A normal mode approach is used to model the behaviour of a linearly-damped, elastic, ice beam floating on a fluid foundation and subjected to a random distributed loading. As an example, two loading regimes are considered to act on the Erebus Glacier Tongue, McMurdo Sound: broad bandwidth (‘white noise’) loading, and an ocean wave-type pressure distribution beneath the tongue. For white noise input, the root mean square (rms) deflexion is found to good accuracy within the first few modes, but the rms bending moment increases with the number of modes included in the summation due to the unlimited frequency content of the forcing. Solutions for the rms deflexion and bending moments quickly converge to their mathematical limit after six modes when the forcing is due to ocean waves. A local maximum in rms bending moment near the end of the beam confirms that waves may be important as a mechanism for iceberg calving.