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Flexural propagation behavior of a metamaterial beam with circular membrane-mass structures is presented. Each cell is comprised of a base structure containing circular cavities filled by an elastic membrane with a centrally loaded mass. Numerical results show that there exist two kinds of bandgaps in such a system. One is called Bragg bandgap caused by structural periodicity; the other is called locally resonant (LR) bandgap caused by the resonant behavior of substructures. By altering the properties of the membrane-mass structure, the location of the resonant-type bandgap can be easily tuned. An analytical model is proposed to predict the lowest bandgap location. A good agreement is seen between the theoretical results and finite element (FE) results. Frequencies with negative mass density lie in the resonant-type bandgap.