Keywords: finite element structures, elastic plastic behaviour, optimal design, elastic shakedown, plastic shakedown, ratchet.

It is known that an elastic plastic structure subjected to a combination of cyclic and steady loads exhibits, after a transient phase, a steady-state response independent of the initial conditions and having the same periodicity features as the cyclic loads [1]. The steady-state structural response can be characterized by: an elastic shakedown behaviour, with plastic strain rates vanishing identically; or a plastic shakedown behaviour, with plastic strain rates non vanishing, but the plastic strain field resulting in the cycle is nought; or, finally, a ratchetting behaviour with non vanishing plastic strain field resulting in the cycle.

In the space of the load parameters , where is the multiplier of the fixed mechanical load and is the multiplier of the cyclic mechanical and/or kinematical load, five different regions can be distinguished representing all the possible different behaviours of the structure under the amplified load conditions and the graphical representation of these zones constitutes the Bree-like diagram related to the given structure/load system.

The optimal design of structures constituted by finite elements with elastic plastic constitutive behaviour, subjected to loads as above described, has been studied taking into account different resistance criteria [2,3,4]. Multicriteria optimal design formulations [5] have been also proposed taking contemporaneously into account different resistance criteria and assuming for each criterion a suitably chosen safety factor. The structures obtained effecting the optimal designs above described exhibit the required behaviour under the prescribed conditions. In particular, the structure obtained by means of the multicriteria optimal design possesses the required safety factor with respect to instantaneous collapse and, at the same time, guarantees a good behaviour in serviceability conditions. Unfortunately, unless further suitable conditions be imposed, also structures so well designed may be exposed to incremental collapse, instead to alternating plasticity, when for a given fixed load the cyclic loads are even slightly above the elastic shakedown limit. As a consequence, in such condition, although the instantaneous collapse is prevented, the structure may be addressed to lose its functionality and/or to collapse in a few cycles.

Aim of the present paper is to propose a formulation of the optimal plastic shakedown design of finite element elastic plastic structures subjected to a combination of steady and cyclic loads, such that ratchet strain be prevented. It is based on a known property of bodies in plastic shakedown conditions [6,7]. Actually, in such a condition, the body can be subdivided into two portions: the first one exhibits an elastic plastic behaviour, the second one persists in a purely elastic behaviour. The proposed formulation is developed imposing that under the amplified cyclic load acting alone the part finds itself in condition of plastic shakedown and the part remains elastic, while under the combination of amplified steady and cyclic loads, the stresses and strains in remain unaltered within the entire cycle and reaches its elastic shakedown limit, i.e. the entire structure finds itself in a condition of impending incremental collapse. A statical approach to the design problem is performed and the related Kuhn-Tucker conditions are deduced. The solution of the proposed formulation provides all the design and bahvioural variables and also the partition of the structure into the two parts and .

An application concludes the paper.

1

J. Zarka, J. Frelat, G. Inglebert, P. Kasnat-Navidi, "A new approach in inelastic analysis of structures", Ecole Polytechnique Palaisau, France, 1990.

2

C. Cinquini, G. Sacchi, "A note on the optimal elastic design for given deflection", Optimization of distributed parameter structures, E. J. Haug and J. Cea, ed., 1, 383-398, 1981.

3

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

4

G. Maier, A. Zavelani Rossi, D. Benedetti, "A finite element approach to optimal design of plastic structures in plane stress", Int. J. Num. Meth. Engrg., 4, 455-473, 1982. doi:10.1002/nme.1620040402

5

F. Giambanco, L. Palizzolo, L. Cirone, "Multicriteria optimal design of elastic plastic structures", (in Italian), Proceedings of the XIV Convegno Nazionale AIMETA, Como (Italy), October 6-9, 1999.

6

C. Polizzotto, "Study states and sensitivity analysis in elastic-plastic structures subjected to cyclic loads", Int. J. Solids Struct., 31, 953-970, 1994. doi:10.1016/0020-7683(94)90005-1

7

A.R.S. Ponter, C. Haofeng, "A minimum theorem for cyclic load in excess of shakedown, with application to the evaluation of a ratchet limit", Eur. J. Mech. A/Solids, 20, 539-553, 2001. doi:10.1016/S0997-7538(01)01161-5

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