The column-to-beam flexural strength ratio (CBFSR) has been used in many seismic codes to achieve the strong column-weak beam (SCWB) failure mode in reinforced concrete (RC) frames, in which plastic hinges appear earlier in beams than in columns. However, seismic investigations show that the required limit of CBFSR in seismic codes usually cannot achieve the SCWB failure mode under strong earthquakes. This study investigates the failure modes of RC frames with different CBFSRs. Nine typical three-story RC frame models with different CBFSRs are designed in accordance with Chinese seismic codes. The seismic responses and failure modes of the frames are investigated through time-history analyses using 100 ground motion records. The results show that the required limit of the CBFSR that guarantees the SCWB failure mode depends on the beam-column connection type and the seismic intensity, and different types of beam-column connections exhibit different failure modes even though they are designed with the same CBFSR. Recommended CBFSRs are proposed for achieving the designed SCWB failure mode for different types of connections in RC frames under different seismic intensities. These results may provide some reference for further revisions of the SCWB design criterion in Chinese seismic codes.