We construct a family of models with negative gravitational mass in the context of Newtonian gravity. We focus, in particular, on a model that reproduces the features of the so-called Dirac-Milne universe, a matter-antimatter symmetric universe that was recently proposed as an alternative cosmological scenario [A. Benoit-Lévy and G. Chardin, Astron. Astrophys. 537, A78 (2012)AAEJAF0004-636110.1051/0004-6361/201016103]. We perform one-dimensional N-body simulations of these negative-mass models for an expanding universe and study the associated formation of gravitational structures. The similarities and differences with the standard cosmological model are highlighted and discussed.