a rough assessment of railway bridges for high speed trains

Engineering Structures 23 (2001) 548–556www.elsevier.com/locate/engstruct

A rough assessment of railway bridges for high speed trains

L. Fryba *

Institute of Theoretical and Applied Mechanics, Academy of Sciences of the Czech Republic, Prosecka´ 76, CZ-190 00 Prague 9, Czech Republic

Received 24 January 2000; received in revised form 11 May 2000; accepted 12 May 2000

Abstract

Resonance vibrations have been observed on railway bridges subjected to high speed trains. An elementary theoretical model ofa bridge was investigated using the integral transformation method which provides an estimation of the amplitudes of the freevibration In addition, the analysis gives the critical speeds at which the resonance vibration may occur. They are caused by tworeasons: repeated action of axle loads and high speed itself. While the first cause of critical speeds has been reached on today’shigh speed lines the second one not yet. The maximum amplitudes of resonance vibration appear at the moment when the last axleleaves the bridge and, therefore, their values were calculated for the deflection, bending moment and acceleration of the bridgedeck. Simple expressions similar to dynamic impact factor were given to these values which enable to assess the railway bridgesfor high speed trains. Comparison of the theory with the experiments is satisfactory. 2001 Elsevier Science Ltd. All rights reserved.

Keywords:Dynamics of bridges; Resonance vibration; Railway bridges

1. Introduction

Intensive vibration of some railway bridges subjectedto high speed trains has been observed. It is suspectedthat this phenomenon could be caused by resonance dueto the repeated action of axle forces.

The classic case of resonance assumes harmonic force(periodically varying in time) which acts on the mechan-ical system. The resonance occurs if the frequency of aninput force coincides with one of the natural frequenciesof the system. Then, the amplitudes of the steady stateforced vibrations grow without all limit if the system isnot damped.

Of course two necessary conditions should be fulfilledin the resonance case: a non-damped system and steadystate vibration after a long time. Actually, these two con-ditions and the type of the input mentioned above cannotoccur on real bridges. The bridges are always dampedand the movement of a train along the bridge is a transi-ent phenomenon. The train possesses a finite number ofaxles and the time for crossing a bridge is also limited

The history of the bridge dynamics [1] has recognizedseveral cases of the resonance vibration of railway

* Tel.: +420-2-838-816-46.E-mail address:[email protected] (L. Fry´ba).

0141-0296/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved.PII: S0141-0296 (00)00057-2

bridges. Firstly the unbalanced counterweights of steamlocomotives resulted in a moving sine force which wasthe reason of the intensive vibration (see Fig. 1). Theeffects of steam locomotives were tackled in the classicbook [3].

Secondly, the resonance vibration of railway bridgesof large spans was mentioned in [4] where the authordescribes several cases (see Fig. 2). The bridges weresubjected to four axle vehicles and the rails were jointedon bridges. The regular impacts of wheels in the gap ofrail joints were the source of resonance vibration. Theauthor called this phenomenon “group impacts”.

While the first two cases are no more actual on mod-ern railways (steam locomotives are in museums and railjoints are not allowed on bridges), the new bridges sufferfrom the high speeds. Fig. 3 shows an example of aFrench bridge [5]. It was the maintenance service whichfirstly signalled that a destabilization of the ballast

Fig. 1. Vibration of a steel railway bridge of span 56.56 m due to asteam locomotive at the critical speed 38.8 km/h [2].

549L. Fryba / Engineering Structures 23 (2001) 548–556

Fig. 2. Deflection–time histories of railway bridges subjected to group impacts of four-axles vehicles [4]: (a)l=66 m, c=40–45 km/h; (b)l=77m (c) l=77 m, c=56 km/h; (d) l=87.83 m; (e) reinforced archl=120 m,c=56.5 km/h.

Fig. 3. Acceleration–time history of a SNCF bridge of span 38 msubjected to a TGV train at speed 192 km/h [5].

occurs on small and medium span bridges on high speedlines. Ballast destabilization results in the deteriorationof passenger comfort, reduction of traffic safety (a possi-bility of derailment of vehicles) and higher maintenancecosts (shorter maintenance intervals).

The importance of the problem have led to the presentstudy, the purpose of which is to analyse the problem,to develop simple expressions for the first, rough andquick assessment of bridges in resonance conditions, thecalculation of critical speeds, comparison with experi-ments and a quantification of interoperability whichenables the international high speed traffic.

2. Movement of a row of forces along a beam

Let us assume an elementary theoretical model usedin the bridge dynamics [1] and consider a simple beamof span l subjected to a row of forcesFn, n=1,2,3, …,N, which are moving with constant speedc from the left

to the right-hand side (see Fig. 4).N is the number ofaxle forces in the train.

The governing Bernoulli–Euler partial differentialequation describes the behaviour of the beam:

EI∂4v(x,t)

∂x4 1m∂2v(x,t)

∂t212mwd

∂v(x,t)∂t

5ONn51

en(t)d(x (1)

2xn)Fn

where it is denoted:v(x, t)=vertical deflection of thebeam at the pointx and timet, E=modulus of elasticity,I=constant moment of inertia of the cross section of thebeam,m=constant mass per unit length of the beam,wd=circular frequency of the damping andFn=the nthaxle force,

en(t)5h(t2tn)2h(t2Tn), (2)

h(t)5H0 for t,0

1 for t$0(3)

Fig. 4. Movement of a row of forcesFn long a beam of spanl atspeedc.

550 L. Fryba / Engineering Structures 23 (2001) 548–556

Heaviside unit function,

tn5dn/c (4)

time when thenth force enters the beam,

Tn5(l1dn)/c (5)

time when thenth force leaves the beam,d(x)=Dirac delta function describing the single con-

centrated force.

xn5ct2dn, (6)

dn=distance of thenth force from the first one (seeFig. 4),

d150, (7)

dn5(n21)d

in the case of equidistant forces.The boundary conditions of a simply supported beam

for x=0 andx=l and the zero initial conditions at the timet=0 when the first forceF1 enters the beam were con-sidered.

Consequently, the natural frequencies,wj, fj, andforms of natural vibrationvj(x) of a non-damped simplebeam are

w2j 5

j 4p4

l4EIm

, fj 5wj

2p, j51,2,3,… (8)

vj (x)5sinjpxl

. (9)

The method of integral transformations will be appliedto the problem. The mutual relations of the Fourier inte-gral transformation are [2]:

V(j,t)5El

0

v(x,t)vj (x) dx,

v(x,t)5O`j51

mVj

V(j,t)vj (x) (10)

where

Vj 5El

0

mv2j (x) dx,

which gives for the simple beam

Vj 5ml/2. (11)

The application of Eq. (10) to Eq. (1) gives

V(j,t)12wdV(j,t)1V2j V(j,t)5ON

n51

Fn

men(t)vj (ct2dn). (12)

The points over the symbols in Eq. (12) denote the dif-

ferentiation with respect to timet. The Laplace–Carsonintegral transformation [2]

V∗(j,p)5pE`

0

V(j,t)exp−pt dt,

V(j,t)51

2pi Ea1i`

a2i`

V∗(j,p)p

exptp dp (13)

is applied to the Eq. (12)

V∗(j,p)5ONn51

jwFn

mexp−pdn/cF(p)[12(21)jexp−pl/c] (14)

where it is denoted:

F(p)5p

(p2+j 2w2)[(p+wd)2+wj 92]

, (15)

w5pcl

, (16)

w92j 5w2

j 2w2d22iwdwj , (17)

V2j 5w92

j 1w2d. (18)

Deriving Eq. (14) the zero initial conditions and thetransformation of the Heaviside unit function appearingin en(t), Eq. (2) and Eqs. (27.6) and (27.10) from [2]were applied.

The inverse Laplace–Carson Eq. (13) and Fourier Eq.(10) transformations present the solution of the problemin the following form

v(x,t)5O`j51

ONn51

v0

Fn

Fjww2

1[f(t2tn)h(t2tn)2(21)j f(t (19)

2Tn)h(t2Tn)] sinjpxl

where the Laplace–Carson inverse transformation of Eq.(15) yields

f(t)51w9

j DFw9

j

jwsin (jwt1l)1exp−wdtsin (w9

j t1j)G (20)

with the notations

v052Fmlw2

1

52Fl3

p4EI<

Fl3

48EI(21)

deflection of a simple beam at its centre due to the forceF=Fn placed at the same point,

D25(V2j 2j 2w2)214j 2w2w2

d, (22)

l5arctan−2jwwd

V2j −j 2w2 , (23)

j5arctan2wdw9j

w2d−w92

j +j 2w2. (24)


Deriving Eqs. (19) and (20) the relation (27.41) from [2]was used.

The first term of the right-hand side of Eq. (20)expresses the forced vibration due to the moving forceswhile the second term denotes the free damped vihration,respectively. The step by step entrance of individualforcesFn to the beam at timetn as well as their departureat timeTn are described by the Heaviside unit functionsh(t) in Eq. (19) as well as by their shifts in timet2tnand t2Tn.

The bending moment of the beam

M(x,t)52EI∂2v(x,t)

∂x2

may be calculated in a similiar way and gives

M(x,t)5O`j51

ONn51

V0

Fn

Fj 3ww2

1[f(t2tn)h(t2tn) (25)

2(21)j f(t2Tn)h(t2Tn)] sinjpxl

,

where

M052Flp2 <

Fl4

(26)

is the bending moment at the centre of a simple beamdue to the forceF=Fn applied to the same point.

The vertical acceleration of the beam

a(x,t)5∂2v(x,t)

∂t2(27)

is also important and yields

a(x,t)5O`j51

ONn51

M0

Fn

Fjww2

1[ f(t2tn)h(t2tn) (28)

2(21)j f(t2Tn)h(t2Tn)] sinjpxl

where

f(t)52w92

j −w2d

w9j DF jww9j

w92j −w2

d

sin (jwt1l) (29)

1exp−wdt sin (w9j t1j9)G

with

j95j1arctan2wdw9j

w92j −w2

d

. (30)

When the derivative Eq. (27) was calculated, the zerotime derivations of the Heaviside function Eq. (3) weretaken into account in Eq. (28). In the previous equationsthe approximate relation

D<w2j 2j 2w2 (31)

may he applied to low damping.

Fig. 5. The axle distancesdi of the trains TGV, THALYS 2 andEUROSTAR.

3. Critical speeds

If the forces are at equidistant distancesd Eq. (7),then a repeated action of axles could cause a resonancevibration. The resonance condition is calculated from thetime necessary for crossing the distanced at speedcwhich is equal to thek-multiple of the period of naturalvibration 1/fj

dc5

kfj, j51,2,3,… k51,2,3,…,1/2,1/3,1/4,…. (32)

The condition Eq. (32) provides the critical speeds

ccr5dfjk

, j51,2,3… k51,2,3,…,1/2,1/3,1/4,…. (33)

The length d denotes a characteristic (many timesrepeated) distance between two axles in one bogie or thedistance between two bogies or a length of the vehicleas can be seen in Figs. 5 and 6.

Eq. (33) provides the speeds that could be reached bypresent trains (see Table 1).

The analysis of the acceleration–time history in Fig.3 shows that the vehicle length is deciding for the reson-ance vibration. Fig. 3 presents the frequencef1=2.8 Hzand the distanced=18.7 m (Fig. 5) for the length of aTGV car which corresponds well to the reality [seeEq. (33)].

Fig. 6. The axle distancesdi of the trains ICE 2 and ETR-Y.


Table 1Critical speedsccr according to Eq. (33) forj=1 andk=1, trains TGV,THALYS 2 and EUROSTAR

Spanl (m) f1 (Hz) di (m) Speedccr (km/h)

5 16 3.0 172.815.7 904.318.7 1077.1

10 8 3.0 86.415.7 452.218.7 538.6

15 5 3.0 54.015.7 282.618.7 336.6

20 4 3.0 43.215.7 226.118.7 269.3

30 3 3.0 32.415.7 169.618.7 202.0

40 2.5 3.0 27.815.7 141.318.7 168.3

The deformation of the beam may increase also fromanother reason:speed. Analysing Eqs. (19) and (22) wesee that for low damping, when

Vj <wj , (34)

see (18) and (17), the deformations become very large if

w2j 5j 2w2, (35)

see the denominatorD (22) in (20).The condition (35) yields another relation for the criti-

cal speed, see Eqs. (8), (16)–(18):

ccr52lfj

j, j51,2,3…. (36)

The condition (36) gives, however, very high speedsin actual cases (see Table 2). The reason is that the beamlooses its stability under moving forces in this case.

Nevertheless the analysis indicates many cases ofcritical speeds that may cause a resonance vibrationand/or its interference.

4. Estimation of maximum amplitudes of freevibration

The beam centrex=l/2 is considered in the calcu-lations which follow. Let us estimate the maximumamplitude of free vibration of the beam which isexpected at the moment when the last forceFN leavesthe beam, i.e. at the time [cf. with Eq. (5)]:

t5TN5l+dN

c. (37)

Table 2Critical speedsccr according to Eq. (36)

Spanl (m) fj, j=1, 2, 3 (Hz) Speedccr (km/h)

5 16 57664 1152

144 172810 8 576

32 115272 1728

15 5 54020 108045 1620

20 4 57616 115236 1728

30 3 64812 129627 1944

40 2.5 72010 144022.5 2160

Due to the properties of the Heaviside unit functionthe components of the forced vibrations vanish in (19)and (20) fort$TN and the maximum amplitude of freevibration at the centre of the beam appears forj=1, k=1,Fn=F andx=l/2 in the following form

v5ONn51

v0

ww1

D[exp−w

d(TN−tn)1exp−w

d(TN−Tn)]. (38)

The expression in the brackets of (38) may be summedfor n=1,2,3…Nbecause it forms a geometric progressionwith the quotient exp−w

dd/c and gives

EN5ONn51

[exp−wd(TN−tn)1exp−w

d(TN−Tn)] (39)

5[1+exp−J/(2a)]·[1−exp−JdN/(2al)]

expJd/(2al)−1.

If the exponential functions in (39) are developed ina series and only the first two terms are taken intoaccount the expression takes approximately the follow-ing form

EN<2S12J4aDN. (40)

It was denoted in the previous equations:

a5c

2f1l(41)

speed parameter, and

J5wd

f15

2pz100

(42)


logarithmic decrement of damping whilez is the damp-ing ratio in %.

For long trains it may be assumed a very large numberof axlesN→` and the expressions (39) and (40) are sim-plified

E`51+exp−J/2(a)

expJd/(2al)−1(43)

or approximately

E`<4alJdS12

J4aD. (44)

The expression (39), (40), (43) or (44) may be put inEq. (38) to receive a rough estimation of the maximumvalue of the amplitude of the beam which could appearafter the last axle leaves the beam. In this way using thesimplest expression (44), we obtain the amplitudes forthe deflection, bending moment and acceleration,respectively, in the following form:

v54a2blJd

v0, (45)

M54a2blJd

M0, (46)

a58a2blJd

FG

g (47)

where

b51−J/(4a)

1−a2 , (48)

G5mlg (49)

permanent load of the bridge,

g59.81 m/s2 (50)

gravitational constant.The expressions (45)–(47) may be applied to the first,

rough and quick assessment of railway bridges subjectedto high speed trains in resonance conditions in a similarway as the dynamic impact factor to the design ofbridges in normal conditions. For that purpose theexpressions could be brought in the formv/v0, M/M0 anda/g. The formulas (45)–(47) include the most importantparameters like the speed, span, natural frequency,damping, length of vehicles, their axle load and perma-nent load of the bridge. As they depend on the squareof the speed it is, thus, explained why the resonancevibration appears at high speeds only. Eqs. (45)–(47)present the highest estimation of the amplitudes and theinterference of natural modes of vibration as well asvarious axle distances could diminish their values.Therefore they remain on the safe side.

The amplitudes of the deflection (45) and acceleration(47) of a row of bridges are demonstrated in Table 3.

Table 3Maximum amplitudesv anda of the free vibration for bridges of vari-ous spansl

l (m) 5 10 15 20 30

G (kN) 350 1000 2250 4000 9000f1 (Hz) 16 8 5 4 3F (kN) 170c (km/h) 350d (m) 15J (1) 0.5 0.3 0.2 0.15 0.1v0 (mm) 0.943 1.320 1.502 1.320 1.043v/v0 Eq. (45) 1.238 4.559 13.364 19.509 30.602a/g Eq. (47) 1.203 1.550 2.020 1.658 1.189

5. Comparison with experiments

The experiments on high speed lines [6,7], haveshown that one of the decisive parameters for the bridgebehaviour is the vertical acceleration of the bridge deck.It was just the acceleration which drew attention to thisproblem. Therefore, the data of 12 French bridges (from[5] and [6]) under the traffic of TGV high speed trainswere summarized in Table 4 and the accelerationa cal-culated according to (47) were compared with experi-mental dataaexp. The acceleration as the function of thespeed is demonstrated in Fig. 7 [6].

Fig. 7 shows the experimental data measured in 1983and 1998 with the application of two filters (open andsolid symbols in Fig. 7). Unfortunately the resultsdepend on the filtration which affects the accelerationamplitudes. The filtration method has not yet been uni-fied and the records without filters can hardly be evalu-ated.

Nevertheless, the comparison of the simplified theorywith experiments is satisfactory and the set of data fromTable 4 provides in the mean about 10% higher calcu-lated values than the measured ones. The dispersion israther high but it is, of course, a quite general sign ofexperiments in the field tests.

6. Interoperability

Interoperability is a technical expression which hasbeen used by many specialists of several professions inrecent time. However its definition is a little vague.

The bridge engineers understand by interoperabilitythe capability of a bridge to carry a particular train orvehicle running at certain speed. On the other hand thevehicle specialists understand interoperability by thetechnical conditions which ascertain that the train couldmove on a given railway line including bridges at thedesigned speed. Of course these conditions must be ful-filled on international lines without respect to the borders(Trans European High Speed System).


Table 4Calculateda and measuredaexp accelerations of SNCF bridges under TGV trains [5,6]

Bridge SNCF l (m) f1 (Hz) J (1) G (kN) c (km/h) a (m/s2) aexp (m/s2)

a 38.00 2.8 0.031 12490 192 4.56 3.0OA01.51 12.83 10.0 0.314 1680 270 1.18 3.0OA71.18 17.43 4.5 0.251 2782 260 4.13 3.5OA71.38 10.90 9.0 0.251 1363 270 3.30 3.9OA71.47 14.40 4.8 0.314 2261 260 4.62 4.2OA71.78 9.70 10.0 0.314 1251 245 1.92 2.0OA89.22b 16.80 6.0 0.188 2582 290 4.18 4.2OA89.49 9.62 13.5 0.251 1178 270 1.65 2.3OA89.79 10.40 16.0 0.251 1430 270 0.80 1.0OA89.101 12.00 11.0 0.188 1920 295 2.14 0.8OA89.104 15.51 7.0 0.251 2275 270 2.14 1.9OA89.142 12.12 11.5 0.188 1673 270 1.76 1.8

a The acceleration–time history (see Fig. 3).b The acceleration–speed diagram (see Fig. 7).

Fig. 7. The accelerations of a concrete SNCF bridge of span 16.8 mas a function of the speed. The parameters of the bridge are given inthe row (2) in Table 4, see [6]. Comparison of the theory with experi-ments carried out in the years 1983 and 1998 using several filters (openand solid symbols).

The experiments, [6,7], have demonstrated that thecritical phenomenon on high speed lines is the destabiliz-ation of ballast on small and medium span bridges. Themaximally acceptable (ultimate) value of the acceler-ation of the bridge deck was estimated as

ault53.5 m/s2 or ault55 m/s2 (51)

for the bridges with ballast or without ballast respect-ively [6].

The decisive criterion will be derived from the con-

dition that the resulting maximum acceleration (47) mustbe lower or equal to the values (51)

ault$a58a2blJd

FG

g. (52)

We define and quantify now the interoperability whichshould be distinguished for both bridges and vehicles.

6.1. Bridge interoperability

Let us separate the vehicle and bridge parameters onthe left and right-hand sides of the inequality (52)respectively. It yields

B5JGault

8a2blg$

Fd

. (53)

If the bridge parametersB are known then the con-dition (53) gives the ratioF/d of the vehicle until thevalues (51) are crossed.

In what follows the bridge parameters from Table 5were applied to the calculation of the bridge interop-erability constantB according to Eq. (53). The resultsare summarized in Table 6. The concrete and steelbridges are distinguished here and the empirical approxi-mate formulas from [1]

Table 5Bridge parameters for various spans

Bridge Parameter Bridge spanl (m)material

5 10 15 20 30

Concrete G (kN) 350 1000 2250 4000 9000J (1) 0.5 0.3 0.2 0.15 0.1

Steel G (kN) 200 600 1500 2200 5000J (1) 0.25 0.15 0.1 0.08 0.05


Table 6Values of the bridge interoperability constantB (kN/m) calculated from Eq. (53)

Bridge material Accelerationault (m/s2) Speedc (km/h) Bridge spanl (m)

5 10 15 20 30

Concrete 3.5 350 17.7 12.4 11.4 10.9 10.5420 10.6 7.8 7.2 7.0 6.8

5.0 350 25.2 17.7 16.2 15.6 15.0420 15.1 11.1 10.3 10.0 9.7

Steel 3.5 350 16.3 12.6 12.7 10.9 9.9420 10.7 8.3 8.8 7.6 6.9

5.0 350 23.8 18.0 18.5 15.9 14.4420 15.3 12.0 12.5 10.8 9.8

f15140/l (54)

for concrete bridges and

f15250/l (55)

for steel bridges were applied to the numerical calcu-lation (f1 is in Hz andl in m). Then Eq. (41) gives (cin m/s) :

a5c/280 ora5c/500, respectively. (56)

Table 6 shows that the ratioF/d decreases withincreasing span and speed but increases with increasingultimate acceleration.

6.2. Vehicle interoperability

Another case arises if the vehicle parameters areknown and we seek the conditions for a bridge. The suit-able separation of parameters in the inequality (52) givesthe condition.

V152Fg

dault

JGf21l

c2b. (57)

The vehicle interoperability constantV1 is calculatedaccording to (57) and the results are shown in Table 7.The valuesF=170 kN and the distancesd=18.7 m forTGV, Thalys 2 and Eurostar whiled=26.1 m for ETR-Ywere used as well as the limits for the acceleration (51).

If the speed parameter (41) is included among the train

Table 7Values of the vehicle interoperability constantV1 (kN/m) calculatedfrom Eq. (57)

Accelerationault (m/s2) Lengthd (m) of the vehicle

TGV ETR-Y18.7 26.1

3.5 51 375.0 36 26

parameters [it could be allowed with respect to theapproximate relations (56)], then another simple con-dition arises

V258a2bFg

dault

#JGl

, (58)

wherebyb has a low effect.The vehicle interoperability constantsV2 were calcu-

lated using the parameters mentioned earlier and aregiven in Table 8 for various bridge materials, acceler-ations, speeds and lengthsd. As above the constantsV1

and V2 seem to be realistic for the distanced equals tothe length of the vehicle.

The suggested definitions of interoperability (53), (57)and (58) are equivalent and it is the question of the timewhich of them will be accepted by technical public.

7. Conclusions

Railway bridges subjected to high speed trains provideintensive vibration similar to the resonance phenomenon.

Table 8Values of the vehicle interoperability constantV2 (kN/m) calculatedfrom Eq. (58)

Bridge Acceleration Speedc Lengthd (m) of thematerial ault (m/s2) (km/h) vehicle

TGV ETR-Y18.7 26.1

Concrete 3.5 350 24.1 17.2420 37.7 27.1

5.0 350 16.8 12.1420 26.4 19.0

Steel 3.5 350 7.0 5.0420 10.4 7.4

5.0 350 4.9 3.5420 7.3 5.2


It arises at speeds higher than 200 km/h on small andmedium spans.

The resonance vibration of railway bridges results inthe deterioration of passenger comfort, reduction oftraffic safety (a possibility of derailment of vehicles) andthe destabilization of ballast (higher maintenance costs).

The simplest elementary theoretical model (simplebeam subjected to a row of forces) was analysed usingthe method of integral transformations. It results in sim-ple formulas (45)–(47) for the maximum amplitudes ofthe free vibration of deflection, bending moment andacceleration of the bridge deck at the moment when thelast axle leaves the bridge. These values transformed inv/v0, M/M0 and a/g — similar to the dynamic impactfactors — may be applied to the assessment or designof bridges if the first form of natural vibration is similarto the sine form (simple beam, continuous beam withequal spans, etc.).

The amplitudes of resonance vibration depend on thesquare of speed and on the span of the bridge andinversely on damping, vehicle length and bridge rigidity.Moreover, the acceleration depends also on the ratio ofthe axle load to the permanent load of the bridge.

Two reasons of resonance vibration of railway bridgeson high speed lines were discovered: repeated action ofaxle loads and loss of stability under moving forces.While the first reason appears actually on high speedlines at today’s speeds the second one is not yet actual.The critical speeds (33) and (36) were derived forboth cases.

The simple formulas (45)–(47) may serve for the first,rough and quick assessment of bridges. For a detaileddynamic analysis, the idealization of a bridge by finiteelements and of the vehicles by a system with lumpedmasses, springs and dampers is recommended [1]. Theother approaches, taking into account the inertial effectsof vehicles, track irregularities, sleeper and cross girdereffects, etc., are also possible. Further information aboutthe problem may be found in [6,8–15].

The interoperability of high speed trains on railwaybridges was defined and quantified in the form of inter-operability constantsB for bridges orV for vehicles.They proceed from the condition that the running traindoes not cross the ultimate vertical acceleration of thebridge deck.

The derived interoperability conditions (53), (57) or(58) are very simple and the entering constants are wellknown or could be easily calculated or measured. Theymay be applied to a quick assessment of bridges orvehicles suitable for high speed traffic.

Acknowledgements

The support of ERRI, the programme KONTAKT ME154/2000 and grants 103/96/K034 and 103/98/1479 fromthe Grant Agency of the Czech Republic is gratefullyacknowledged.

References

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[4] Kazej II. Dynamic analysis of railway bridges. Moscow:Transzheldorizdat, 1960 (In Russian).

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[8] Matsuura A. Dynamic behavior of bridge girder for high speedrailway bridge. Quarterly Reports, Railway Technical ResearchInstitute, Tokyo 1979;20(2):70–6.

[9] Yang et al. Vibration of simple beams due to trains moving athigh speeds. Engineering Structures 1997:19:936–44.

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[15] Naprstek J. Resonance speed of axle forces row moving along abeam. In: Proceedings: Colloquium Dynamics of Machines 2000.Prague: IT ASCR, 2000:145–52. (in Czech).

a rough assessment of railway bridges for high speed trains

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