Curved-beams　can　be　used　to　design　modular　multistable　metamaterials　(MMMs)　with　reprogrammable　material　properties,　i.e.,　programmable　curved-beam　periodic　structure　(PCBPS),　which　is　promising　for　controlling　the　elastic　wave　propagation.　The　PCBPS　is　theoretically　equivalent　to　a　spring-oscillator　system　to　investigate　the　mechanism　of　bandgap,　analyze　the　wave　propagation　mechanisms,　and　further　form　its　geometrical　and　physical　criteria　for　tuning　the　elastic　wave　propagation.　With　the　equivalent　model,　we　calculate　the　analytical　solutions　of　the　dispersion　relations　to　demonstrate　its　adjustability,　and　investigate　the　wave　propagation　characteristics　through　the　PCBPS.　To　validate　the　equivalent　system,　the　finite　element　method　(FEM)　is　employed.　It　is　revealed　that　the　bandgaps　of　the　PCBPS　can　be　turned　on-and-off　and　shifted　by　varying　its　physical　and　geometrical　characteristics.　The　findings　are　highly　promising　for　advancing　the　practical　application　of　periodic　structures　in　wave　insulation　and　propagation　control.