A weakly nonlinear fully dispersive model equation is derived which describes the propagation of waves in a thin elastic body overlying an incompressible inviscid fluid. The equation is nonlocal in the linear part, and is similar to the so-called Whitham equation which was proposed as a model for the description of wave motion at the free surface of an inviscid fluid. Steady solutions of the fully nonlinear hydro-elastic Euler equations are approximated numerically, and compared to numerical approximations to steady solutions of the fully dispersive but weakly nonlinear model equation. The bifurcation curves for these two different models are compared, and it is found that the weakly nonlinear model gives accurate predictions for waves of small to moderate amplitude. For larger amplitude waves, the two models still agree on key qualitative features such as the bifurcation points, secondary bifurcations, and the number of oscillations in a given fundamental wave period.