P-Delta effect in structural optimization

The present paper is devoted to the optimal design of structures constituted by elastic perfectly plastic material. In particular, reference will be made to the minimum volume design of multi-storey steel frames taking into account the P-Delta effect. For the sake of generality, the structure is thought as discretized into compatible finite elements and subjected to fixed and quasi-statically acting (wind) loads as well as to dynamic (seismic) loads. The effects of the dynamic actions are studied by utilizing a classical modal technique. The relevant optimal design problem is formulated on the grounds of a statical approach and different resistance limits are considered: in particular, it is required that the optimal structure simultaneously satisfies the elastic shakedown limit and, alternatively, prevents the instantaneous collapse or does not overpass the plastic shakedown limit, imposing for each different condition a suitably chosen safety factor. In such a way a multicriterion limit optimal design or a shakedown elastic/plastic optimal design is reached. The proposed treatment is referred to the most recent Italian Codes related to the structural analysis and design: all the loads are considered as quasi statically acting and suitably given as a combination of fixed loads and perfect cyclic loads. The cyclic loads describe the effect of the wind actions as well as of the seismic actions, the latter deduced by utilizing a modal technique and, therefore, depending on the (unknown) dynamic properties of the structure. Following the referenced Italian Codes the optimal design is at first formulated as the search for the minimum volume structure with simultaneous constraints on the elastic shakedown behaviour, related to appropriate serviceability conditions, and on the instantaneous collapse, related to high intensity loads (wind effects or seismic actions). Subsequently, the same minimum volume problem is performed but imposing an elastic shakedown limit, for serviceability conditions, and an alternating plasticity limit, for high intensity loads. The obtained optimal structures are compared in order to evaluate the different behavioural features and it is found that the optimal structure obtained by the solution of the latter problem possesses more adequate safety features cause the ratchetting failure is prevented. Furthermore, both the optimal minimum volume problems are also studied taking into account the P-Delta effect, particularly influent in the case of multi-storey frames. The numerical applications are related to flexural and cross braced steel frames.