Computational Technology Resources - CCP

Keywords: Bedas software, bridge, dynamic amplification factor, dynamic stiffness, moving load, high frequencies, TGV, vibrations.

Summary

In this paper, we present a recent new efficient and practical method implemented in the Bedas software, to model the dynamic and transient vertical vibrations of trussed and multi-span railway bridges, induced by moving multi-axle high-speed train. The speeds of train varying from 150 to 400 km/h are considered. The influence of the speed of moving multi-axle train on the dynamic response especially in the resonance zones is studied on three existing multi-span two-beams bridges (concrete-steel section). These three bridges are located in France and they were designed for the French high-speed train TGV.

This study presents a specially developed Bridges Exact Dynamic Analysis Software program called BEDAS, which can be routinely used to analyse the static and dynamic behaviours of common type bridge structures. The advantages of BEDAS are:

the program uses an exact dynamic stiffness method [1] and gives exact frequencies and mode shapes with only one element per span;
finite dynamic model based on exact dynamic expression leads to higher precision of internal forces at any location of the bridge;
the program is applicable for studying the dynamic behaviour of common type of bridge structure including slab on I-beam bridges

BEDAS has been developed in such a way that it is applicable to any bridge structure discretized by beam elements (continuous beams bridge, frames, trusses, etc.). The dynamic stiffness method treats the structure as an infinite degrees-of- freedom system with distributed mass and stiffness. Similar to the finite element method, stiffness matrices of each super element are first computed and then assembled to form the global dynamic stiffness matrix of the structure. The method is equally very suitable for personal computer implementation more than finite element package, because it provides an exact solution in frequencies and dynamic analysis with few elements [1,2]. In this dynamic stiffness approach, the modal technique is used coupled with the FFT algorithm [3,4,5] to obtain the dynamic response of continuous bridges [1,6] under multiple moving and fixed loads. A summary of different dynamic solution techniques and the procedure used in BEDAS are presented in [1]. After assembling elementary dynamic stiffness $[k(\omega)]$ and elimination of rows and columns corresponding to the boundary conditions, the free vibration problem becomes: $[K_d (\omega)]{X_n}={0}$ . The solution of this non-linear eigenvalue problem is performed using Wittrick & Williams algorithm [2]. Projection of the dynamic equation in the modal space is easily obtained as follows:

$\displaystyle \ddot{y}_j+2\xi_j \omega_j\dot{y}_j(i)+\omega_j^2 y_j (t)=p_j(t)$

(85.1)

where

is the generalised load. The dynamic solution can be obtained by transforming the modal equation into frequency domain, using FFT technique:

(85.2)

To obtain a very good approximation of the displacements, velocities and accelerations, we must evaluate the generalized loads

in exact procedure, because the transfer function between loading

and displacement

is generated analytically. The fast Fourier transform (FFT), which allows very efficient and accurate response, is based on an algorithm developed in [3]. For more details on the implementation of this algorithm in dynamics structural analysis, see [5]. The dynamic responses for these studies are the displacement, velocity, acceleration and the internal forces at any point of the bridge. Some of the interesting results are the DAF, the maximum of the dynamic response on the static one in function of the train speed. For the BANCEL bridge analysis, the maximum is obtained for a speed of 300 km/h and the DAF is 1.58. The French code gives the bounded value of 1.67.

References

1: K. Henchi., M. Fafard, G. Dhatt, M. Talbot, "Dynamic behaviour of multi- span beams under moving loads", J. of sound and vibration,199(1), pp. 33-50, 1996. doi:10.1006/jsvi.1996.0628
2: F.W. Williams, W.H. Wilttrick, "A general algorithm for computing natural frequencies of elastic structures", Quarterly Journal of Mechanics and Applied Mathematics, 24, pp. 263-284, 1971. doi:10.1093/qjmam/24.3.263
3: J.W. Cooley, J.W. Tukey, "An algorithm for the machine calculation of complex fourier series", Mathematical Computation, 19, pp. 297-301, 1965. doi:10.2307/2003354
4: J.L. Humar, "Dynamics of Structures". Prentice-Hall, Englewood Cliffs, New Jersey, 1990.
5: K. Henchi, "Analyses dynamiques des ponts par éléments finis sous la sollicitation des véhicules mobiles", Thèse de Doctorat, Division MNM, Université de Technologie de Compiègne, France, 1995.
6: K. Henchi, G. Dhatt, M. Talbot, M. Fafard, "Comportement dynamique des poutres continues sous les chargements mobiles. Approches semi-analytiques et éléments finis", Revue européenne des éléments finis, 6(2), pp.165-195, 1996.

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