On the provenance of hinged-hinged frequencies in Timoshenko beam theory

•Timoshenko theory is modified to allow for a tensile, zero or compressive static axial load.•Both Timoshenko spectra can be determined from a single, exact quadratic equation.•‘Precise’ Timoshenko theory can be used to eliminate the possibility of a second spectrum.•Every Bernoulli-Euler frequency maps uniquely to a single Timoshenko frequency.•One of these frequencies is mapped to the cut-off frequency. Its modal number can be found a priori.•Each Timoshenko frequency corresponds to a single independent mode shape. An exact differential equation governing the motion of an axially loaded Timoshenko beam supported on a two parameter, distributed foundation is presented. Attention is initially focused on establishing the provenance of those Timoshenko frequencies generated from the hinged-hinged case, both with and without the foundation being present. The latter option then enables an exact, neo-classical assessment of the ‘so called’ two frequency spectra, together with their corresponding modal vectors, to be undertaken when zero, tensile or compressive static axial loads are present in the member. An alternative, ‘precise’ approach, that models Timoshenko theory efficiently, but eliminates the possibility of a second spectrum, is then described and used to confirm the original eigenvalues. This leads to a definitive conclusion regarding the structure of the Timoshenko spectrum. The ‘precise’ technique is subsequently extended to allow, either the full foundation to be incorporated, or either of its component parts individually. An illustrative example from the literature is solved to confirm the accuracy of the approach, the nature of the Timoshenko spectrum and a wider indication of the effects that a distributed foundation can have.