Frequency-Domain Analysis of An Elevated Rail Bridge Using A Periodic Method

Y.S. Yang (speaker)National Center for Research on Earthquake Engineering

Y.J. LeeNational I-Lan Institute of Technology

T.W. LinNational Taiwan University

The 9th International Conference on Computing in Civil and Building EngineeringApril 3-5, 2002Taipei, Taiwan

National Center for Research on Earthquake Engineering

Objective

Ground vibration response Induced by an elevated rail bridge

The elevated rail bridge Consists of hundreds of spans

The 9th International Conference on Computing in Civil and Building EngineeringApril 3-5, 2002Taipei, Taiwan

National Center for Research on Earthquake Engineering

Analysis methods (1/2)

N-span time-domain analysis Advantages:

Complicated structure configuration Nonlinear response

1-span frequency-domain analysis (periodic method) Advantage:

Needs fewer degrees of freedom

The 9th International Conference on Computing in Civil and Building EngineeringApril 3-5, 2002Taipei, Taiwan

National Center for Research on Earthquake Engineering

Analysis methods (2/2)

N-span time-domain analysis

Periodic method Limitation:

Linear analysis (freq. domain) Identical spans A large number of spans

The 9th International Conference on Computing in Civil and Building EngineeringApril 3-5, 2002Taipei, Taiwan

National Center for Research on Earthquake Engineering

F = F ( x , t )

F = F ( x, t )

ru lu

Periodic method (1/4) Time phase of response

The 9th International Conference on Computing in Civil and Building EngineeringApril 3-5, 2002Taipei, Taiwan

National Center for Research on Earthquake Engineering

F = F ( x, t )

ru lut

ul

t

ur

t’= S/VS: span lengthV: train speed

ur (t)= ul (t - t’)

pr (t)= - pl (t - t’)

Fourier transform

Ur (w)= Ul (w)

Pr (w)= - Pl (w)V

Sie

μ

Periodic method (2/4) Transfer everything to frequency domain

The 9th International Conference on Computing in Civil and Building EngineeringApril 3-5, 2002Taipei, Taiwan

National Center for Research on Earthquake Engineering

r

l

r

l

i

r

l

i

rrrlri

lrllli

irilii

P

P

F

F

F

U

U

U

KKK

KKK

KKK 0

ˆˆˆ

ˆˆˆ

ˆˆˆF = F ( x, t )

ru lu

Ur (w)= Ul (w)

Pr (w)= - Pl (w) VSi

eμ

Periodic method (3/4)

Using Lagrange’s method

The 9th International Conference on Computing in Civil and Building EngineeringApril 3-5, 2002Taipei, Taiwan

National Center for Research on Earthquake Engineering

0

)(

)(

)(

)(

)(

)(

0 0

ˆˆˆ

ˆˆˆ0ˆˆˆ

1r

l

i

r

l

i

rrrlri

lrllli

irilii

F

F

F

U

U

U

II

IKKK

IKKK

KKK

Original equations

Constraint equations

Lagrange multiplier

Periodic method (4/4)

Unit moving load

The 9th International Conference on Computing in Civil and Building EngineeringApril 3-5, 2002Taipei, Taiwan

National Center for Research on Earthquake Engineering

)(),( tVxtxf

V

xiti e

VdtetVxxF

1

)(),(

Fourier transform

V

xiti e

VdtetVxxF

1

)(),(

Finite element model

The 9th International Conference on Computing in Civil and Building EngineeringApril 3-5, 2002Taipei, Taiwan

National Center for Research on Earthquake Engineering

f = f( x, t )

ru lu

Column

EA= N1110373.2

E I= 21410205.2 Nm

3 90.273

mkg

EA= N910345.6 E I = 0

m

kgm 3102.417

Girder

EA= N1110425.2

E I= 21510214.2 Nm

3 53.529

mkg

T r a c k

28101.8

mkgkB

30 m

8.8m

1 .8 5 m

Track: BC elements

Horizontal springs

Girder: BC elements

Periodic constraint nodes

Foundation: 6x6 stiffness matrix (by FE/BE method)

Foundation model

The 9th International Conference on Computing in Civil and Building EngineeringApril 3-5, 2002Taipei, Taiwan

National Center for Research on Earthquake Engineering

Periodic analysis result (1/2)

Frequency range: 0.02 Hz ~ 15 Hz, df = 0.02 Hz Ground response (horizontal X direction)

The 9th International Conference on Computing in Civil and Building EngineeringApril 3-5, 2002Taipei, Taiwan

National Center for Research on Earthquake Engineering

Spectrum of ground reaction X

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12

Hz

N

X reaction force

-0.4

-0.2

0.0

0.2

0.4

- 2. 52 - 2. 16 - 1. 80 - 1. 44 - 1. 08 - 0. 72 - 0. 36 0. 00 0. 36 0. 72 1. 08 1. 44 1. 80 2. 16 2. 52

Time (sec.)

N

Inverse Fourier transform

Time history analysis result (1/2)

N-span finite element model N=6 to 40

Dynamic analysis method HHT dynamic time integration Using ABAQUS Time interval=0.005 sec.

The 9th International Conference on Computing in Civil and Building EngineeringApril 3-5, 2002Taipei, Taiwan

National Center for Research on Earthquake Engineering

Comparison of analysis results

N=6

N=40

X reaction force

-0.6

-0.4

-0.2

0.0

0.2

0.4

- 2. 52 - 2. 16 - 1. 80 - 1. 44 - 1. 08 - 0. 72 - 0. 36 0. 00 0. 36 0. 72 1. 08 1. 44 1. 80 2. 16 2. 52

Time (sec.)

N

6-span

Periodic

X reaction force

-0.6

-0.4

-0.2

0.0

0.2

0.4

- 2. 52 - 2. 16 - 1. 80 - 1. 44 - 1. 08 - 0. 72 - 0. 36 0. 00 0. 36 0. 72 1. 08 1. 44 1. 80 2. 16 2. 52

Time (sec.)

N

40-span

Periodic

Summary and possible future work

Summary A periodic method for an elevated rail bridge

Frequency domain A large number of identical spans Linear analysis

Compare to a time-domain dynamic analysis (N spans) The results tend to consistent when the N is larger Vibration of higher frequency differs FE model for the periodic method is much smaller

Summary and possible future work

Possible future work The foundation can be modeled:

Foundation-foundation interaction can be considered

Thank you very much