The multistable mechanical metamaterials have the multiple stable status and geometric reconstruction characteristics, which are of great research value in deformation energy absorption and tuning of the band gap. In this study, a multi-stable mechanical metamaterial is proposed,which is composed of bistable curved rods and related solid support structures. Combining the finite element method and Bloch’s theorem, the band structure of the structure is calculated, the band gap characteristics of the structure with different steadystates are analyzed, the effects of geometric parameters on the band gap are obtained, and the transmissions of the array composed of 5×5 unit cells are studied when part of the cells are deformed. It is found that the deformation status of the bistable connecting rod can significantly change the band gap of the structure, especially in the low-frequency range. The elastic wave propagation can be attenuated within the corresponding band gap frequency range by deformation of a row of cells in the structure.