© 2020, Korean Society of Civil Engineers. In this paper, an approach is proposed to obtain fragility curves of reinforced concrete structures which can accumulate damage due to inelastic behavior when they are exposed to seismic load sequences in a given instant of time. The cumulative damage caused by earthquakes is quantified considering the maximum drift. This study is mainly focused on the stochastic uncertainty and variability of seismic occurrence, which are considered the main sources that can influence the performance of the structural system. The approach is validated with the help of a numerical study using a reinforced concrete continuous bridge considering the effect of cumulative damage given by seismic sequences for different time instants of 0, 50, 100, 150 and 200 years, after bridge construction. The bridge is located in the transition part of the soil in Mexico City. Different drift thresholds are studied to evaluate the probability of being exceeded. The proposed approach demonstrates the importance of considering the effect of cumulative damage under seismic sequences. In addition, it is assumed that it could be possible to design bridge structures that develop a certain pre-established drift threshold.