In modern engineering, physical processes are modelled using advanced computer simulation tools (e.g. finite element models) to assess and optimize their performance. Moreover, awareness is growing on concepts like structural reliability and robust design, hence making the efficient quantification and propagation of uncertainties a key challenge. A major part of those analyses is the characterization of uncertainty in the input, which is typically done with probabilistic variables. In the case of sparse data sets, however, probability theory is often not the optimal choice. Imprecise probabilities provide a more general framework accounting for both aleatory and epistemic uncertainties. The uncertainty propagation of imprecise probabilities leads to imprecise responses. In this context, an algorithm for solving imprecise structural reliability problems is presented. The algorithm transforms the imprecise problem into two precise structural reliability problems, which opens up possibilities for using traditional structural reliability analyses techniques. An adaptive experimental design algorithm based on Polynomial-Chaos-Kriging is used to efficiently estimate imprecise failure probabilities. The capabilities of the framework are illustrated through an application using an analytical function and a realistic engineering problem setting.