We study a new rate splitting (RS)-based hybrid beamforming scheme for multi-user downlink cellular networks in which the base station having a hybrid beamforming structure serves multiple users each having conventional multiple antennas. To maximize the potential of RS, we propose a general RS-enabled hybrid beamforming framework that can be applied to both fully-connected and sub-array hybrid beamforming structures, allowing for the assignment of a flexible number of streams for each user. Our proposed RS method divides each stream into an arbitrary number of common and private sub-streams, where private sub-streams are only recoverable by a dedicated user, whereas common streams can be recovered by all users. We propose a low-complexity analog and digital beamforming design suitable for the proposed RS and optimize the number of allocated common and private sub-streams for all users through a low-complexity genetic algorithm to maximize the achievable sum rate or minimum rate over multiple users. Numerical results demonstrate that the proposed scheme achieves a near-optimal sum rate close to that of dirty paper coding with low computational complexity and outperforms the benchmark approaches without considering RS.