Current seismic codes specify design earthquake loads as single events. The structure, however, may experience multiple ground accelerations in a short period of time. The evidence from recent earthquakes confirms this scenario. For instance, the 2004 Niigata earthquake consisted of two acceleration sequences. An earthquake of repeated sequences can cause more damage to the structure than a single ordinary event, due to the accumulation of inelastic deformations. However, information on repeated acceleration sequences is currently limited. This paper proposes a simple stochastic model for representing repeated acceleration sequences. Subsequently, the model is used in investigating the response of nonlinear single-degree-of-freedom (SDOF) structures to random earthquakes of repeated sequences. The ground acceleration is represented as a stationary Gaussian random process modulated by an envelope function of repeated character. The structural response is quantified in terms of the input and hysteretic energies, ductility demand, damage indices and failure probability. Numerical demonstrations of the response of nonlinear SDOF systems to acceleration sequences are provided.