Keywords: moving loads, dynamics of bridges, high-speed railway bridges, multi-span beams, nonuniform beams, semi-analytic solutions, flexural vibrations.

During the last decade the amount of research devoted to the assessment of the dynamic behaviour of beams under moving loads has increased steadily. This is largely due to the extensive construction of new high-speed railway lines in many developed countries. Bridges located on those lines undergo the most demanding dynamic excitation among the civil engineering structures subjected to moving loads, specially because of the appearance of resonance phenomena when the frequency of the moving forces is equal to one of the natural frequencies of the structure.

The behaviour of continuous beams under loads moving at high speeds makes them particularly suited for use in bridge engineering. Nevertheless, only approximate methods are nowadays available for the dynamic analysis of continuous nonuniform beams, which in turn are a very common alternative for the construction of the real structures. Analytic solutions have been presented by Hayashikawa and Watanabe [1], Chen and Li [2], Dagush and Eisenberger [3] , and Henchi et al. [4].

The present paper constitutes a step forward in the analysis of continuous nonuniform beams under moving loads. The proposed approach is semi-analytic: it is based on a spatial discretization of the beam using Euler-Bernoulli finite elements, but the modal equations of motion obtained from the spatial mesh are solved in closed-form. Viscous modal damping is considered in the model.

Three main steps can be identified in the semi-analytic approach. First, the natural frequencies and mode shapes of the discretized structure are computed using standard finite element procedures. Second, since the moving load is represented by a unit Dirac Delta function, the equivalent modal loads are obtained in terms of cubic hermitian polynomials, which leads straightforwardly to the closed-form solution for a unit load in the time domain. Third, the response for a train of moving forces is built simply by adding the contribution of each one.

The closed-form is expressed in terms of 10 coefficients per element and per mode. Furthermore, the values of these coefficients are independent of the speed of the moving loads, which proves very convenient for carrying out parametric analyses over a range of speeds.

The main advantages of the proposed methodology are the following:

1. The only approximation involved in the procedure comes from the finite element discretization of the beam, which is the most widespread and renowned technique in modern structural analysis.
2. Numerical integration errors are totally eliminated because the solution is obtained in closed-form. These entails a great benefit in terms of robustness and accuracy.
3. The method is applicable to the analysis of nonuniform multi-span beams.
4. The semi-analytic solution is very fast. Several numerical tests have proved that even the response of long viaducts subjected to a number of trains can be obtained within a few minutes.

1

T. Hayashikawa, N. Watanabe, "Dynamic Behavior of Continuous Beams with Moving Loads", Journal of the Engineering Mechanics Division, ASCE, 107, 229-246, 1981.

2

Y.H. Chen, C.Y. Li, "Dynamic Response of Elevated High-Speed Railway", Journal of Bridge Engineering, ASCE, 5, 124-130, 2000. doi:10.1061/(ASCE)1084-0702(2000)5:2(124)

3

Y. A. Dagush, M. Eisenberger, "Vibrations of Non-Uniform Continuous Beams Under Moving Loads", Journal of Sound and Vibration, 254, 911-926, 2002. doi:10.1006/jsvi.2001.4135

4

K. Henchi, M. Fafard, G. Dhatt, M. Talbot, "Dynamic Behaviour of Multi-Span Beams Under Moving Loads", Journal of Sound and Vibration, 199, 33-50,1997. doi:10.1006/jsvi.1996.0628

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