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Dynamic analysis of railway bridges by means of the spectral method

Giuseppe Catania, Silvio SorrentinoDIEM, Department of Mechanical Engineering,

University of Bologna, Viale del Risorgimento 2, 40136 Bologna, Italy

ABSTRACT

This study investigates the dynamic behaviour of railway bridges crossed by travelling trains. A simplified formulation was

adopted in order to perform a direct analysis of the effects of the parameters involved in the problem. The bridge is modelled

as a rectangular plate, while the trains are modelled as travelling inertial distributed loads. The formulation is accomplished

by the use of the Rayleigh-Ritz method, yielding a low order model with time-dependent coefficients. Several numerical

examples are presented and discussed, aimed at investigating the effects of each of the model-governing parameters.

Keywords

Railway bridges, moving loads, Rayleigh-Ritz method.

Nomenclature

C damping matrix u unit step distributionD stiffness of the plate V potential of applied loadsE Youngs modulus v velocity of the loadg gravity acceleration w displacementH velocity response function x spatial coordinate

h thickness of the plate y spatial coordinate

K stiffness matrix D velocity ratioL length E frequency parameterl width H relative error

M mass matrix ) modal matrixn modal index P length ratiop load per unit area Poissons ratio

r mass ratio U mass per unit area

q modal coordinate n modal natural frequency

t time [ moving coordinate

U potential of strain energy ] damping factor

1 Introduction

In the analysis of dynamic effects of railway vehicles on bridges, simplified models are suggested and usually used, taking

into account only some aspects, such as deterministic, vertical effects, and the influence of moving forces and masses [1-2].Inertial effects of both bridge and vehicle can be influential, and not negligible, since the mass of the external load introducesa coupling effect between the load and the structure. Other important aspects, such as dynamic properties of travellingvehicles and track irregularities, are not considered in the present study.The railway bridge model most commonly used is a continuous Euler Bernoulli beam [2], or a Timoshenko beam [3],traversed by either concentrated [4] or distributed moving loads [5]. Possible applications of lumped vibration absorbers have

also been investigated [6]. However, in the present study a homogeneous Kirchhoff plate is considered, allowing the analysisof lateral vibrations due to trains travelling on double-track bridges. Structure damping is included in the model, as it may

play an important role.

DOI 10.1007/978-1-4419-9316-8_2, The Society for Experimental Mechanics, Inc. 2011

21T. Proulx (ed.), Civil Engineering Topics, Volume 4, Conference Proceedings of the Society for Experimental MechanicsSeries 7,

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The train is simply modelled by means of a continuous load in the form of a moving strip, an idealization which can beadopted when the span of the bridge is large in comparison with the distance between axles [2]. A constant speed of motion

of vehicles along the bridge is assumed.The formulation is accomplished by the use of the Rayleigh-Ritz method [7], and the solution is expressed in terms of a linear

combination of functions, which in the present study are selected as tensor products of eigenfunctions of prismaticpinned-pinned and free-free beams in flexural vibration. This approach yields a reduced order model with time-dependentcoefficients, allowing a parametric analysis of plates loaded by travelling distributed masses.

Different example cases are presented and discussed in detail, analyzing the effects of velocity, mass and length of the trainon the plates dynamic response with respect to the mass, stiffness and damping of the plate itself.

2 Theoretical model

A homogeneous isotropic Kirchhoff plate is considered, simply supported on two opposite sides, free on the other two sidesand crossed by a travelling distributed load. The load per unit areap over the plate may be expressed as:

2

2

( , , )( , , ) ( , ) , vt x

d w x y t p x y t f y g x t

dtU [ [

, (1)

where w is the vertical displacement of a point of the plate or of the load (Fig.1),Ut is the equivalent mass per unit area of the

load,gis the gravity acceleration, vx is the travelling speed in thex direction, [ is a moving coordinate in the same direction[1] andfmodels the translating strip representing the instantaneous position of the load:

> @> @( , ) ( ) ( ) ( ) ( )t tf y u L u u y u y l[ [ [ G G . (2)

Note that within the present studyUt is assumed to be constant; however, piecewise-constant or other distributionsUt([) maybe considered and adopted in the following developments.

Equation 2 containsthe unit step distribution u(),Ltand ltare the length and the width of the strip modelling the train, and Gis the distance between the side of the strip and the edge y = 0 of the plate, as shown in Fig. 1. The second term on theright-hand side of Eq. 1 describes the inertial action of the load. The total acceleration may be expressed in the followinggeneral form:

2 2 2 2 2 2 22 2

2 2 2 2v 2v v v 2v 2v a ax x y y x y x y

d w w w w w w w w w

x y x t y t x ydt t x yw w w w w w w w

w w w w w w w ww w w(3)

where vx, vy, ax, ay express the velocities and accelerations of the travelling load in the x and y directions respectively [1].Considering a train travelling at constant speed v in thex direction, Eq. 3 reduces to:

2 2 2 22

2 2 22v v

d w w w w

x tdt t x

w w w

w ww w(4)

The first term of the right-hand side of Eq. 4 expresses the influence of vertical acceleration of the moving load, the secondterm the influence of Coriolis acceleration, and the third term the influence of track curvature [1].

y

x

Lt

lt

lb

Lb

G

vx

0

Fig. 1 Model scheme

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The functional of the total potential energy of the coupled system can be written as the sum of a term Udue to the strainenergy plus a term Vrepresenting the potential of all applied loads (including the inertial forces):

U V3 (5)

The potential of the strain energy can be written in terms of second order derivatives of the out-of-plane displacement w:

32 2 2

2

0 0

12 2(1 ) ,

2 12(1 )

b bl L

xx yy xx yy xy

EhU D w w w w w dxdy DQ Q

Q (6)

where the subscripts denote differentiation with respect to the spatial variables and D is the flexural stiffness of the plate,

expressed as a function of Youngs modulusE, Poissons ratio Q and thickness h [7]. In the adopted formulation the inertialforces are included in the potential of applied loads Vas follows:

b

0 0

( )b bl L

V ww wp dxdyU (7)

where Ub is the mass per unit area of the plate and p is the load in Eq. 1. The out-of-plane displacement w is expressed by

means of a linear combination of shape functions, selected as products of homogeneous uniform prismatic beameigenfunctions I:

1

( , )N

n n

n

w q wI [ K 7

q (8)

where q is the generalized coordinate vector. Introducing the displacement expansion in the quadratic functional 3, andimposing its stationarity, yields the following algebraic eigenproblem:

2 2[ ] 2 [ ] [ ]r r r rg ' ' ' M M q C q K K q f (9)

with:

2

v 1, ,

t

b b bb

Dr

L L

U

U U (10)

where E is a frequency parameter and D is a dimensionless parameter depending on the speed v. The matrices in squarebrackets in Eq. 9 can be regarded as dimensionless quantities, and they can be computed according to the following integrals:

1 1

0 0

1 1

0 0

T

0 0

4 T T T T T

0 0

2 T

( ) , ( ) , ( )

[ ( ) 2(1 )( )]

2 ( ) ,

b b t t

b b

t t

l L l l x x

b x

x x

l L

b xx xx yy yy xx yy yy xx xy xy

l lx x

b xx

x x

dxdy dxdy L dxdy

L dxdy

L dxdy dxdy

Q Q

7 7

' '

'

M M C

K

K f

II II II

I I I I I I I I I I

II I

(11)

In Eq. 11 the integration interval [x0, x1] is time-dependent.Introducing the ratio between the lengthsLtandLb:

t

b

L

L (12)

thenx0 andx1 vary according to Tab. 1.

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Table 1 Time-dependent interval of integration

case 1 case 1 case 1!

1

0

v0 v

0t

x tt L

x

-d

1

0

vv

vt b

t

x tL t L

x t L

-d

1

0

vv

b

b b t

t

x LL t L L

x t L

-d

1

0

v0 v

0b

x tt L

x

-d

1

0

v 2v

b

b b

b

x LL t L

x t L

-d

1

0

v0 v

0b

x tt L

x

-d

1

0

v0

b

b t

x LL t L

x

-d

1

0

vv

b

t t b

t

x LL t L L

x t L

-d

To model energy dissipation within the structure, a dimensionless damping matrix C may be defined by means of the plate

modal matrix ) (mass normalized) and eigenvalues Zn (computed from the M and Kmatrices), and considering a modaldamping ratio ] equal for all modes:

T 1 T 12 2 2

n ndiag diag Z Z C (13)

Introducing Eq. 13 in Eq. 9 yields:

2 2[ ] 2[ ] [ ]r r r rg ' ' ' M M q C C q K K q f (14)

Equation 14 is a reduced order discretized model with time-dependent coefficients, which can be solved numerically.

3 Numerical results

Some numerical examples are presented for studying the dynamic behaviour of the model described in section 2. The

influence of parameters v, r, P, E, ] governing Eq. 14 is highlighted by studying time responses w(t) and dynamic responsefunctionsHof the dimensionless frequency D (playing the role of frequency response functions) defined according to:

> @

, ;

max ( )( , ; )

w

t

s x y

w tH x y

D

D (15)

where wsis the static deflection due to the load centered inLb/2.

Numerical solutions of Eq. 14 are computed using the Runge-Kutta algorithm, expanding the solution w (Eq. 8) with 4 u 2beam eigenfunctions (4 pinned pinned eigenfunctions alongx direction and 2 free free eigenfunctions alongy direction).

Realistic values for parameterE are computed by means of the empirical expression:

2

1

2 baL JSO

(16)

based on large collections of experimental data [2], where a and Jare parameters depending on the kind of bridge considered,

as reported in Tab. 2. The values (in Hz) of the first natural frequency and of parameterE for different kinds of bridges arereported as functions of the lengthLbin Fig.2.

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10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

Lb

[m]

f1[H

z]

General bridges

Steel truss bridges

Steel plate girder bridges with ballast

Concrete bridges with ballast

Concrete bridges without ballast

10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

12

14

16

18

Lb

[m]

E[

rad

/s]

General bridges

Steel truss bridges

Concrete bridges with ballast

Fig. 2 First natural frequency f1 [Hz] (left) and frequency parameterE [rad/s] (right)as functions of the lengthLb for different kinds of bridges

Table 2 Parameters in Eq.16, as reported in [2]

Kind of bridge a J

General bridges (average case) 133 0.9

Steel truss bridges 307 1.1

Steel plate girder bridges with ballast 59 0.7

Steel plate girder bridges without ballast 208 1

Concrete bridges with ballast 190 1.1

Concrete bridges without ballast 225 1.2

Effect of the speed of the load

As a reference case study, the following values for the parameters are assumed:

Plate:Lb = 50 m, lb = 10 m, E = 5 rad/s, ] = 0.05.Moving load: P = 1.4, lt= 2.5 m, G = 1.5 m, r= 0.5.Time responses w(t) are computed at coordinatex =Lb/2,y = lb/2 with speed v varying from 30 m/s to 50 m/s (108 Km/h to180 Km/h), as shown in figure 4.

Maximum deflection at different points

Parameter values are assumed as in the reference case. Response functionsH(D) are computed at different points (x,y) alongthe structure, as reported in Fig. 3. The frequency parameter varies from 0.1 up to 1, i.e. v varies from 40 Km/h up to 400

Km/h. FunctionsH(x,y; D) show a peculiar undulating trend, not significantly affected by the choice of coordinatex.

Effect of the mass of the load

Parameter values are assumed as in the reference case with v = 40 m/s = 144 Km/h, varying r from 0.1 to 1. Response

functions w(t) andH(D) are computed inx =Lb/2,y = lb/2, as reported in Fig. 4.

Effect of the length of the load

Parameter values are assumed as in the reference case with v = 40 m/s = 144 Km/h, varying P from 0.1 to 2. Responsefunctions w(t) and H(D) are computed in x = Lb/2, y = lb/2, as reported in Fig. 5. ParameterP (related to Lt) is able tosignificantly affect the behaviour ofH(D). Note that the plots ofH(D) in the case Pt 1 are superimposed.

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0 1 2 3 4 5-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5x 10

-3

t [s]

w[m

]

v = 40 m/s

v = 50 m/s

v = 30 m/s

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 111

1.02

1.04

1.06

1.08

1.1

D

H

x = Lb/2; y = l

b/2

x = Lb/4; y = l

b/2

x = 3Lb/4; y = l

b/2

x = Lb/2; y = 0

x = Lb/2; y = l

b

Fig. 3 Effect of parameter v on w(t) (left);H(D) at different points (x,y) (right)

0 1 2 3 4 5-7

-6

-5

-4

-3

-2

-1

0

1 x 10-3

t [s]

w[m]

r= 0.5

r= 1.0

r= 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.05

1.1

1.15

D

H

r= 1.0

r= 0.8

r= 0.5

r= 0.1

Fig. 4 Effect of parameterron w(t) (left); effect of parameterronH(D) (right)

0 1 2 3 4 5-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5x 10

-3

t [m]

w[m]

P = 0.2

P = 0.6

P = 1.0

P = 1.4

P = 1.8

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

D

H

P = 2.0

P = 1.0

P = 0.5

P = 0.1

Fig. 5 Effect of parameterP on w(t) (left); effect of parameterP onH(D) (right)

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.02

1.04

1.06

1.08

1.1

D

H

E = 50.5

rad/s

E = 30.5

rad/s

E = 80.5

rad/s

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.02

1.04

1.06

1.08

1.1

D

H

] = 0

] = 0.01

] = 0.05

] = 0.10

] = 0.20

] = 1.00

Fig. 6 Effect of parameterE onH(D) (left); effect of parameter] onH(D) (right)

0 0.5 1 1.5 2 2.5 3 3.5-2.5

-2

-1.5

-1

-0.5

0

0.5 x 10-3

t [s]

w[m]

F = 1/480

F = 1/48

F = 1/6

F = 1/2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.05

1.1

1.15

1.2

1.25

1.3

1.35

D

H

F = 1/2

F = 1/6

F = 1/48

F = 1/480

Fig. 7 Effect of partially distributed load on w(t) (left); effect of partially distributed load on onH(D) (right)

Effect of structural stiffness

Parameter values are assumed as in the reference case, varying E from 3 rad/s to 8 rad/s. Response functions H(D) arecomputed inx =Lb/2,y = lb/2, as reported in Fig. 6. The plots are almost superimposed.

Effect of structure damping

Parameter values are assumed as in the reference case, varying ] from 0 to 1. Response functionH(D) is computed inx =Lb/2,y =lb/2, as reported in Fig. 6. Raising ] reduces the amplitude of oscillation ofH(D), until its behaviour becomes monotonic (howeverthis is not the case for real bridge structures).

Effect of partially distributed load

Parameter values are assumed as in the reference case, with Lt = 24 m and v = 40 m/s = 144 Km/h. Different loadingdistributions are compared: the continuous one (as represented in Fig. 1) and partial distributions consisting of two shorter

sections in which the load is distributed.The assumed partial distributions are given by:

0 andt tt tL L

L L[ [F F

d d d d

(17)

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with F 0.5 (F= 0.5 yields the continuous distribution). Since for the continuously distributed load it is assumed r0= 0.5,for the partially distributed load described by Eq. 17 r0 increases to r= 1/(2F) ur0. Response functions w(t) andH(D) arecomputed inx =Lb/2, y = lb/2 for different values ofF (1/6, 1/48, 1/480) as reported in Fig. 7. Load distribution variations

such as that described in Eq. 17 may dramatically affect the behaviour of the response functionH(D).

Effect of time dependent matrices

Parameter values are assumed as in the reference case, with v = 40 m/s = 144 Km/h. The effect of neglecting the timedependent matrices 'M, 'C, 'Kon the solution w(t) is evaluated by introducing a relative error, according to:

> @M

( ) ( )

max ( )t

w t w t

w tH '

M 0 (18)

where [w(t)]'M = 0 refers to the solution computed assuming 'M = 0 in Eq. 14.Similarly, HC and HKcan be defined, considering 'C = 0 and 'K= 0.

0 0.5 1 1.5 2 2.5 3 3.5 4-10

-8

-6

-4

-2

0

2

4

6x 10

-3

t [s]

H

HC

HK

HM

HTot

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.02

1.04

1.06

1.08

1.1

D

H

'M = 'C = 'K = 0

Exact

'C = 'K = 0

'C = 0

Fig. 8 Effect of neglecting time dependent matrices 'M, 'C, 'Kon w(t) (left) and onH(D) (right)

The error functions HM,HC and HKare plotted versus time in Fig. 8, where HTot represents the total error, assuming 'M, 'C and'Kequal to 0 at the same time. The smallest, and negligible contribution to the error HTot appears to be HC, while the maincontribution is due to HM. Fig. 8 also shows the effect on H(D) of neglecting the time dependent matrices 'M, 'C, 'K.Again, the effect of neglecting 'C is very small. The total error, on the contrary, can be significant.

4 Discussion

FunctionH(x, y; D) appears to be an effective tool for studying the dynamic behaviour of a structure crossed by travellingloads with constant speed, in some way equivalent to a frequency response function for time-varying coefficient systems.This function shows peculiar undulating trends (Fig. 3), influenced by the parameters governing Eq. 14. The response can be

evaluated at any coordinate point (x, y) of the plate, making it possible to study the variation of structural deflection alsoalong they coordinate (Fig. 3). Mass parameterrcan produce important shifts in magnitude, but not in shape (Fig. 4). On the

contrary, length parameterP controls both shape and magnitude ofH(D) (Fig. 5), but only in the case 0 < Pd 1. The dampingparameter] has the effect of progressively smoothing the oscillation ofH(D), until it becomes monotonic (Fig. 6, though thelatter limit case is not realistic for actual bridge structures): in general, the reduction in amplitude becomes particularly

significant at high speed. Frequency parameterE, within the range of real bridges, scarcely affects the behaviour ofH(D)(Fig. 6), soHmay be considered independent from E. Changes in the spatial distribution of the load can produce dramaticvariations inH(D) (Fig. 7): this result should highlight the importance of properly modelling the ballast, directly influencingthe load distribution on the actual structure.

The contribution to the solution of the time dependent matrices 'M, 'C and 'Kis globally not negligible (Fig. 8), howeverthe effect of'C is usually very small in comparison with the contributions of'K, and especially of'M.

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5 Conclusions and future work

In this study, the dynamical behaviour of railway bridges crossed by travelling trains was investigated by adopting asimplified model, i.e. a plate loaded by a travelling distributed mass, solved by means of the Rayleigh-Ritz method.

The effects of each of the model governing parameters was studied introducing a dynamic function of the travelling speed,equivalent to a frequency response function for time-varying coefficient systems. This function can be an effective tool forstudying the dynamic behaviour of a structure crossed by travelling loads, since the travelling speed is the most important

parameter influencing the dynamic stresses in railway bridges, which in general increase with increasing speed.In particular, it was shown how different spatial distributions of the load can deeply influence the dynamic response of thestructure, highlighting the importance of properly modelling the ballast. Future work will thus concern this significantproblem.

Acknowledgments

This study was developed within the INTERMECH laboratory with the contribution of the Regione Emilia Romagna -Assessorato Attivit Produttive, Sviluppo Economico, Piano telematico, PRRIITT misura 3.4 azione A Obiettivo 2.

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[3] Lin Y.H., Vibration analysis of Timoshenko beams traversed by moving loads, Journal of Marine Science and

Technology 2 (4), pp. 25-35, 1994.

[4] Stancioiu D., Ouyang H., Mottershead J.E., Vibration of a continuous beam excited by a moving mass and

experimental validation,Journal of Physics, Conference series 181, 1999.

[5] Adetunde I.A., Dynamical Behavior of Euler-Bernoulli Beam Traversed by Uniform Partially Distributed Moving

Masses,Reasearch Journal of Applied Sciences 2(4), pp. 476-483, 2007.

[6] Lin Y.H., Cho C.H., Vibration suppression of beam structures traversed by multiple moving loads using a damped

absorber,Journal of Marine Science and Technology 1 (1), pp. 39-48, 1993.

[7] Timoshenko S., Young D.H., Weaver W., Vibration problems in engineering, 4th edition, Wiley, 1974.

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