This paper proposes reliability and availability models for ageing systems subject to failures detected on-line and (hidden) failures only revealed by proof tests, such as safety-related systems. Both types of failures are modelled by Weibull distributions with three parameters: shape, scale, and (virtual) age. For on-line detected failures (e.g. self-revealed or quickly detected by frequent automatic diagnostic tests), a corrective maintenance is assumed with a constant repair rate. For the other failures, a preventive maintenance consists in periodic proof tests that reveal the failures and, if applicable, a corrective maintenance is also performed. While the elements are assumed as bad as old (ABAO) after corrective maintenance, arithmetic reduction of age (ARA) with memory is considered for preventive maintenance. Analytical formulas are proposed for the time-dependent (virtual) age, failure rate, and unavailability of an element, for each type of failure. These formulas are implemented in a fault tree analysis tool for application to different architectures of systems with elements subject to different types of failures. For safety-related systems, results show that the use of the proposed models may prevent underestimating the unavailability of ageing systems (and therefore the risks) and enables definition of more appropriate maintenance policies.