Keywords: moving loads, modal contributions, modal superposition, railway bridges, high-speed train bridges, bridge dynamics, double-track railway bridges.

The vibration of railway bridges due to the passing of vehicles is an issue of main concern especially since the appearance of high-speed trains. Many approaches have been adopted to the problem: from the moving point load case exposed by Timoshenko [1] or Biggs [2] crossing an Euler-Bernoulli beam to complicated vehicle-interaction models in double track bridges as those presented by Yau et al. [3], Zhang et al. [4] and Wu et al. [5] among others. During the passed years, many authors have addressed this problem applying numerical algorithms of different nature: Yang and Fonder [6] applied iterative algorithms, Blejwas et al. [7] worked on Lagrange's multipliers methods, Yang and Lin [8] based their work on condensation procedures, etc.

Due to the non-linear nature of the vehicle-bridge interaction problem, modal analysis has not been recalled quite often. Although, in many practical situations a linear elastic approach may lead to accurate solutions with a very low computational effort when compared to time-history analyses as in Memory et al. [9] or Marchesiello et al. [10].

In this paper, modal analysis of simply supported beams subjected to moving point loads is addressed. The objective of the study is to approximate the number of modes that should be taken into account in a modal superposition analysis in order to compute the dynamic amplification of the response with sufficient accuracy. It is known that high frequency modes contribution to the dynamic response of a bridge is negligible. However, there is not a firm criterion regarding the number of modes that should be considered in a superposition modal analysis depending on the velocity of the vehicles, the span of the bridge and the structural damping. The analysis and conclusions that derive from this work are not only restricted to pure flexural modes of the beam but also to torsional ones.

The proposed approach is based on obtaining analytically the dynamic response of a single degree of freedom system subjected to excitations composed by a series of half-cycle sine pulses. The amplification of the response depends on the damping ratio and the duration and number of input half-cycle sine waves. A parametric numerical study is performed in terms of these three variables. From the obtained results, the first mode which contribution to the dynamic response may be considered as negligible can be approximated.

1

S.P. Timoshenko, "Vibration of bridges", Transactions, American Society of Mechanical Engineers, 53, 1928.

2

J.M. Biggs, "Introduction to structural dynamics", McGraw-Hill, 1964.

3

J.D. Yau, Y.B. Yang, S.R. Kuo, "Impact response of high speed rail bridges and riding comfort of rail cars", Engineering Structures, 21 (9), 836-844, 1999. doi:10.1016/S0141-0296(98)00037-6

4

Q.L. Zhang, A. Vrouwenvelder, J. Wardenier, "Numerical simulation of train-bridge interactive dynamics", Computers and Structures, 79 (10), 1059-1075, 2001. doi:10.1016/S0045-7949(00)00181-4

5

Y.S. Wu, Y.B. Yang, J.D. Yau, "Three-Dimensional analysis of train-rail-bridge interaction problems", Vehicle System Dynamics, 36 (1), 1-35, 2001. doi:10.1076/vesd.36.1.1.3567

6

F. Yang, G.A. Fonder, "An iterative solution method for dynamic response of bridge-vehicle 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

7

T.E. Blejwas, C.C. Feng, R.S. Ayre, "Dynamic interaction of moving vehicles and structures", Journal of Sound and Vibration, 67 (4), 513-521, 1979. doi:10.1016/0022-460X(79)90442-5

8

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

9

T.J. Memory, D.P. Thambiratman, G.H. Brameld, "Free vibration analysis of bridges", Engineering Structures, 17 (10), 705-713, 1995. doi:10.1016/0141-0296(95)00037-8

10

S. Marchesiello, A. Fasana, L. Garibaldi, B.A.D. Piombo, "Dynamics of multi-span continuous straight bridges subject to multi-degrees of freedom moving vehicle excitation", Journal of Sound and Vibration, 224 (3), 541-561, 1999. doi:10.1006/jsvi.1999.2197

11

A.K. Chopra, "Dynamics of structures. Theory and applications to Earthquake Engineering". Prentice Hall, 1995.

12

Union Internationale des Chemins de Fer. Code UIC 774-1R, "Recommandation pour le dimensionnement des ponts-rails en beton arme et preconstrainst", Union Internationale des Chemins de Fer, 1984.

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