The present study proposes a simple and relatively complete displacement field, which, with the finite element formulation, can be employed in the analyses of the free vibration of circular and annular plates, the vibration with preloading in the radial direction, which is studied quite often, and the vibration with preloading under twisting, which has not yet appeared in the literature. The present results of the free vibration and the vibration under compressive stresses for circular and annular plates with different boundary conditions and various aspect ratios will be compared with those by ANSYS and other research. From the comparisons, we know that the present method does not need a fine mesh to get accurate natural frequencies of the plates. As to the vibration for the bar or tube with pretorsional loading, coupled vibration modes in the r-z plane and £c direction, it can be revealed by the present method, and the appearance of these vibration modes may be due to the coupling terms from the proposed displacement field. Also, these vibration modes have not been shown before.