Institute of Fundamental Technological Research

This paper deals with the application of mathematical programming in shakedown problems. After recalling results of some recent works in this field, the authors propose a finite element formulation of the load domain of shakedown. The static approach is utilized together with the yield criterion of the mean. The latter assumption has already given good numerical results in limit analysis applied to plane problems [1]The assumptions on the discrete residual stress field are adopted exactly in the same manner as for elastic asymptotic stress fields in pure equilibrium elements which guarantee the total stress transmission to be perfectly continuous. In this way, nonlinear programming aspects of the shakedown problem are pointed out. Accordingly, it appears that the use of hybrid elements is recommended. A simple and convenient kinematic finite element formulation concludes the work.

In many practical problems the need arises to consider bodies whose non-homogeneity is described by non-continuous functions. The subject of the paper is the determination of the carrying capacity of simply supported circular plates, loaded by uniform pressure and composed of concentric annuli with different mechanical properties. The above jump of mechanical properties may be caused by jump of plate thickness or by change of material properties. Depending on the values of non-homogeneity and the division parameters there exist six different solutions of the considered problem. In each particular case the moment field, velocity field and limit load are given. The ranges of validity of all solutions are established. Also, the problem of optimum design in the above class of plates is considered. In the second part an orthotropic plate is discussed. The special case concerning the uniform distribution of circumferential reinforcement in the case of a reinforced concrete plate is considered. Here again, depending on different parameters, there may be seven solutions. The above analysis has allowed some qualitative conclusions to be drawn concerning the design of isotropic and orthotropic plates with jump non-homogeneity.