Justification of the bridge span vertical stiffness on high-speed railways

When designing bridges on high-speed railways, special attention should be paid to ensuring the safety of train traffic and the comfort of passengers. Excessive structure deformations (both elastic and non-elastic) result in unfavorable irregularities in the train movement pattern on the bridge and so can lead to violation of the traffic safety requirements as well as to vibration and acceleration of the train body, which is inadmissible due to its effect on the human body or the transported goods. In this paper, based on numerical simulation, the results of the study of the motion of a high-speed train along bridge structures of the dynamic bridgetrain interaction was performed with respect to various models of high-speed trains running along the bridges. The obtained dependences help to provide a practical assessment of high-speed passenger car dynamics and passenger comfort under the most unfavorable conditions, when the train is running along a multi-span bridge. For these purposes, the dependences of the admissible value of the relative vertical deflection are presented, based on the envelope curves that show the typical dynamic passenger car parameters (natural frequency of car oscillations) and Corresponding with their oscillations on the multi-span girder bridges with various lengths


Introduction
Even though the required bearing capacity of the bridge elements is provided, it does not necessarily imply that all terms for its normal operation are observed. Excessive structure deformations (both elastic and non-elastic) result in unfavorable irregularities in the train movement pattern on the bridge and so can lead to violation of the traffic safety requirements as well as to vibration and acceleration of the train body, which is inadmissible due to its effect on the human body or the transported goods [1]. To standardize the bridge span vertical stiffness values means to establish the elastic deformation limits complying with the operational requirements to the bridge structure interacting with the high-speed train. Subsequent paragraphs, however, are indented.

Problem statement
As part of the study, the numerical calculation of the dynamic bridge-train interaction was performed with respect to various models of high-speed trains running along the bridges [2][3][4][5][6][7]. In calculations, the train wheels movement patterns were identified as well as the vertical accelerations with-train car bodies.
The most unfavorable and widespread (especially on high-speed lines) case is considered, when the train moves along a multi-span bridge with spans having the same length. The movement patterns of passenger units will consist of tandem irregularities (deflections) caused by the live load stress type.
Thus, as a structural model a chain of spans was taken having the same length of spans ( Fig.1) whose value varied in the course of calculation [8].
We introduce the concept of the coach resonant speed equal to the span length -coach vibration frequency product. The speed at which the kinematic effect has a frequency equal to the coach natural frequency [7]: , where 1,2,3...

Analysis of train-bridge dynamic interaction
From the practical viewpoint, the dependence of the highest railway car vertical acceleration values upon various lengths of the multi-span bridge superstructure is especially interesting. The maximum bouncing accelerations determine the factor of passenger comfort. As far as the admissible vertical accelerations are limited by standard requirements, one can build up a diagram showing the dependence of the vertical deformations of the bridge span (relative deflections) upon various span lengths considered in the passenger comfort test. The kind of this dependence typical of the train moving at a speed of 350 km/h is given below. Here and elsewhere in the diagram we show the relative deflections recalculated for the standard load C8 (according to the relevant Russian standards and norms) with respect to the dynamic factor complying with the requirement of limiting the maximum vertical accelerations to the value of 1.0 m/sec 2 . The limit value of 1.0 m/sec 2 is selected because the car bouncing is caused not only by the span deflections, but also by track irregularities and wheel defects that lead to accelerations of not more than 0.5 m/sec 2 .    Vertical accelleration, m/sec² Time, sec and the last car have an asymmetrical movement pattern against the middle of the superstructure, which is definitely the reason why the bouncing amplitudes increase. Identifying the type of the bridge-train interaction and analysing their common oscillations, we have defined the linear dependence of the maxi-mum vertical accelerations in the car body upon the train speed. Based on this as well as on a series of calculations, envelope curves characterizing the dependences of the relative superstructure deflection upon various span lengths were devised; these dependences will serve as a foundation for the passenger comfort test [9,10]. It should be noted that for the majority of trains with distributed load, the comfort condition of passengers is carried out with a large margin. These trains, as a rule, have a "soft" suspension and small values of the natural frequencies of oscillations of the car body. For trains with a locomotive load (for example, TGV), the suspension is more rigid and the requirements for limiting vertical deflections of span structures are higher.

Conclusion
The obtained dependences help to provide a practical assessment of high-speed passenger car dynamics and passenger comfort under the most unfavourable conditions, when the train is running along a multi-span bridge. For these purposes, the dependences of the admissible value of the relative vertical deflection are presented, based on the envelope curves that show the typical dynamic passenger car parameters (natural frequency of car oscillations) and corresponding with their oscillations on the multi-span girder bridges with various lengths. The difference of the given recommendations from the similar requirements of European Norms is the fact that the curve is not limited to the span length of 120 m, which lets to assess the long-span structure against the passenger comfort factor without making complicated dynamic calculations with respect to the train-bridge interaction. Unlike European Norms [12], the suggested limits describe both fundamental and multiple resonance modes of car oscillations on the superstructure. The described dependences are associated with the Russian standard [11] live load СК (К=8) rather than the European LM71.