Keywords: high-speed railway bridges, resonance phenomena, fluid viscous dampers, moving loads, torsional oscillations, orthotropic plates.

The dynamic performance of railway bridges due to the passing of high-speed trains has become an issue of main concern for many scientists and engineers in recent last decades. The main reason is the extensive construction of new high-speed lines and the use of old lines for higher operating train velocities. Higher design velocities may lead to resonant phenomena, especially in short simply supported bridges. In a railway bridge, resonance occurs when the time interval between two axles of the train passing over a certain section of the bridge is a multiple of one of its natural periods. As the train velocity approaches the resonant one, a dynamic amplification of the structural response may be expected and in particular inadmissible vertical accelerations occur on the bridge deck that may cause passenger discomfort, a reduction of the service life of the bridge, ballast deconsolidation, and the subsequent risk of derailment. This kind of behaviour has been reported by some members of the D-214 Committee of the European Rail Research Institute [1,2].

Therefore, it becomes essential to reduce the excessive vibration in such bridges from the passage of trains. For this purpose, the possibility of increasing the overall structure damping with passive energy dissipation devices is evaluated. In particular, the retrofit of railway bridges with linear fluid viscous dampers (FVD) to be installed underneath the bridge deck by means of an auxiliary steel structure was already proposed by the authors [3]. As it was demonstrated, the inclusion of these devices enabled the structure to dissipate enough energy to reduce the bridge dynamic amplification below acceptable performance levels. The authors' past publications regarding this issue were based on two-dimensional models of the bridge. Therefore, only the flexural modes contributions were accounted for and conclusions were limited to non-skewed single-track bridges. Under these circumstances, a three dimensional study has been performed in order to realistically prove the proposed retrofitting system efficiency when three-dimensional modes of the deck may not be neglected. This is typical in bridges with close flexural-torsional first natural frequencies, under eccentric traffic or in skewed configurations. As short simply supported bridges (12m-25m span) may experience quite high deck vertical accelerations under resonance conditions, and since these structures are built in many practical situations by means of prestressed concrete slabs, T-beam decks and slabs stiffened with ribs and other topologies where the dynamic behaviour is quite close to orthotropic, an orthotropic plate model has been programmed based on the linear varying curvature triangular plate finite element (Felippa [4], Clough and Tocher [5]) and is used for this purpose.

The retrofitted bridge is firstly analysed under a sinusoidal excitation in order to capture the variables governing the resonant behaviour. The optimal parameters for the dampers which minimise the bridge dynamic response governed by individual modes of vibration are obtained in closed form as well. Afterwards the adequacy of these optimal expressions to real bridges subjected to railway traffic is shown over a wide range of velocities.

From the investigations presented here it is concluded that (i) the resonant vibrations that may appear in simply supported bridges when subjected to moving loads can be drastically reduced with the retrofit design proposed herein; (ii) for a particular auxiliary beam, there is an optimum value of the FVD constants that minimises the bridge response associated with an individual modal contribution; the optimal constraints for minimising the displacement are slightly different from the optimal ones for minimising the acceleration; (iii) the retrofit strategy should be to select the smallest auxiliary beam such that, when associated with its corresponding optimal FVDs, leads to the desired system performance; (iv) analytical expressions for the optimal damper constants are provided which lead to quite accurate results as long as the maximum response of the bridge is in the range of evaluated velocities that occur at resonance.

1

L. Frýba, "Dynamic behaviour of bridges due to high-speed trains", Workshop Bridges for High-Speed Railways, Porto, 137-158, 2004.

2

F. Mancel, "Cedypia: Analytical software for calculating dynamic effects on railway bridges", Proceedings of the Fourth European Conference on Structural Dynamics 2, 675-680, 1999.

3

M.D. Martinez-Rodrigo, P. Museros, "A numerical assessment of the use of fluid viscous dampers to reduce the resonance response of high-speed railway bridges", Proceedings of the Tenth International Conference on Civil, Structural and Environmental Engineering Computing, Ed. B.H.V. Topping, Civil-Conmp Press, Stirling, UK, 2005. doi:10.4203/ccp.81.238

4

C.A. Felippa, "Refined finite element analysis of linear and nonlinear two-dimensional structures", PhD Thesis, University of California Berkeley, 1966.

5

R.W. Clough, J.L. Tocher, "Finite element stiffness matrices for analysis of plate bending", Proceedings of the Conference on Matrix Methods in Structural Mechanics, AFFDL-TR66-80, 515-545, 1966.

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