We consider a quadratic nonlinear Bernoulli-Euler beam equation supported by a distributed-spring elastic (Winkler) foundation, which is a simple model for flexural waves on an ice sheet floating on water. The linear dispersion relation of the quadratic nonlinear beam equation features a minimum at a nonzero wavenumber. Below this minimum, there exist two kinds of solitary wave solutions; elevation and depression solitary waves, which are numerically computed by a modified Petviashvili method. By assuming slowly varying envelope solutions in the small-amplitude weakly nonlinear limit, it is predicted that elevation solitary waves are unstable and depression solitary waves are stable, which agrees with the existing stability analysis. In numerical computations, however, for given initial perturbations, depression solitary waves turn out to be unstable in that they show periodic breathing behaviors, i.e., breather solutions. Furthermore, based on the Poincaré-Lindstedt method, it is numerically found that the period of a breather solution decreases as the magnitude of the initial perturbation increases. This work was supported by National Research Foundation of Korea (NRF-2023R1A2C1003600).