Abstract
Abstract
The resonances of railway bridges have often been analysed for short bridges under periodical high-speed trains, for simply supported one-span bridges, for the fundamental bridge mode, and by time-domain analyses. Many time-consuming calculations have been performed to establish simplified rules for standards. In this contribution, the passage of different (existing, new and hypothetic) trains over different (simply supported, integral, multi-span, continuous) bridges will be analysed in frequency domain by using three separated spectra with the purpose to get a better physical insight in the phenomena. At first, the excitation spectrum of the modal forces is built by the mode shape and the passage time of the train over the bridge. The second spectrum is the frequency response function of the bridge which include the modal frequency, damping and mass. The third part is the spectrum of the axle sequence of an arbitrary train which is not limited to periodical or specific (conventional, articulated, regular or standard) trains and which does not include any bridge parameters. The final solution in frequency domain is obtained as the product of these three complex, strongly varying spectra for the dominating bridge mode or in general as the sum of these products over all relevant bridge modes. The time domain solution is obtained via the inverse Fourier transform, and the resulting time histories have been successfully compared with some measurement results. The method is applied to the vertical and torsional modes of a mid-long single-span bridge on elastomeric bearings under standard train speeds, to a short two-span bridge under high-speed traffic, and to a long three-span integral bridge under long periodical freight trains. Different resonance and cancellation effects have been found for systematically varied train speeds according to the axle sequence of the whole train which is dominated by the two locomotives in that case. To be more specific, the first torsional mode of the mid-span bridge is excited for a train speed of 100 km/h whereas the second bending mode is excited for a train speed of 160 km/h. In both cases, the other mode is suppressed by the minima of the axle-distance spectra. In addition, the case of the German high-speed train ICE4, a Maglev train on a viaduct, and the very high-speed hyperloop case will be discussed briefly. In general, it is shown that resonance effects are also worth to be studied for freight and passenger trains with lower speeds.