Due to the low power consumption and low cost nature of transmissive reconfigurable intelligent surface (RIS),in this paper, we propose a downlink multi-user rate-splitting multiple access (RSMA) architecture based on the transmissive RIS transmitter, where the channel state information (CSI) is only accquired partially. We investigate the weighted sum-rate maximization problem by jointly optimizing the power, RIS transmissive coefficients and common rate allocated to each user. Due to the coupling of optimization variables, the problem is nonconvex, and it is difficult to directly obtain the optimal solution. Hence, a block coordinate descent (BCD) algorithm based on sample average approximation (SAA) and weighted minimum mean square error (WMMSE) is proposed to tackle it. Numerical results illustrate that the transmissive RIS transmitter with ratesplitting architecture has advantages over conventional space division multiple access (SDMA) and non-orthgonal multiple access (NOMA).