Time-variant reliability analysis aims at assessing the probability of failure of a time-variant system within a given time horizon. We illustrate in this paper the computation of the reliability kernel which is the set of initial states for which the probability of failure remains under a threshold within the considered time horizon. This paper supposes that the time-variant system is discrete in time and space with given probabilities of transition between space states. We use a recursive relation for computing the cumulative probability of failure of the system, linking the probability of failure at time t with the probability of being at a given state x (for all possible states) at time t ' 1. Applying this relation, it is possible to compute the probability of failure at any starting point in the state space and hence to derive the reliability kernel. The computation of this kernel gives informations about the system which can be further helpful in reliable design. The approach is illustrated on an example of a steel beam under corrosion.