Abstract : We study a generic class of models for ultra-high energy cosmic ray (UHECR) phenomenology, in which the sources accelerate protons and nuclei with a power-law spectrum having the same index, but with different values for the maximum proton energies, distributed according to a power-law. We show that, for energies sufficiently lower than the maximum proton energy, such models are equivalent to single-type source models, with a larger effective power law index and a heavier composition at the source. We calculate the resulting enhancement of the abundance of nuclei, and find typical values of a factor 2-10 for Fe nuclei. At the highest energies, the heavy nuclei enhancement ratios become larger, and the granularity of the sources must also be taken into account. We conclude that the effect of a distribution of maximum energies among sources must be considered in order to understand both the energy spectrum and the composition of UHECRs, as measured on Earth.