Dynamic excitationsof transport structures

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

UNC 2008

Institute of Theoretical and Applied Mechanics, v.v.i.Academy of Sciences of the Czech Republic, Prague

• Introduction• Movement of loads• Rails• Random vibration of structures• Degradation of structural materials under

dynamic loads• Conclusions

Outline

Introduction

Movement of the load

Transport structures – moving load– earthquake– wind

Moving load

[ ] [ ]2

2

( ),( , ) ( )

d v u t tf x t x u t F m

dtδ

⎧ ⎫⎪ ⎪= − −⎨ ⎬⎪ ⎪⎩ ⎭

m

v(x,t)

x u t= ( )

[ ] 22 2 2 2 2

2 2 2 2

( ),2

d v u t t v du v du v d u vdt u dt u t dt u dt t

∂ ∂ ∂ ∂⎛ ⎞= + + +⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

Uniform movement at velocity c2

2( ) , 0du d ux u t ct cdt dt

= = = =

2 2 2 22

2 2 2

( , ) ( , ) ( , ) ( , )2x ct

d v ct t v x t v x t v x tc cdt t x t x

=

⎡ ⎤∂ ∂ ∂= + +⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦

Motion in (x,y) plane x = u(t), y = v(t)load f(x,y,t) – m(x,y,t) d2w / dt2

2 22 2 2 2 2

2 2 2 2 2d w w du w dv w w du dvdt x dt y dt t x y dt dt

⎡∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞= + + + +⎢ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎢⎣2 2 2 2

2 2( )( )

2 2x u ty v t

w du w dv w d u w d vx t dt y t dt x dt y dt =

=

⎤∂ ∂ ∂ ∂+ + + + ⎥∂ ∂ ∂ ∂ ∂ ∂ ⎦

Rails

Diferencial equation

0( , ) ( ),v x t v v s=

( , ) ( , ) 2 ( , ) ( , ) ( )IVbEIv x t v x t v x t kv x t x ct Pμ μω δ+ + + = −&& &

Steady state vibration

0 38 2P PvEI k

λλ

= =

1/ 4

4kEI

λ ⎛ ⎞= ⎜ ⎟⎝ ⎠

P

v(x,t)

x t= c

-x +xk

s

0

Moving coordinate:

Speed parameter:

Damping parameter:

Critical speed:

( )s x ctλ= −1/ 2

2cr

c cc EI

μαλ

⎛ ⎞= = ⎜ ⎟⎝ ⎠

1/ 2( / )b kβ ω μ=1/ 2

2crEIc λμ

⎛ ⎞= ⎜ ⎟

⎝ ⎠

Ordinary differential equation:2( ) 4 ( ) 8 ( ) 4 ( ) 8 ( )IV II Iv s v s v s v s sα αβ δ+ − + =

Solution

1 1 2 12 21 1 2

2( ) ( cos sin )( )

bsv s e D a s D a sa D D

−= ++

3 2 4 22 22 3 4

2( ) ( cos sin )( )

bsv s e D a s D a sa D D

= ++

for s > 0

for s < 0

Random vibration of a beam on elestic foundation under moving force

Equation of the beam

[ ]( , ) ( , ) ( , ) 2 ( , ) ( ) ( , )IVbL v x t EIv x t v x t v x t k x v x tμ μω≡ + + + =&& &

( ) ( )x ct F tδ= −

Foundation ( ) ( )k x k k xε= +o

[ ]( )k E k x=

Force 1ε( ) ( )F t F F tε= +

o

Moving coordinate

Static deflection and bending moment underforce F

[ ]( )F E F t= 1/ 4

( ) ,4ks x ctEI

λ λ ⎛ ⎞= − = ⎜ ⎟⎝ ⎠

0 03 ,8 2 4

F F Fv MEI k

λλ λ

= = =

Coeficient of variation VV and VM as a function of speed α,β = 0.2, γi = 0, σ = 0.2, D = 10/λ

COVARIANCEPOWER SPECTRAL DENSITY

Random vibration of a beamBernoulli-Euler equation

Normal mode analysis

Generalized deflection

Generalized force

( , ) ( , ) 2 ( , ) ( , )IVbEI v x t v x t v x t p x tμ μω+ + =&& &

1( , ) ( ) ( )j j

jv x t v x q t

∞

=

= ∑

1( , ) ( ) ( )j j

jp x t v x Q tμ

∞

=

= ∑2( ) 2 ( ) ( ) ( )j b j j j jq t q t q t Q tω ω+ + =&& &

0

1 ( , ) ( ) 0( ) for

00

L

jjj

p x t v x dxMQ t t

⎧≥⎧⎪= ⎨ ⎨<⎩⎪

⎩

∫

Solution

( ) ( ) ( ) ( ) ( )j j j j jq t h t Q d h Q t dτ τ τ τ τ τ∞ ∞

−∞ −∞

= − = −∫ ∫

Impulse function (weighting function)1( ) ( )

2i

j jh H e dτωτ ω ωπ

∞

−∞

= ∫

Frequency response function

( ) ( ) ij jH h e dωτω τ τ

∞−

−∞

= ∫Load

[ ]( , ) ( , ) ( , )p x t E p x t p x t= +o

Stationary vibration of a beam undermoving continuous random load

Load ( , ) ( ) 1 ( )p x t p p s r tε ε⎡ ⎤ ⎡ ⎤= + ⋅ +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

o o

[ ]( , )p E p x t=2( , ) ( ) ( ) ( ) ( )p x t p s p r t p s r tε ε ε= + +

o o o o o

Covariance of the moving load

/s t x c= −1ε

2 2 2 21 2 0 1( , , ) ( ) ( ) ( / )pp pp rr prC x x C p C pC x cτ ε τ τ ε τ ε τ= + + + + +

2 3 3 3 3 42( / )rp ppr prr ppr prr pprrpC x c C pC C pC Cε τ ε ε ε ε ε+ − + + + + +

1 20

x xc

τ −=

Non-stationary vibration of a beamunder moving random force

Load( , ) ( ) ( )p x t x ct P tδ= − ( ) ( )P t P P t= +

o [ ]( )E P t P=

Covariance of the force

Covariance of moving load

Covariance of generalized deflection

1 2 1 2( , ) ( ) ( )ppC t t E P t P t⎡ ⎤= ⎢ ⎥⎣ ⎦

o o

1 2 1 2 1 1 2 2 1 2( , , , ) ( ) ( ) ( , )pp ppC x x t t x ct x ct C t tδ δ= − −

, ,

1 2 1 1 2 2 1 2 1 2 1 21( , ) ( ) ( ) ( ) ( ) ( , )

j kq q j k j k ppj k

C t t h t h t v c v c C d dM M

τ τ τ τ τ τ τ τ∞ ∞

−∞ −∞

= − −∫ ∫

Speed parameterDamping parameter

Simple beam

1/(2 )c f lα =

1/ /(2 )bβ ω ω ϑ π= =3

0( ) sin ,48j

j x Plv x vl EIπ

= =

Degradation of structural materialsunder dynamic loads

Dynamic loads

Fatigue tests

Wöhler curve

Concept of fatigue assessment

Static system

Material properties(Wöhler curve)

“Real” LoadsP (t)

Stress curveσ (t)

Spectrumni (σo, σn)

Calculation valuefor damage DSd

ProofDSd ≤ Dlim

Simulation oftrain passages

Counting procedure(Rainflow)

Damage hypothesis(Palmgren-Miner)Δσ = σmax - σmin

ρ = σmin / σmax

Strategy of maintenance, repairs and reconstructions,

yes – the conditionsare fulfilled,

not – the conditionsare not fulfilled

Effect of traffic loads

Stress-time recordsstatistical evaluation

estimation of residual life

yes

yes

yes

yes

yes

Modal analysis

criteria of dereriorationnot

not

not

not

Monitoring of cracks

criteria of cracks

Inspections

Technical economic decision

Maintenance

Repair

Reconstruction

Conclusions

The dynamic loads, i.e. in time varying loads, increase the stresses in transport structureswith increasing speed

The dynamic loads are varying in time eitherregularly or randomly

The response of structural materialsdeteriorates their properties in time