Keywords: moving loads, train-bridge interaction, high speed bridges, resonance in railway bridges, ballast instability, similarity formulas.

Train-bridge interaction effects are of great importance for the simulation of the dynamic behaviour of high-speed railway bridges. The usual Moving Loads Models are fast and require short computer times, but sometimes overestimate the maximum displacements and accelerations of the deck, which surpass those computed by means of Interaction Models in as much as 30% or 40%. These pessimistic predictions of the Moving Loads Models occur in a significant number of situations, mainly in medium and short simply supported bridges. This kind of behaviour arises from the fact that train-bridge interaction effects become relevant typically in resonance situations, which are much more likely to happen in simply supported structures. Out of resonance the response of the bridge is not governed by the cumulative effects of successive loads but by the passage of the heaviest axles, the response predicted by the Moving Loads Models and the Interaction Models resulting very similar in such conditions.

In this paper an analysis of the influence of several fundamental parameters on the reduction of the response caused by train-bridge interaction is presented.

First, a dimensionless formulation of the equations of motion of the Simplified Interaction Model is derived, from which seven different kinds of fundamental parameters can be identified. These parameters govern the dynamic behaviour of the train-bridge system and are the following: dimensionless speed, frequency ratio, mass ratio, vehicle mass ratio, structural damping, vehicle suspension damping and, last, the ratio, where symbolises the span of the bridge and the characteristic distance between repeated groups of loads. In addition to structural damping, which directly influences the overall response, the most influential parameters are the frequency ratio (quotient between the primary suspension frequency and structural fundamental frequency), mass ratio (quotient between the semi-sprung mass and the total mass of the bridge) and the ratio.

The proposed dimensionless formulation provides a mathematical basis for the Generalized Similarity Formulas, which establish a relation between the dynamic responses of two similar systems (i.e., two train-bridge systems that possess the same values of the fundamental parameters).

Second, a parametric study is presented which purpose is analysing the influence of the most relevant among the fundamental parameters defined previously. Realistic ranges of variation for the fundamental parameters are defined, and the reduction of the response due to train-bridge interaction evaluated identifying the basic trends caused by such variations. The most important conclusions that can be extracted from the studies presented in the paper are the following:

1. Train-bridge interaction effects can be beneficial in a significant number of situations, mainly when the behaviour of medium-short span bridges at resonance is of concern.
2. The reduction of the response that the more sophisticated Interaction Models predict with respect to the Moving Loads Models at resonance is identical in similar systems.
3. The maximum reduction of the response is obtained for a particular value of the frequency ratio, this value becoming smaller and approaching unity as the ratio increases.
4. The reduction of the response always increases with the mass ratio.
5. The reductions of displacements and accelerations are very similar provided that the ratio is equal to unity or smaller. For greater values of this parameter the reduction of the accelerations exceeds the reduction of the displacements.

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