For　uncertain　structures　with　the　coexisting　random　and　interval　inputs,　effectively　estimating　the　lower　and　upper　bounds　of　failure　probability　is　always　a　challenge.　To　address　this　issue,　this　paper　first　proposes　a　two-stage　adaptive　radial-based　importance　sampling　(TARBIS)　method,　where　two　optimal　spheres　are　sought　successively　in　two　stages　to　estimate　the　bounds　of　failure　probability.　Then,　by　replacing　the　true　limit　state　function　using　the　Kriging　model,　a　Kriging-assisted　TARBIS　(K-TARBIS)　is　further　developed　to　improve　the　computational　efficiency.　In　the　first　stage,　the　training　points　mostly　contributing　to　the　estimation　of　two　bounds　of　failure　probability　are　identified　by　a　system　reliability　theory-based　U　(SYSU)　learning　function　to　update　the　Kriging　model.　In　the　second　stage,　the　Kriging　model　is　updated　only　on　sample　points　contributing　to　the　estimation　of　the　upper　bound　of　failure　probability.　Throughout　the　active　learning　process,　the　Kriging　model　is　sequentially　updated　in　a　series　of　small　sub-candidate　sample　pools　of　TARBIS,　which　greatly　reduces　the　computational　cost.　The　accuracy　and　efficiency　of　the　proposed　method　are　demonstrated　through　four　representative　examples.　©　2023,　The　Author(s),　under　exclusive　licence　to　Springer-Verlag　GmbH　Germany,　part　of　Springer　Nature.