RESONANCE VIBRATION OF RAILWAY BRIDGES SUBJECTED TO PASSING VEHICLES RESONANCE VIBRATION OF RAILWAY BRIDGES SUBJECTED TO PASSING VEHICLES

N , with identical space intervals d v (the length of the vehicle) passing the bridge with a constant speed c [4] as is shown in Fig. 2. In this paper the dynamic behaviour of the simply supported railway bridge with the span L b = 38 m, subjected to the successive identical moving loads is solved. The close form solution is obtained by means of the modal superposition method [1-2, 5-6]. The presented parametric study is focused on the dynamic deflection of the bridge at the mid-span / ,t w L 2 c 2 ^ ^ h h due to loading of the conventional train with the Slovak IC-coaches for the resonance speed c 2 res = 65.03 m/s = 234.11 km/h, [7]. This paper discusses some issues related to dynamic effects in railway bridges focussed on the simulation of the resonant vibration for the small and medium span simply supported railway bridges subjected to a series of moving vehicle. Presented parametric study is concerned with the dynamic deflection of the simply supported railway truss bridge of the span L b = 38 m, due to the series of the ten moving loads representing a conventional train with the IC-coaches employed in Slovakia, with the impact to the resonance speed c 2 res = 65.03 m/s = 234.11 km/h. The deflection amplitude / increase with an increasing number of load forces n = 1, 2, ... N forces moving along the bridge.


Introduction
The dynamic response of railway bridges, subjected to moving trains is influenced by a number of factors such as the speed of load, the bridge span, natural frequencies of the bridge and railways vehicles, the inertia and damping of the two interaction systems (vehicles and the bridge), the distance between the vehicles, and arranging axles of vehicles. At present, the actual question for the bridge loading follows from high speed trains, which may consist of a number of identical cars connected together moving with the speed c. In those cases, the resonance caused by configuration of the train, consisting of a number of vehicles of similar types (Fig. 1), may occur, especially at high speed ranges.
To solve indicated problems one needs to apply the special dynamic analysis depending on the type of the bridge structure with regards to the static determination of the structure. For statically indeterminate structures, like continuous deck bridges or frame structures, more sophisticated methods of analysis (FEM) must be applied. For the simple bridges, the solution is based on the modal superposition method [1][2][3].
Railway vehicles travelling along the bridge are modelled as a series of identical moving loads and assuming that the vehicle/ bridge mass ratio is small m v <<m b and loads move along the bridge, the close form of solution can be obtained. The dynamic displacement , w x t h and acceleration , w x t p^h of the bridge are governed at different extents by the two sets of frequencies: In this paper the dynamic behaviour of the simply supported railway bridge with the span L b = 38 m, subjected to the successive identical moving loads is solved. The close form solution is obtained by means of the modal superposition method [1][2][5][6]. The presented parametric study is focused on the dynamic deflection of the bridge at the mid-span / ,t w L 2 c 2 h h due to loading of the conventional train with the Slovak IC-coaches for the resonance speed c 2res = 65.03 m/s = 234.11 km/h, [7].

RESONANCE VIBRATION OF RAILWAY BRIDGES SUBJECTED
m -is a fundamental mode shape and for the simply beam is of the sinusoidal type (the first mode is an one-half cycle; the second mode is a full cycle). Thus, the dynamic deflection may be represented by the summation of modal components. When the first and last moving load on the bridge span be P 1 and P M at a time t, Equation (1) can be expressed in terms of the generalized coordinates as

The single-mode analytical solution
The vertical deflection ,t w x h in Equation (1) for a simply supported beam for a moving load problem can be well simulated by considering the first mode of vibrations 1^h only. The corresponding modal coordinate from Equation (4), taking into account M-vehicles on the bridge, is given by superposition of a forced response (a quasi-static response) due to the moving load and a transient response (the dynamic part of the response), can be expressed as follows:

Formulation of the theory for the bridge response induced by moving load series
Consider a simply supported beam (without damping) subjected to a series of concentrated constant loads P, which are moving at a uniform speed c, in the meaning of where: q (j) (t) -are the generalized coordinates that define the amplitude of vibration with time t,

Condition of the resonance
When all the moving loads have passed the beam, the force part of the vibration terminates immediately. However, the free vibration part continues to exit until it is eventually damped out. Our parametric studies for different train speeds show that both the phenomena of resonance and cancellation are related to the free vibration component of the beam vibrations only. Thus, the sum of the free vibration response components by each moving load can result in the resonance if the total number N of the moving loads is large enough.
If the forces act on the bridge at equal distances d v then their repeated action can cause a resonant vibration. The resonant condition follows from the well know condition [5][6][7]: The resonant condition (9) is calculated from the time necessary for crossing the distance d v at the speed c which is equal to the k-multiple of the natural beam vibration f (1) . From Equation (9) results the resonant speed under the condition Critical speeds c , i reŝ h according to Equation (10) are given in Table 1.
Critical speeds for train with IC-cars (d v = 24.5 m) and for L b = 38 m Table 1 k The vertical deflection / ,

Displacement response at the mid-span of the beam
The important practical significance is just the dynamic deflection / , for loads of a moving series (P 1 , P 2 ,... P N ). If the number of the loads, moving out of the span, is K and the number of moving forces moving just on the span is M at the time t, the displacement response of the beam can be generalized considering the superposition of the next loading effects [2,[5][6]:  for n = 1,2,3,… N = M+K, and for / sin The first term in Equation (7) represents the component corresponding to each moving load P n travelling over the beam -it produces the force vibration response (the quasi-static part of the response). The second and the third term in Equation (7) produce the free vibration response (the dynamic part of the response) considering forces moving just on the beam and forces moving away of the span, respectively. The fourth term in Equation (7)  The quasi-static component of the response includes the effect of forces moving direct over the beam (without damping) and is defined by the first term in Equation (7). Because of drawing the displacement w in the downward direction in Equation (11), the minus sign is applied. The result is plotted in Fig. 4.  The dynamic component of the beam deflection includes the response defined by the second term in Equation (7). This dynamic response is plotted in Fig. 5.   The steel truss bridge L b = 38 m is subjected by the series of 10 IC-cars (P 1 +P 2 +...+P 10 ) in the sense of Fig. 3. The load of equal weights P = 524 kN is spaced at the car interval d v = 24.5 m, the train speed is c 2,res = 65.03 m/s, see Table 1. Evaluation of all time dynamic response waveforms were applied from a programming system Wolfram Mathematica 8.0.1 [8].

Components of the beam deflection due to the sequence of load IC-cars (P 1 +P 2 +…P 10 )
Components of the beam response due to the sequence of load IC-cars (P 1 +P 2 +…P 10 ) are defined by expressions (7).

• Components of the total response
The components of the total deflection response / , , belong to the load (P 1 , P 2, ….P 10 ) that move directly over the beam and to the loads that have passed the beam are depicted in Fig. 8. The components are described by Equations (9), (10) and (12) can be symbolically written as the superposition of effects: The result is plotted in Fig. 9. Fig. 9 The total beam displacement response  Fig. 6.    involving also the effect of the residual free vibration of the beam caused by moving loads that have passed the beam, demonstrates the strong increase of the DAF at 54.93 %.

Conclusions
The dynamic response of bridges of small and medium spans, is markedly influenced just by composition of the periodic load of moving vehicles, due to the loading interval of vehicles, the rate of the moving loads, and the bridge length. The total dynamic deflections / , w L t 2 , , , c P P P due to load by the series of IC-cars (P 1 +P 2 +…P N ) is defined by superposition of the quasistatic and the dynamic components described by expressions (11), (12) and (14). For small and medium spans and for low