When considering a single set of external loads, monotonically increasing with a load parameter, the safety factor of an elastic-plastic structure can be effectively evaluated by numerical implementations of the classical theorems of limit analysis or, even more efficiently, by recovering the equilibrium path by means of path-following algorithms, well suited to implementation in general purpose FEM codes. It is known, however, that the collapse multiplier doesn't furnish a reliable safety index when the structure is subjected to a combination of loads that can vary, cyclically or in a generic way, inside a given load domain. In this case, in fact, other different failure modes are possible. A continuous increase in plastic deformations, along successive plastically admissible load cycle, could lead to a loss of the functionality or produce the collapse due to fatigue. Then, a further requirement has to be met, that the rising amount of plastic deformations be confined only to an initial phase after which the structure behavior, for any combination of load contained in the load's domain, is purely elastic. If this happens [1,2,3] we say that shakedown occurs.

The lack of a procedure well suited to implementation in FEM codes, has limited the diffusion of this method. An iterative method for shakedown analysis, using a path-following iterative scheme similar to that used in limit analysis, has been proposed recently by Casciaro and Garcea [4]. In this work, the application was limited to plane frames.

In the present work, using the theoretical framework of [4] the method is extended for the analysis of two-dimensional structures (plates subjected to in-plane loads) using finite element discretization and a piecewise linearization of the elastic domain. The purpose is to show that even in more complex FEM contexts the methods described can be easily implemented in a robust and efficient way.

The finite element formulation for the two-dimensional problems has been kept as simple as possible and is based on a mixed triangular finite element.

The proposed strategy appears to be an alternative to direct methods [6] and more efficient than other numerical methods proposed in literature [7], especially for large dimension problems where the matrix operation prevails.

The implementational differences with respect to the standard path-following algorithms currently used for evaluating the equilibrium path of an elastic-plastic structure, are minimal. This fact should make it very easy to modify commercial codes aimed at elastic-plastic analysis into codes able to perform shakedown analysis in technically relevant applications.

1

E. Melan, "Zur Plastizität des raümlichen continuum", Ing. Arch, 9, 116-126, 1938. doi:10.1007/BF02084409

2

W.T. Koiter, "General theorems for elastic-plastic solids", in Progress in solids mechanics, ed I. N. Sneddon and R Hill, 165-221, North-Holland, Amesterdam, 1960.

3

J.A. König, G. Maier, "Shakedown analysis of elastic structures: a review of recent developments ", J. Nuclear Engng. Design, 66, 81-95, 1981. doi:10.1016/0029-5493(81)90183-7

4

R. Casciaro, G. Garcea, "An iterative method for shakedown analysis", Computer Methods in Applied Mechanics and Engineering, 191, 5761-5792, 2002. doi:10.1016/S0045-7825(02)00496-6

5

N. Zouanin, L. Borges, J.L. Silveira,"An algorithm for shakedown analysis with nonlinear yeld function", Comp. Meth. Appl. Mech. Engng. 191, 2463-2481, 2002. doi:10.1016/S0045-7825(01)00374-7

6

Maier, G., Carvelli, V., Cocchetti, G., "On direct methods for shakedown and limit analysis">, Plenary Lecture, 4th Euromech Solid Mechanics Conference, Metz, June 2000, European Journal of Mechanics A/Solids, 19, Special Issue, S79-S100, 2000.

7

A.R.S. Ponter, K.F. Karter, "Shakedown state simulation tecniques based on linear elastic solutions", Comp. Methods Appl. Mech. Engng., 140, 259-279, 1997. doi:10.1016/S0045-7825(96)01105-X

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