The rapid advancement of the next generation of mobile communication technologies has set increasingly stringent demands on transmission quality, capacity, and rate. In this paper, we discuss the weighted sum rate (WSR) using intelligent reflecting surface (IRS) assisted rate-splitting multiple access (RSMA) for downlink multiuser multiple-input single-output (MISO) systems with consideration of phase quantization of IRS. The aim is to propose a scheme to obtain the maximum WSR by optimizing the equivalent beamforming matrix, quantized IRS phase shift matrix, and the common rate splitting scheme. To this end, an equivalent optimization scheme is designed on the basis of weighted minimum mean square error (WMMSE) algorithm, difference of convex sets (DCS) theory, Lagrange penalty and Taylor approximation, where the non-convex mixed continuous and discrete optimization problem is converted to three continuous convex optimization sub-problems and then worked out iteratively with the alternating optimization algorithm. We compare the maximum WSR achievable by quantized IRS-aided 1-layer-RSMA with three other multiple access (MA) technologies: dirty paper coding (DPC), non-orthogonal multiple access (NOMA), and multiuser linear precoding (MULP) under different signal-to-noise ratio (SNR) scenarios. It is also found that the proposed method provides an improvement in WSR with the increase of the number of base station antennas, IRS reflective elements and phase quantization bits.