffarelli, "Computational procedures for plastic shakedown design of structures", to appear on Journal of Structural and Multidisciplinary Optimization, 2004. doi:10.1016/j.compstruc.2003.10.015

7

C. Polizzotto, "Shakedown analysis and design in presence of limited ductility behaviour", Engineering Structures, (6), 80-87, 1984. doi:10.1016/0141-0296(84)90065-8

8

I. Kaneko, G. Maier, "Optimum design of plastic structures under displacement constraints", Computational Methods in Applied Mechanics Engineering, (27), 369-391, 1981. doi:10.1016/0045-7825(81)90139-0

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