Catalytic ammonia synthesis is believed to proceed via dissociation of N-2 and H-2 with subsequent stepwise addition reactions from an adsorbed nitrogen atom to NH3. The first step, N-2 dissociation, has been thoroughly studied. However, little is known about the microscopic details of the stepwise addition reactions. To shed light on these stepwise addition reactions, density functional theory calculations with the generalized gradient approximation are employed to investigate NHx (x=1,3) formation on Ru(0001). Transition states and reaction barriers are determined in each elementary step. It is found that the reaction barriers for stepwise addition reactions are rather high, for example, the barrier for NH hydrogenation is calculated to be 1.28 eV, which is comparable with that of N-2 dissociation. In addition, one of the stepwise addition reactions on a stepped surface is also considered. The reaction barrier is found to be much higher than that of N-2 dissociation on the same stepped surface, which indicates the importance of stepwise addition reactions in ammonia synthesis.