Keywords: railway bridge, running train, railway track, dynamic interaction, substructures, component-mode synthesis.

Owing to the construction of an increasing number of high-speed railways worldwide, the problem of train-bridge interaction has received much attention in the last two decades. A comprehensive review of the history and literature on this topic can be found in Reference [1]. In early studies, the bridge has been modelled as a beam-like structure and the train as a series of moving forces or moving masses. Such simplified models in some cases allow to recover closed-form solutions for the dynamic response of the bridge (see e.g. Reference [2]). However, when the riding comfort or response of rail cars are of concern, more sophisticated vehicle models comprising also the suspension units are needed. In References [3,4] an interaction element consisting of a beam element and the masses and suspension units of the car-body directly acting on it has been developed. Such element, however, accounts only partially for the track structure modelling the ballast stiffness by continuously distributed springs, while neglecting the flexural stiffness of the rail. This limitation has been partially overcome in References [5,6] by introducing a new element called bridge-track-vehicle element, which allows to investigate the interactions between a moving train and its supporting railway track and bridge structure.

This paper presents an alternative procedure based on substructure technique, which takes into account the effects of both vehicle suspension units and railway track structure. Following an approach recently proposed for analysing the vibration of railway suspension bridges [7], the train, rails and bridge deck are here regarded as three substructures. The train is idealized as a sequence of identical vehicles moving at constant speed, each of which comprises a car-body, two bogies and axles. Both the rails and bridge deck are modelled as linear elastic Bernoulli-Euler beams, while the rail bed is characterized as a Winkler elastic foundation. According to substructure technique, the equations ruling the dynamic response of the compound train-rails-bridge system are obtained by assembling the equations of motion of the three subsystems separately taken and imposing the compatibility and equilibrium conditions at the contact points between wheels and rails. In particular, by applying a variant of the component-mode synthesis method [8,9], the axle degrees-of-freedom are condensed into those of the rails in contact, and a set of reduced ordinary differential equations with time-dependent coefficients in the generalized coordinates of the bridge and rails, and physical displacements of the railway vehicles is obtained. So operating, the dynamic responses of the three subsystems can be simultaneously handled, taking into account the interaction effects. Numerical results concerning a single-span simply supported prestressed concrete bridge of length 20 m [5] are presented in the paper. The dynamic magnification factors for displacement and bending moment at bridge deck midspan are computed and compared with those reported in Reference [5], finding good agreements. By analysing an analogous bridge of length 40 m, the influence of span length on the dynamic amplification factors is also examined. Furthermore, through appropriate comparisons with the results obtained modelling the bridge as a beam-like structure and applying the "improved series" expansion proposed in Reference [10], it is shown that to obtain a realistic prediction of bending moment and shear force distributions along bridge deck, the railway track has to be accounted for.

1

Frýba, L., "Dynamics of Railway Bridges", Thomas Telford, London, 1996.

2

Yang, Y.B., Yau, J.D., Hsu, L.C., "Vibration of Simple Beams due to Trains Moving at High Speeds", Engineering Structures, 19(11), 936-944, 1997. doi:10.1016/S0141-0296(97)00001-1

3

Yang, Y.B., Yau, J.D., "Vehicle-Bridge Interaction Element for Dynamic Analysis", Journal of Structural Engineering, 123(11), 1512-1518, 1997. doi:10.1061/(ASCE)0733-9445(1997)123:11(1512)

4

Yau, J.D., Yang, Y.B., Kuo, S.R., "Impact Response of High Speed Rail Bridges and Riding Comfort of Rail Cars", Engineering Structures, 21, 836-844, 1999. doi:10.1016/S0141-0296(98)00037-6

5

Cheng, Y.S., Au, F.T.K., Cheung, Y.K., "Vibration of Railway Bridges under a Moving Train by Using Bridge-Track-Vehicle Element", Engineering Structures, 23, 1597-1606, 2001. doi:10.1016/S0141-0296(01)00058-X

6

Wu, Y.S., Yang, Y.B., "Steady-State Response and Riding Comfort of Trains Moving over a Series of Simply Supported Bridges", Engineering Structures, 25, 251-265, 2003. doi:10.1016/S0141-0296(02)00147-5

7

Biondi, B., Muscolino, G., Sofi, A., "Analysis of Dynamic Interaction between Suspension Bridges and Running Trains", Structural Dynamics, EURODYN2002, Grundmann & Schuëller (eds.), 1041-1046, 2002.

8

Biondi, B., Muscolino, G., "Component-Mode Synthesis Method Variants in the Dynamics of Coupled Structures", Meccanica, 35(1), 17-38, 2000. doi:10.1023/A:1004730011796

9

Muscolino, G., "Dynamic Analysis of Structural Systems Using Component Mode Synthesis", in "Computational Structures Technology", (Edited by B.H.V. Topping and Z. Bittnar), Saxe-Coburg Publications, Stirling, Scotland, Chapter 11, 255-282, 2002. doi:10.4203/csets.7.11

10

Pesterev, A.V., Bergman, L.A., "An Improved Series Expansion of the Solution to the Moving Oscillator Problem", Journal of Vibration and Acoustics (ASME), 122, 54-61, 2000. doi:10.1115/1.568436

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