Nonlinear numerical simulations on the interaction of waves with a vertical elastic plate are performed by a newly developed computer code based on the MEL (Mixed Eulerian Lagrangian) method. The time-dependent elastic deflection of a vertical plate is described by the mode-expansion method. An implicit boundary-condition method is developed to calculate the acceleration of the plate without evaluating the hydrodynamic force acting on it. The 4-th order Runge-Kutta scheme with uniform time step is applied for time marching in solving the coupled motions of an elastic plate and the fluid. A hybrid wave absorbing beach is installed to prevent the wave reflection from the end of the wave tank. With an initial condition that a plate is bent, illustrative results of numerical computation for the elevation of the free surface are compared with corresponding linear analytical solution. The hydroelastic vibration of a vertical plate induced by an initial pulse-type elevation of the free surface is also simulated. Finally, the effect of the plate stiffness and the edge condition at the top of plate on hydroelastic behavior is studied systematically.