A closed-form solution to a viscoelastically supported Timoshenko beam under harmonic line load

This study aims to formulate a closed-form solution to a viscoelastically supported Timoshenko beam under a harmonic line load. The differential governing equations of motion are converted into algebraic equations by assuming the deflection and rotation of the beam in harmonic forms with respect to time and space. The characteristic equation is biquadratic and thus contains 14 explicit roots. These roots are then substituted into Cauchy's residue theorem; consequently, five forms of the closed-form solution are generated. The present solution is consistent with that of an Euler-Bernoulli beam on a Winkler foundation, which is a special case of the present problem. The current solution is also verified through numerical examples.