An exact spectral formulation for the wave characteristics in an infinite Timoshenko-Ehrenfest beam supported by periodic elastic foundations

Periodic structures have been widely used for vibration reduction and wave control. In the present work, an exact spectral formulation (ESF), being capable to evaluate the dispersion spectra for the bending and shear waves in an infinite Timoshenko-Ehrenfest beam supported by periodic elastic foundations, is proposed, considering the finite length and periodic interval of the distributed elastic supports. First, the frequency-domain displacement function of a unit cell is solved analytically by considering the continuity and equilibrium conditions at the interface between the supported and unsupported parts. Such displacement is formulated as the interpolation of the displacement responses at the end points of the unit cell, and in particular, a compact form is obtained by employing the Hadamard product. Based on this, the spectral matrix for the entire unit cell, involving a total number of 4 degrees of freedom, is derived from the Hamilton's principle, where the referred matrices are all explicitly given. The dispersion spectra can then be evaluated by solving the eigenvalue problem formulated from the Bloch-Floquet theorem. From a benchmark example, the bandgap characteristics indicated from the proposed ESF have been validated by comparing to the frequency response function of a spectral element model including ten unit cells. Lastly, the distributions of the bandgaps, with respect to the length, the transverse and rotational stiffnesses of the elastic foundation, are investigated based on the defined dimensionless parameters, having potential value to the optimal design of periodic supports for long-span beam structures.