Intensive vibration of bridges due to high speed trains

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1 2002 WT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Paper from: Computers in Railways V, J Allan, RJ Hill, CA Brebbia, G Sciutto and S Sone (Editors). SBN ntensive vibration of bridges due to high speed trains L. Frjba nstitute of Theoretical and Applied Mechanics, Czech Academy of Sciences, Prague, Czech Republic Abstract The railway bridges may be intensively excited due to the high speed trains. Therefore, a simple theoretical model of a bridge was subjected to a row of axle forces which are moving along it. The integral transformation method, the Heaviside and Dirac functions expressed the solution in a comprehensive form. The numerical solution brings the deflection-time, bending momenttime and vertical acceleration-time histories of a bridge. The paper shows the effect of the speed up to 500 km/h and span (5 to 50 m) on the dynamic increments of bridge deflections due to three types of high speed trains, CE 2, and Talgo AV 2. 1 ntroduction The maintenance service of French and German R.ailways have reported in recent years that a destabilization of ballast appears on bridges of small and medium spans on high speed lines. t is a serious problem because it could cause the derailment, troubles in passenger comfort and higher maintenance costs. t arose a suspicion that the intensive vibration of bridges due to high speed trains is caused by the resonance at speeds higher than about 200

2 2002 WT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Paper from: Computers in Railways V, J Allan, RJ Hill, CA Brebbia, G Sciutto and S Sone (Editors). SBN Computers in Railway V km/h. The first study [l] discovered the speed ranges in which the resonance vibration may occur. n the second 011e [2], the maximum amplitudes of deflect,ions, bending moments arid vertical accelerations of bridges were est,imat,ed. At last [3], the time hist,ories of all important values were calculat,ed. The present paper brings a summary of the calculations and studies t,he effect, of t,he speed and span of both the concrete and steel bridges due t,o t,hree t,ypes of high speed trains, CE 2, (TGV) and Talgo AV 2. 2 Movement of a row of forces along a beam Let us a,ssume an element,ary theoretical model widely used in the bridge dynamics, [4], and consider a simple beam of span 1 subjected to a row of forces F,,, n = 1, 2, 3,..., AT, which are moving with a constant speed e, N being the number of axle forces in the t,rain. The governing Bernoulli-Euler part,ial differential equation describes the behaviour of the beam : N n= 1 where it, is denoted : 7>(z, t) - verical deflectlion of t,he beam at z and t,ime t, E - modulus of elasticit,y of the beam, - constant, moment, of inertia of t,he cross sect,ion of the beam, p - constant, inass per unit length of t,he beam, ud ~ circular freqencg of damping, d = ucl/fi ~ logarithmic decrement, of damping, ETL(t) = h(t - t,) - h(t - T,,), 0 for t < 0 ll(t) = i 1 for t Heavisitle unit funct,ion, f,, = d,2/c ~ time when the n-th force enters t,he beam, T,, = (1 + &)/c - time when tjhe 71-th force leaves t,he beam, &(::) - Dirac function,

3 2002 WT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Paper from: Computers in Railways V, J Allan, RJ Hill, CA Brebbia, G Sciutto and S Sone (Editors). SBN Computers in Railway V 597 x,, = et - d,, - distance of the 71,-th force from t,he first one, dl = 0. The application of t,he Fourier and Laplace-Carson integral transformations, [Z], arid suitable forniulations of Heaviside and Dirac functions present a closed solut,ion [a] assuming the zero init,ial conditions and the boundary condit,ions of a simple beam : where : 7T c 2F 2F13 F13 WO = - %- p1u; = 48E' Due t,o Heaviside furdon, the equation (4) is valid for both the forced and free vibrations at, the tinie when the n-t,h force at,s on t,he beam as well as aft,er leaving it. A similar approach yielded the bending moment M(x, t) = -Eld2w(z, t) /dz2 and vertical acceleration of the heani u(n:,t) = d'u(z, t)/3t2, [a],[3]. Besides, the bending moment-time histories were evaluat,ed using the rainflow method. t, counts the number of stress ranges in each stress class

4 2002 WT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Paper from: Computers in Railways V, J Allan, RJ Hill, CA Brebbia, G Sciutto and S Sone (Editors). SBN Computers in Railways V 1.8, Dynamic increments of a concrete bridge, = 5 m, for various trains._ ~ b - v, CE (1) ' / i, Q -M,CE2 - -v, 4- M, c (kmlh) Figure 1 :... v, Talgo M, Talgo Dynamic impact factor as a function of the speed c for the deflection w, bending moment M and various trains. Concrete bridge, 1 = 5 m. Dynamic increments of a concrete bridge, 20 m, for various trains - v, CE 2 M, CE 2 - -v, 4- M,... v, Talgo - 0- M, Talgo c (kmlh) Figure 2 : Dynamic impact factor as a function of the speed c for the deflection U, bending moment M and various trains. Concrete bridge. 1 = 20 m.

5 , 2002 WT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Paper from: Computers in Railways V, J Allan, RJ Hill, CA Brebbia, G Sciutto and S Sone (Editors). SBN Computers in Railways V 599 Dynamic increments of a steel bridge, = 5 m, for various trains nt *,- v, CE ,' 1,, --,CE2 2.2 ~,e,', e, * * - -v,,q'," (1) 0 io c (kmlh) -0- M,... v, Talgo M, Talgo Figure 3 : Dynamic impact factor as a function of the speed c for the deflection U, bending moment M and various trains. Steel bridge, 1 = 5 m. Dynamic increments of a steel bridge, 20 m, for various trains -.~ ~- / --,CE2, i ' --v, -0-M, v, Talgo B c (kmlh) Figure 4 : Dynamic impact factor as a function of the speed c for the deflection w, bending moment M and various trains. Steel bridge, 1 = 20 m.

6 2002 WT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Paper from: Computers in Railways V, J Allan, RJ Hill, CA Brebbia, G Sciutto and S Sone (Editors). SBN Computers in Railways V 1.4 Dynamic increments of concrete bridges under high speed trains at c = 350 kmlh - v, CE 2 --,CE2 (1 ) v, 4'- M, (m).-- v, Talgo - 0- M,Talgo Figure 5 : Dynamic impact factor of concrete bridges of various spans 1 under high speed trains at c = 330 km/h, v = deflection, A4 = bending moment. Dynamic increments of steel bridges under high speed trains at c = 350 kmlh - v. CE (1) l -% M, CE 2 - -v, -0- M, _ v, Talgo (m) M, Talgo Figure G : Dynamic impact factor of steel bridges of various spans 1 under high speed trains at c = 330 km/h, 'U = deflection, M = bending moment.

7 2002 WT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Paper from: Computers in Railways V, J Allan, RJ Hill, CA Brebbia, G Sciutto and S Sone (Editors). SBN Computers in Railways V 601 Dynamic ncrements of concrete and steel bridges under CE 2 at c = 350 kmlh \\ v, steel 4- M. *tb* Figure 7 : so (m) Dynamic impact factor of concrete and steel bridges of various spans 1 under CE 2 at c = 350 km/h. Dynamic ncrements of concrete and steel bridges under at c = 350 kmlh (1) M. concrete - -v..tee M, sted (m) Figure 8 : Dynaniic impact factor of concrete and steel bridges of various spans 1 under at c = 350 km/h. (1 ) Dynamic increments of concrete and steel brldges under Tala0 AV 2 nt G = 350 kmlh - M. concrete - - V. steel M, steel 1 Figure 9 : so (m) Dynamic impact factor of concrete and steel bridges of various spans 1 under Talgo AV 2 at c = 350 kni/h.

8 2002 WT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Paper from: Computers in Railways V, J Allan, RJ Hill, CA Brebbia, G Sciutto and S Sone (Editors). SBN Computers in Railways V and gives t,he stress spectra that, are import,ant for calculation of inspection intervals and for t,he est,imat,ion of fatigue life of bridges, [3], [4]. 3 Numerical calculation For paramet,ric study, both t,he concret,e and steel bridges of short and medium spans from 5 t,o 50 m were used. Their paramet,ers may be seen in the Table 1. The bridges differ by their spans 1, natural frequencies fi, damping characteristics 6 and masses (weight, G). Their values were taken over from the empirical formulas in [4]. Material Span 1 (ni) fi (Hz) Concret,e 6 (1) G (kn) fi (H7,) StJeel 6 (1) G (kn) Three types of high speed trains, CE 2, (TGV) and Talgo AV 2, were considered. They differ by the lengths of cars, axle distances nad many other parameters. Their most, important data are summarized in tjhe Table 2. Charact,erist,ics axle force riurriber of axle forces dominant car length Table 2. Train parameters notatlion unit CE 2 Talgo AV 2 F (kn) "i (1) d (m) The time histories were calculated for all bridge and train parameters for eleven speeds : 3, 50, 100, 150, 200, 250, 300, 350, 400, 450 and 500 km/h. Several hundred cases were considered altoget,her. The speed of 5 km/h represents a quasistatic case to which the other cases are re1at)ed. n this way, we received t,he dynamic impact factjors :

9 2002 WT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Paper from: Computers in Railways V, J Allan, RJ Hill, CA Brebbia, G Sciutto and S Sone (Editors). SBN Computers in Railways V 603 niax w(2/2,t) at speed c 6= niax u(2/2,t) at speed 5 km/h' (13) The dynamic impact factor (13) cannot be confused with that one applied to the standard load model LC Effect of some parameters From a large study we slion here only t,he effect of t,wo important paramet,ers, speed and span: The speed of the t,rain movement is the most, important parameter for the design of high speed lines. The bridges have been recently constructed for the speed 350 km/h with a 20 % reliability, i.e. 420 km/h. The indispensable ext,rapolat,ion amount t,o 500 km/h and: therefore, all the calculations mere performed up t,o tjhis value. The effect of the speed c on t,he dynamic impact, factor 6 for concrete bridges of spans 5 and 20 m are shown in Figures 1 and 2, respectively, for all considered trains. The increnient,s of deflection are depicted by U and t,hat ones of the bending moment, by M. The same appears in Figures 3 and 4, however for steel bridges of the sanie spans. The resonance vibrations with t,he highest dynamic impact factors occur at bridges of about 20 m span at, speeds near to 500 km/h (see Figures 2 and 4). The dynamic impact factors of concrete and steel bridges of various spans are summarized in Figures 5 and 6, respectively, again for all assumed high speed trains at the speed of 350 km/h. The comparison of concrete and steel bridges of various spans subjectled tjo individual t,rains, CE 2, Eurost,ar and Talgo AV 2 at the speed 350 ltm/h, can be seen in Figures 7; 8 a,nd 9, respectively. t must be pointed out that, the concret>e and st,eel bridge of t,lie sanie span form two different dynamic systems as t>liey provide different natural frequencies, damping charact,eristics and masses. 5 Conclusions t has shown t,hat, the high speed t,rains substantially affect the dynamic heha\-iour of railway bridges. The dynamic impact, fact,ors roughly increase with increasing speed. Some peaks appear at t,hese figures and their values and positlions depend on the complex dynamic syst,em bridge + train. Only t,he first) approximation of the train system as a row of axle forces was considered in the paper and it cannot explain all the details of the problem.

10 2002 WT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Paper from: Computers in Railways V, J Allan, RJ Hill, CA Brebbia, G Sciutto and S Sone (Editors). SBN Computers in Railways V Nevertheless, even the simple theoretical model confirms the possibility of t,he resonance vibration. t, is caused by the sequence of a long row of axle forces distribut,ed in almost, regular distances. The resonance occurs at speeds over 200 km/h and, of course, is not pure due to some irregularities of axle distances, short duration of the run of a high speed train and damping. The length of bridges (span) has an unimpressive effect on their dyrianiic behaviour (at speeds of 350 km/h!). Only in some cases, a diminishing of dynamic effect,s with increasing span can be observed. The dynamics of steel bridges is greater than that, of concrete bridges in most investigated cases. The dynamic impact factors of bending moments (stresses) are a lit,tle higher than that ones of deflection. The other interesting conclusions. e.g. about the vertical acceleration of bridges, st,ress spectra: damping, numerical calculation et,c., are mentioned in [3]. Acknowledgement ThP support of grants GX AS CR and GA CR 103/01/0243 are gratefully acknowledged. References Fr$ba: L. & Nliprstelc,.J., Appearance of resonance vibration on railway bridges. Advances in Civil and Structural Engineering Co~rnputiriy for Practice, ed. B.H.1; Topping, Civil-Comp Press: Edinbiirgh, pp , Friha, L., A rough assessment, of railway bridges for high speed trains. En,gineering Structures, 23 (5) pp , Friba, L. & Fischer, C. & PospiSil, S., Dynamic effects of high speed trains on bridges. Structural Dynamics, ed. H. Grundmann, Balkema: Rott,erclam, in press, Frfba: L., Dyrmnics of Railwuy Bridges. 2-nd edit>ion, Thomas Telford : London, Frfba, L., Vihrations of Solids and Structure,r Under Moving Loads. 3-rd edit,ion: Thomas Telford: London, 1999.

Dynamic analysis of railway bridges by means of the spectral method

Dynamic analysis of railway bridges by means of the spectral method Giuseppe Catania, Silvio Sorrentino DIEM, Department of Mechanical Engineering, University of Bologna, Viale del Risorgimento, 436 Bologna,