Theoretical and computational analyses of bootstrapped tokamaks with oscillating field current drive (OFCD) have been developed using a nonlinear magnetic helicity model. Assuming a rigid current profile, the conditions for optimal tokamak steady-state operation are derived and are shown to agree with the results of computer calculations. Generalized limit formulae for the toroidal plasma beta and bootstrap current fraction are also obtained. The results of the analyses indicate that steady-state bootstrapped tokamak operation with high toroidal plasma beta can best be achieved in tight aspect ratio reactors with OFCD supplying as much as a quarter of the total toroidal plasma current.