A general case of an axially-loaded beam with ends elastically restrained against rotation is analyzed for either flexural or torsional vibration. The general frequency equation is numerically evaluated for various end fixities. The results are graphically presented for the first three modes and are of interest in bridge and machine designs. It is believed that this is the most general case analyzed for the end conditions as stated. Many of the other end conditions in the literature can be obtained by degenerating the frequency equation into special cases.