In this paper, we take all users' secrecy rate requirements into consideration and propose a rate-splitting multiple access (RSMA)-based secure beamforming approach to maximize the weighted sum-rate (WSR), where multi-user interference (MUI) is partially decoded and partially treated as noise. User messages are split and encoded into common and private streams. Each user not only decodes the common stream and the intended private stream, but also eavesdrops other users' private streams. To solve the introduced non-convex security constraints in the formulated problem, a successive convex approximation (SCA)-based approach is first proposed to maximize the instantaneous WSR under perfect channel state information at the transmitter (CSIT). When only imperfect CSIT is available, to ensure robustness, a weighted ergodic sum-rate maximization problem is formulated. Then, a joint weighted minimum mean square error and SCA-based alternating optimization algorithm is proposed to optimize the precoder. Numerical results demonstrate that, compared with baseline schemes, the performance advantage of the proposed RSMA-based secure beamforming design has been shown in terms of WSR and robustness to channel errors while ensuring all users' security requirements thanks to its powerful interference management capability.