In　order　to　investigate　the　propagation　and　isolation　mechanisms　of　elastic　waves,　we　propose　a　programmable　curved　beam　periodic　structure　(PCBPS).　The　PCBPS　is　assembled　by　a　unit　cell　containing　bistable　curved　beams,　which　is　equivalent　to　a　spring　oscillator　system,　and　is　used　to　analyze　the　principle　of　bandgap　formation.　We　employ　the　finite　element　method　(FEM)　and　theoretical　analysis　to　validate　the　proposed　equivalent　model.　By　equating　the　spring　oscillator　system,　we　compute　closed-form　solutions　to　demonstrate　the　accuracy　and　predictability　of　the　dispersion　relation.　Our　results　show　that　the　spring-oscillator　model　can　accurately　predict　the　structural　bandgap　of　PCBPS　while　obtaining　the　effect　of　the　geometrical　parameters　of　the　unit　cell　on　the　structural　bandgap.　The　ideas　presented　and　the　results　obtained　have　significant　potential　for　designing　functional　structures　and　facilitating　the　practical　application　of　periodic　structures　for　wave　insulation　and　propagation　control　in　different　frequency　ranges.　©　The　Author(s),　under　exclusive　license　to　Springer　Nature　Singapore　Pte　Ltd.　2024.