The analysis of VBI basically requires the writing of two sets of motion equations, one for the bridge and the other for each vehicle. Due to the interaction forces existing at the contact points between the wheels and rail surface (railway bridge) or pavement surface (highway bridge), the two sets of equations are coupled and characterised by time-dependent coefficients. The problem is usually solved by adopting iterative procedures (see e.g., [2]), whose efficiency, however, worsens when dealing with the more realistic case of bridges travelled by a large number of vehicles. In the literature, several approaches based on the condensation method are also available (see e.g., [3]).

Recently, the authors [4] analyzed the dynamic interaction between a running train, the track structure and the supporting bridge by applying an alternative approach, based on the substructure technique. It consists of treating the train, track and bridge deck as three substructures. The running train is idealized as a sequence of identical vehicles. The track system lying on the bridge is characterized as a couple of infinite rails supported by a continuous and homogeneous foundation (rail bed). The substructure approach is also able to model the rail bed as a viscoelastic foundation and to take into account the irregularities which can be represented either as deterministic functions or stochastic processes.

In this review paper, first the philosophy of substructure approach for the analysis of VBI is presented with reference to railway bridges by adopting a plane model, then the advantages over traditional finite element procedures are demonstrated through numerical results.

A recently proposed variant [5] of the Component-Mode Synthesis (CMS) method is applied to couple the continuous (rails and bridge) and discrete (train) substructures. A set of ordinary differential equations with time-dependent coefficients in the generalised coordinates of the bridge and rails, and physical displacements of the railway vehicles is obtained. One of the main features of the present approach is the capability to analyse simultaneously the dynamic responses of the train, track structure and bridge, taking into account the interaction among the three sub-systems.

For comparison purpose, the single-span simply supported prestressed concrete bridge analysed in Reference [6] by using bridge-track-vehicle elements is taken as a case study. The results provided by the substructure technique turn out to be in good agreement with those obtained by the finite element method, despite the computational burden being substantially reduced.

[1]

Y.B. Yang, J.D Yau, Y.S Wu, "Vehicle-bridge interaction dynamics", World Scientific, London, 2004.

[2]

F. Yang, G.A. Fonder, "An iterative solution method for dynamic response of bridge-vehicles systems", Earthquake Engineering and Structural Dynamics, 25, 195-215, 1996. doi:10.1002/(SICI)1096-9845(199602)25:2<195::AID-EQE547>3.0.CO;2-R

[3]

Y.B. Yang, B.H. Lin, "Vehicle-bridge interaction analysis by dynamic condensation method", Journal of Structural Engineering (ASCE), 121, 1636-1643, 1995. doi:10.1061/(ASCE)0733-9445(1995)121:11(1636)

[4]

B. Biondi, G. Muscolino, A. Sofi, "A substructure approach for the dynamic analysis of train-track-bridge system", Computers & Structures, 83, 2271-2281, 2005. doi:10.1016/j.compstruc.2005.03.036

[5]

B. Biondi, G. Muscolino, "Component-mode synthesis method for coupled continuous and FE discretized substructures", Engineering Structures, 25, 419-433, 2003. doi:10.1016/S0141-0296(02)00183-9

[6]

Y.S. Cheng, F.T.K. Au, Y.K. Cheung, "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

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