Free vibrations of simply-supported beam bridges under moving loads: Maximum resonance, cancellation and resonant vertical acceleration

The advent of high-speed railways has raised many concerns regarding the behaviour of bridges. Particularly, the analysis of the free vibrations generated by each load is of great interest because they can possibly accumulate and create resonance phenomena. Regarding simply supported beams, earlier contributions showed that the free vibrations created by a single moving force are of maximum or zero amplitude (cancellation) for certain speeds. In the present paper new closed-form expressions are given for the cancellation speeds of a generic mode, as well as for the most representative points of maximum amplitude. Similar new results are provided for elastically supported beams as well. A simpler, closed-form approximate expression of the cancellation condition for an elastically supported beam is also derived from the analysis of a single passing load; this approximate formula is in good agreement with the exact results. Knowing a priori the speeds of maximum free vibrations or cancellation is of great interest for experimental tests on bridges, particularly as regards the evaluation of amplitude-dependent magnitudes such as structural damping. Regarding the resonance phenomena, if the resonance speeds coincide with either a maximum free vibration or a cancellation speed, then a maximum resonance or a cancellation of resonance will occur. The most relevant cases thereof have been investigated, and new expressions which allow predicting them for a generic mode are given. Finally, a new approximate formula is proposed for estimating the maximum acceleration of simply supported bridges caused by resonances of the fundamental mode. After extensive numerical testing, the formula has proved to be a useful tool for a first assessment of simply supported bridges according to building codes such as Eurocodes.