In　time-dependent　structural　reliability　analysis　it　is　not　always　the　case　that　the　limit　state　functions　are　continuous　and　differentiable.　A　commonly　encountered　case　is　the　structures　subjected　to　instantaneous　loads　with　high　intensity,　such　as　seismic　actions.　This　paper　intends　to　propose　an　efficient　method　for　time-dependent　reliability　analysis　of　structures　with　non-differentiable　limit　state　functions.　A　new　formulation　is　proposed　for　outcrossing　rate　based　on　the　minimum　value　of　limit　state　functions　at　the　time　of　outcrossing.　The　developed　method　also　considers　the　degradation　of　structural　resistance.　It　is　found　in　the　paper　that　the　instantaneous　discrete　load　with　high　intensity　can　induce　non-differentiable　limit　state　functions　and　that　the　probabilistic　information　of　such　instantaneous　loads　significantly　affects　the　time-dependent　failure　probabilities.　The　paper　concludes　that　the　proposed　method　can　predict　the　time-dependent　failure　probability　of　structures　for　non-differentiable　limit　state　functions　accurately　and　efficiently.　The　proposed　method　contributes　to　the　body　of　knowledge　of　time-dependent　reliability　with　wider　practical　applications.　©　2025　Elsevier　Ltd