Seismic shakedown design of frames based on a probabilistic approach

The present study concerns the optimal design of elastic perfectly plastic structures subjected to a combination of fixed and seismic loads. In particular, plane frames are considered and suitable measures of the beam element cross sections are chosen as design variables. The optimal design is required to behave in a purely elastic manner when subjected just to the fixed load and to have the capability to eventually shakedown when simultaneously subjected to fixed and seismic loads. Due to the natural uncertainness related to the definition of the seismic load history, a new probabilistic approach is proposed, consisting into two subsequent search steps. At first a suitably chosen large number of minimum volume designs are obtained for as many random seismic load histories deduced by a suitably chosen reference power spectral density function, determining a probabilistic distribution of optimal volumes. Subsequently, the volume obtained with probability 1 is assigned as the optimal structural volume, and a new optimal design problem is solved in order to obtain the material optimal distribution. The latter is a minimum elastic strain energy one for fixed volume. The performed applications confirm the effectiveness of the proposed procedure.