The dynamic response of eccentrically prestressed viscoelastic Timoshenko beams under a moving harmonic load is studied by using Lagrange equations. In the study, for using the Lagrange equations, trial functions denoting the deflection of the beam and the rotation of the cross-sections are expressed in polynomial forms. The constraint conditions of supports are taken into account by using Lagrange multipliers. The effects of the value of the eccentricity of the compressive load, the excitation frequency, the constant velocity of the transverse moving harmonic load and viscous damping of the material of beams are studied in detail. Convergence studies are made. The validity of the obtained results is demonstrated by comparing them with exact solutions based on the Euler-Bernoulli beam theory obtained for the special cases of the investigated problem. (c) 2006 Elsevier Ltd. All rights reserved.