Ratio of Hysteretic and Input Energy Spectra for Nonlinear Structures under Seismic Sequences

In the last few years, several mainshock–aftershock sequences have demonstrated their effects on structures; especially after the occurrence of an earthquake with high magnitude, the number of aftershocks tends to increase. For this reason, several studies have been oriented to estimate the influence of aftershocks on the structural response, most of them in terms of peak or residual displacement; however, energy plays an important role for long-duration earthquakes. In this paper, several bilinear SDOF systems with different post-yielding stiffness are subjected to long-duration seismic sequences, considering different levels of intensity of the aftershocks with the aim to compute constant-ductility spectra for input energy (EI), hysteretic energy (EH) and the ratio between them (EH/EI). The results suggest that although the energy demands tend to increase as the intensity of the aftershock increases, it is observed that the ratio of input and hysteretic energy is very similar for the selected aftershock intensity levels; hence, analytical equations to predict this ratio are proposed. The equations can be used toward earthquake-resistant energy-based design of buildings.