Aftershocks have the potential to cause collapse of a structure damaged by the preceding main shock and thus the conventional single event-based seismic design may not guarantee the desired safety against earthquake ground motions. A systematic investigation is carried out in this study on the relative contribution of aftershock ground motions to cumulative structural damage in the case of three Indian code-designed reinforced concrete bare frames having different fundamental periods. Assuming that the response spectrum and time-frequency characteristics of the aftershock motions may be related to these characteristics of the main shock motion, a variety of seismic scenarios (in terms of earthquake magnitude, representative distance, focal depth and soil type), and response spectrum and time-frequency characteristics of the anticipated main shock motion are considered, and corresponding main shock and aftershock ground motions are generated. The damage analysis shows that short-period structures are more vulnerable to the aftershock-induced damage compared to the long-period structures, unless the main shock motion has a significant long period component or is associated with relatively shallower aftershocks. Further, the severity of aftershock events (in comparison with that of the main shock event) increases significantly for the longer main shock and shallower aftershock events, and for the high-frequency main shock motions in the case of short-period structures. It is also found that the relative aftershock severity can be estimated in terms of the relative ground motion parameters like the ratios of duration and spectral ordinates for the aftershock to those for the main shock.