Mathematical models for the optimization problems of metal structures at shakedown definitely contains strength (ULS – ultimate limit state) and stiffness (SLS – serviceability limit state) constraints. Residual displacements determined by plastic deformations appear in stiffness constraints. The residual displacements developing during shakedown process of ideally elastic-plastic structures under variable repeated load can vary non-monotonically. Therefore, it is important to determine variation bounds of residual displacements (particularly in the cases when only variation bounds of loading rather than a particular history of a load are known). Thus, numerical methods, extremum energy principles, the assumption of a small displacement and mathematical programming theory have been used in the paper to develop mathematical models for the analysis problem of residual displacements. Using the proposed scanning method, it is acceptable to define the revised bounds of residual displacements without analysing the detailed history of loading. However, scanning technique is approximate method comparing by comprehensive analysis of elastic-plastic structure, when to exact values of displacement are obtained. The application of the proposed scanning method is illustrated by a benchmark of a plane multi-supported beam.
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