In this paper, a closed-form approximate formula for estimating the maximum resonant response of beam bridges on viscoelastic supports (VS) under moving loads is proposed. The methodology is based on the discrete approximation of the fundamental vertical mode of a non-proportionally damped Bernoulli-Euler beam, which allows the derivation of closed-form expressions for the fundamental modal characteristics and maximum amplitude of free vibration at the mid-span of VS beams. Finally, an approximate formula to estimate maximum resonant acceleration of VS beams under passage of articulated trains has been proposed. Verification studies prove that the approximate closed-form formula estimates the resonant peaks with good accuracy and is a useful tool for preliminary assessment of railway beam bridges considering the effect of soil-structure interaction at resonance. In combination with the use of full train signatures through the Residual Influence Line (LIR) method, the proposed solution yields good results also in the lower range of speeds, where resonant sub-harmonics are more intensely reduced by damping.