During　the　service　life　of　civil　infrastructures,　their　performance,　functionality,　and　serviceability　are　essentially　time-dependent　reliability　problems　as　both　structural　resistance　and　load　effect　are　related　with　time.　With　respect　to　the　practical　engineering,　the　performance　function　is　usually　complicated,　multi-dimensional　and　implicit　and　involves　both　correlated　and　non-correlated　random　variables;　thus　the　computational　process　of　time-dependent　reliability　analysis　is　relatively　challenging.　To　address　this　issue,　the　time-dependent　failure　probability　of　structures　with　complicated,　multi-dimensional　and　implicit　performance　functions　involving　correlated　random　variables　is　mathematically　formulated　in　this　paper.　A　novel　and　efficient　approach　is　then　proposed　to　compute　the　time-dependent　failure　probability.　In　the　proposed　method,　the　integral　of　time-dependent　failure　probability　with　respect　to　time　domain　and　random-variate　space　are　estimated　by　Gauss-Legendre　quadrature　and　point-estimate　method　based　on　bivariate　dimension-reduction　integration,　respectively.　Accordingly,　the　time-dependent　probability　of　failure　within　a　given　period　is　decomposed　into　a　series　of　probabilities　of　failure　with　time-dependent　random　variables　at　specified　time　points　conditioned　on　the　estimating　points　of　the　time-independent　random　variables,　which　can　be　evaluated　from　state-of-the-art　techniques　for　reliability　assessment.　The　efficiency　and　accuracy　of　the　proposed　method　are　demonstrated　by　three　numerical　examples　with　either　explicit　or　implicit　performance　functions　involving　correlated　random　variables,　in　which　Monte　Carlo　simulations　are　employed　for　comparison.