Additional　damping　ratio　(ADR),　which　plays　a　significant　role　in　damped　structure　design,　is　broadly　utilized　to　evaluate　damping　effect　of　the　dampers.　In　current　method,　the　ratio　of　energy　dissipated　by　the　dampers　in　one　complete　cycle　to　maximum　strain　energy　of　the　structural　system　is　adopted　as　the　value　of　ADR.　In　the　computation　of　damper　energy　dissipation,　the　bilinear　model　is　commonly　used　to　characterize　hysteresis　behavior　of　displacement-dependent　dampers　such　as　BRB.　However,　the　accuracy　of　the　bilinear　model　is　not　precise　enough.　In　the　light　of　the　Bouc-Wen　model　can　accurately　characterize　hysteresis　behavior　of　various　dampers,　the　analytical　solution　of　hysteresis　energy　is　derived.　The　ADR　calculation　methods　based　on　the　strain　energy　and　the　energy　dissipated　by　the　inherent　damping　of　the　main　structure　are　proposed,　respectively.　The　bi-objective　design　method　of　the　RC　frame　with　BRBs　is　put　forward.　The　damping　effect　of　the　dampers　is　included　in　the　design　objectives,　which　makes　up　the　deficiency　of　single　objectives　with　the　story　drift　ratio.　The　design　objectives　are　that　the　story　drift　ratio　of　the　frame　does　not　exceed　the　limit,　and　the　damping　ratio　reaches　the　expected　value.　The　layout　and　force　of　the　dampers　are　taken　as　design　variables.　Based　on　the　finite　element　method,　a　RC　frame　with　BRBs　is　designed　iteratively.　The　reasonable　time　period　length　of　time-varying　energy　is　determined.　Meanwhile,　the　strain　energy　method　and　modal　damping　energy　dissipation　method　are　compared　and　analyzed.　As　a　result,　the　calculation　result　of　strain　energy　method　is　larger　than　that　of　modal　damping　energy　dissipation　method.　The　strain　energy　method　is　hazardous　to　structural　safety,　and　the　modal　damping　energy　dissipation　method　is　suggested.　The　bi-objective　design　method　is　effective,　which　can　further　improve　the　structural　performance.