The a priori multi-expansion modal (MEM) hyper-reduction method for nonlinear structural dynamics finite element (FE) problems is described, extended, and applied to the dynamic nonlinear FE simulation of a car tire rolling over a rough road surface, including geometrical, material, follower force, and contact nonlinearities. Rather than using time-domain training simulation data, a priori calculated static nonlinear contact configurations and eigenmodes are used as a reduction basis and to perform the hyper-reduction element sampling. The hyper-reduction element sampling is performed by solving an L1 optimization problem subject to a set of equality constraints. This yields a reduced set of elements with an a priori known cardinality, which depends on the amount of constraints taken into consideration and the reduction basis dimension. It is shown that care has to be taken during the hyper-reduction process when considering distributed contact constraints, as is the case for, e.g., a tire rolling over a rough road surface. Large speedup factors can be obtained while still retaining a relatively high accuracy, making application of the MEM method suitable to, for instance, industrial design optimization cases.