[Abridged] Context: Radiation-driven mass loss plays a key role in the life-cycles of massive stars. However, basic predictions of such mass loss still suffer from significant quantitative uncertainties. Aims: We develop new radiation-driven, steady-state wind models for massive stars with hot surfaces, suitable for quantitative predictions of global parameters like mass-loss and wind-momentum rates. Methods: The simulations presented here are based on a self-consistent, iterative grid-solution to the spherically symmetric, steady-state equation of motion, using full NLTE radiative transfer solutions in the co-moving frame to derive the radiative acceleration. We do not rely on any distribution functions or parametrization for computation of the line force responsible for the wind driving. Results: In this first paper, we present models representing two prototypical O-stars in the Galaxy, one with a higher stellar mass M/Msun=59 and luminosity log10(L/Lsun)= 5.87 and one with a lower M/Msun= 27 and log10(L/Lsun)= 5.1. For these simulations, basic predictions for global mass-loss rates and velocity laws are given, and the influence from additional parameters like wind clumping and microturbulent speeds discussed. A key result is that our mass-loss rates are significantly lower than those predicted by the mass-loss recipes normally included in models of massive-star evolution. Conclusions: Our results support previous suggestions that Galactic O-star mass-loss rates may be overestimated in present-day stellar evolution models, and that new rates thus might be needed. Indeed, future papers in this series will incorporate our new models into such simulations of stellar evolution, extending the very first simulations presented here toward larger grids covering a range of metallicities, B supergiants across the bistability jump, and possibly also Wolf-Rayet stars.
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