This paper investigates the dynamic characteristics of a beam axially moving over multiple elastic supports. The spectral element matrix is derived first for the axially moving beam element and then it is used to formulate the spectral element matrix for the moving beam element with an interim elastic support. The moving speed dependance of the eigenvalues is numerically investigated by varying the applied axial tension and the stiffness of the elastic supports. Numerical results show that the fundamental eigenvalue vanishes first at the critical moving speed to generate the static instability.