A challenge for developing acoustic metamaterials (AMMs) is considering the application of broadband muffling and load bearing capacity simultaneously. In this paper, a honeycomb based graded AMM muffler is proposed, which can widen the attenuation band and improve the structural stiffness without any external device by means of integrated design. Firstly, the acoustic and mechanical characteristics of the muffler unit cell are theoretically and numerically studied, and the graded muffler is designed based on these characteristics. The numerical results show that the graded muffler widens the attenuation bandwidth of the unit cell, and the simulation also shows that the graded muffler has greater stiffness than the uniform one. The stiffness driven muffler provides new possibilities for the design of advanced metamaterial with simultaneous sound insulation and load bearing performances.