The present study deals with the hydroelastic vibration analysis of structures in contact with fluid via coupled fluid-structure interaction (FSI) embedded with a finite element method (FEM) such that a structure displacement formulation is coupled with a fluid pressure-displacement formulation. For the preliminary study and validation of FEM based coupled FSI analysis, hydroelastic vibration characteristics of a rectangular plate in contact with fluid are first compared with the elastic vibration in terms of boundary condition and mode frequency. Numerical results from coupled FSI analysis have been shown to be rational and accurate, compared to energy method based theoretical solutions and experimental results. The effect of free surface on the vibration mode is numerically studied by changing the submerged depth of a rectangular plate. As a practical application, the hull structural vibration of 4,000 twenty-foot equivalent units (TEU) container ship is considered. Hydroelastic results of the ship hull structure are compared with those obtained from the elastic condition.