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Dynamics of Bridges

Under Moving Loads

Prof. Ing. Ladislav Frba, DrSc., Dr.h.c.

UNC 2008

Institute of Theoretical and Applied Mechanics, Academy of Sciences

of the Czech Republic, Prague

(Past, Present and Future)

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Contents:

1. Introduction

2. Past3. Present

4. Future5. Conclusions

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1. Introduction

First railway bridges in England

First experimental and theoretical papers

by Stokes (1849) and Willis (1849) Important progress by Timoshenko,

Inglis and Kolouek

steam locomotives F (t) = F0 sin t

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theory

experiments theory

Steel bridge, l= 56.56 m

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2. Past

International investigations by ORE, ERRIand OSD

Dynamic characteristics of bridges

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First natural frequencies of bridges

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Logarithmic decrements of damping

steel concrete

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Cross girder effect

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Sleeper effect

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Theoretical model[ ]

2 2

21

d ( )( 1) ( ) ( ) 0,

d

=

+ + =i

i

i i biIt

- D Z t Z t t

[ ]2 2

3

3 21

d ( )( ) ( ) 0

d = + = i bi

i

v tm Z t Z t

t

2

23

d ( )( )

d

( ) ( ) 0; 1, 2,

+ + +

= =

ii i i i

bi i

v tP P m Z t

t

+Z t R t i

4 2 2

4 2i =1

( ) ( , ) ( , )2 ( ) ( ).

+ + =

b i i iv x,t v x t v x t

EJ x x R t

x t t

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Effect of the speed

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ORE experiments (DB, SNCF)

DB steel bridge, l= 19.6 m, 200 km/h

SNCF composite bridge, l= 26.4 m, 241 km/h

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Stochastic concept

( ) ( ),= + &t F F t

[ ][ ]( , ) ( ) 1 ( )= + +& &f x t p p s r t

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3. Present

Resonant vibration

SNCF bridge, l= 38 m, TGV, 192 km/h

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Theoretical idealization

( )4 24 2

1

, ( , ) ( , )2 ( ) ( )

N

d n n n

n

v x t v x t v x tEI t x x F

x t t

=

+ + =

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Vibration of a steel bridge at low and resonant speed

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Effect of the speed on deflection

and bending moments

Effect of the speed

on vertical accelerationsteel bridge, l= 5 m concrete bridge, l= 10 m

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Stress spectra

max min =

stress range

rain-flow

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Stress spectra for steeland concrete bridges

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concrete bridge, l= 5 m

Effect of the speed

Stress ranges

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Critical speeds

, 1, 2, 3,..., 1, 2, 3,...,1 2,1 3,1 4,...j

cr

dfc j k

k

= = =

2, 1, 2, 3,...

j

cr

lfc j

j= =

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4. Future

Elastic supports Triangular falsework

Double beam Double string

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( ) ( )

( ) ( )

4 2 2

1 11 1 1 24 2 2

=

1(x,t) (x,t) (x,t)EI N + + k x,t x,tx x t

= t x - c t F

( ) ( )( ) ( )

2 2

2 2

2 2 2 12 2 0

x,t x,tN + + k x,t x,t =

x t

Beam coupled with a string

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= 0

= 0.2

= 1

= 2

beam deflection

string deflection

= dimensionless speed

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v1(x) < 1 for B12 < A12 A22

v2(x) < 1 for B22 < A12 A22At = 0:

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5. ConclusionsDynamic effects on bridges rise with increasing speeds

of trains

The dynamic response of concrete bridges is a littlelower than steel ones due to their different mass anddamping ratios

Stress spectra present important data for theassessment of bridges at fatigue

Vertical accelerations of bridges may be a limit state for

the design of new high speed linesThe diminishing of bridge dynamic response could be

achieved with dampers or by interconnection of

prestressed beams with pretensiled strings

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