## 1 Introduction  
The actual application of structural analysis deals with complex constructions subjected to randomly variable loads. Generally, the load intensity bounds are prescribed by standards and authorities and are evaluated by starting from probabilistic and statistical considerations. It is almost impossible, in practical cases, to know actual load history and their distribution all over the structure, considering that only their intensity range is estimated. Under these hypotheses, evaluating structural safety is a matter of overall consideration rather than the specific pointwise calculation of the response. Under the above-quoted circumstances, the main strategy for structural safety assessment is the calculation of load collapse limit, both under proportionally and randomly variable cases [1; 2]. The former case cited above concerns the evaluation of the usual collapse limit, whereas the latter, besides the collapse limit, requires the calculation of the shakedown limit. Under variable loads, whenever the load reaches the collapse limit, the structure can undergo a sudden collapse. Moreover, it is possible that although at no time collapse occurs, the structure is invested by plastic deformation that during load path increases in time. Now three phenomena can present on the structure: first, the displacement increases in time indefinitely and the structure fails at the end for excessive strain (ratcheting); second, the plastic strain changes sign indefinitely during load path hence, although no excessive displacement occurs the structure fails for plastic fatigue; third, the plastic strain produces self-equilibrated stress that prevents the structure from accumulating further plastic strain, and it starts to definitely behave as elastic. This latter case is called shakedown. The shakedown limit is the load level that the structure shakes down, whilst over it, the structure suffers ratcheting or plastic fatigue [2]. In this situation, some specification should be done about the velocity of the phenomenon to distinguish between elastic and plastic shakedown propagation, but here only statics will be considered [3; 4; 5]. Generally, the knowledge of the structural limit load is pursued using step by step or limit analyses. This latter method is based on the two approaches coming from the upper bound and of the lower bound theorems [1; 6]. These theorems are based on kinematic and static approaches, respectively. The first is aimed at calculating the limit load as the maximum among any statically admissible loads and the second as the minimum among any kinematically admissible loads. Both calculations are independent of the actual load histories, and although this is useful in the real-life analysis where the loads are known only in terms of their intensity bounds, it constitutes a limitation for the use of the method since it cannot evaluate the dissipation during ultra-elastic deformation. It is possible, therefore, that the permanent deformation, required for the ultra-elastic load path to develop, attains its limit before the collapse. Such an eventuality causes the limit analysis to fail since the collapse occurs before the complete permanent deformation develops. This happens at an actual load level lower than the theoretical limit. This disadvantage is only apparently overcome by the step-by-step analysis since this approach is linked to the knowledge of the actual load history and gives results that cannot be generalized. In the modern application of structural engineering, the numerical formulation of the limit analysis allows the application of the methods to a wide range of structures. In particular, the finite element method (FEM) has wide application to two- and three-dimensional structures. In [7; 8], shakedown analysis for elastic-perfect plastic frames is discussed, and an incremental-iterative solution is proposed. A general procedure dealing with finite element application for lower bound determination of collapse and ratcheting load is in [9].