This paper studies a dual-hop multiple-access relay network where two independent source nodes transmit information to a common destination node with the aid of multiple single-antenna amplify-and-forward relays. Each relay node is subject to an individual power constraint. We focus on the design of distributed beamforming schemes for the relays to support the transmission rate requirements of the two sources. To this end, we first characterize the achievable rate region for this network via solving a sequence of corner point optimization problems proposed in this paper. We also develop several low-complexity suboptimal schemes in closed form. Two inner bounds of the achievable rate region are theoretically shown to be approximately optimal in two special scenarios. Finally, numerical results demonstrate the effectiveness of our proposed approaches.