Keywords: resonance in railway bridges, high-speed bridge, second mode contribution,.

This paper deals with the assessment of the contribution of the second flexural mode to the dynamic behaviour of simply supported railway bridges. Alluding to the works of other authors, it is suggested in some references [1,2] that the dynamic behaviour of simply supported bridges could be adequately represented taking into account only the contribution of the fundamental flexural mode. On the other hand, the European Rail Research Institute (ERRI) proposes that the second mode should also be included whenever the associated natural frequency is lower than 30 Hz [1]. This investigation endeavours to clarify the question as much as possible by establishing whether the maximum response of the bridge, in terms of displacements, accelerations and bending moments, can be computed accurately not taking account of the contribution of the second mode.

To this end, a dimensionless formulation of the equations of motion of a simply supported beam traversed by a series of equally spaced moving loads is presented. This formulation brings to light the fundamental parameters governing the behaviour of the beam: damping ratio, dimensionless speed =VT/L, and L/d ratio (L stands for the span of the beam, V for the speed of the train, T represents the fundamental period of the bridge and d symbolises the distance between consecutive loads).

Assuming a damping ratio equal to 1%, which is a usual value for prestressed high-speed bridges, a parametric analysis is conducted over realistic ranges of values of and L/d. The results can be extended to any simply supported bridge subjected to a train of equally spaced loads in virtue of the so-called Similarity Formulae. The validity of these formulae can be derived from the dimensionless formulation mentioned above.

In the parametric analysis the maximum response of the bridge is obtained for one thousand values of speed that cover the range from the fourth resonance of the first mode to the first resonance of the second mode. The response at twenty-one different locations along the span of the beam is compared in order to decide if the maximum can be accurately computed with the sole contribution of the fundamental mode. The most important conclusions obtained from the investigation are the following:

1. At speeds lower than 420 km/h (upper limit of the high-speed range recommended by ERRI), for sixteen different values of L/d ratio comprised in the usual interval 0.3 <= L/d <= 4.0, no case has been found such that the maximum displacement and bending moment are not obtained at mid-span. Consequently, in all those cases the contribution of the first mode would suffice for an accurate computation of the maximum values of these two variables. \item On the contrary, it has been found that in several cases the maximum acceleration does not correspond to the mid-span section. This is due to the development of resonance situations associated with the second flexural mode, and consequently the contribution of this mode can not be disregarded. In these cases the maximum acceleration can be computed accurately either at the first quarter of the span or at three quarters of span since there is very little difference between the results at these two sections. It is of interesting to mention that, since that the maximum value at mid-span is not exceeded by a large amount, it is likely that the maximum accelerations would take place at this section providing that the damping ratio assigned to the second mode was higher than the one assigned to the first mode.
2. If the speed range is extended in order to encompass the first four resonances of the first and second mode, the maximum displacement is still obtained at mid- span for all sixteen values of L/d. On the contrary, several cases can be identified such that the maximum bending moment is obtained at other sections along the bridge. In addition, only in a few cases the maximum acceleration corresponds to the mid-span section: this is a consequence of the high acceleration produced by the first resonance of the second mode.
3. The maximum displacements obtained at the first quarter of span and at three quarters of span have been found to be very similar (difference less than 5%) for all speeds up to the first resonance of the second mode. The same behaviour has been observed for the accelerations at speeds lower than 420 km/h. Conversely, some cases have been identified such that the maximum bending moments at both sections differ by more than 5%, and the same applies to the accelerations when speeds up to the first resonance of the second mode are considered.

1

ERRI D-214 Committee, "Ponts-Rails pour vitesses > 200 km/h. Rapport Final", 1999. ("Railway Bridges for Speed > 200 km/h. Final Report", Technical Report, 1999, written in French).

2

J. Li, M. Su, "The Resonant Vibration for a Simply Supported Girder Bridge under High-Speed Trains", J of Sound and Vibration, 224(5), 897-915, 1999. doi:10.1006/jsvi.1999.2226

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