Ped Bridge Vibration

8/18/2019 Ped Bridge Vibration

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Dynamic Response of Pedestrian

Bridges/Floor Vibration and Various

Methods of Vibration Remediation

Chung C. Fu, Ph.D., P..


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Presentation

! Brief o"er"ie# of structural "ibration

! $nderstanding ho# people percei"e and

react to un#anted "ibration! %eneral response of pedestrian bridges to

"ibration

! Various design guidelines! Damping

! Bridge case study


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&tructural Vibration

! %eneral e0uation of motion

( ) ( ) ( ) ( )t F t kxt xct xm e=+′+′′


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&tructural Vibration! Free Vibration

! &olution

( ) ( ) ( ) 0=+′+′′ t kxt xct xm ( ) 00 = x ( ) 00 =′ x

( ) ( ) ( )

−

′++= − t

x xt xet x

d

n

oon

d o

t n ω ζ ω

ζω ω ζω sin

1cos

2

( ) ( ) ( )

−′+−′=′ − t x xt xet x d oon

d ot n ω

ζ ζ ω ω ζω sin

1

cos2

m

k

n =

2ω m

cn =ζω 2

21 ζ ω ω −= nd


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&tructural Vibration! Forced Vibration

! &olution

( ) ( ) ( ) ( )t F t kxt xct xme

=+′+′′

( ) ( ) ( ) +

−′++= −− t e

x xt e xt x d

t

n

oond

t

onn ω

ζ ω

ζω ω ζω ζω sin

1

cos2

( ) ( ) ( ) ( ) ( )

( )

−

′+−− −− t e

x xt e xt x d

t

n

p pn

d

t

p pnn ω

ζ ω

ζω ω ζω ζω sin

1

00cos0

2

( ) ( ) ( ) +

−

′+−′=′ −− t e

x xt e xt x d

t oon

d

t

onn ω

ζ

ζ ω ω ζω ζω sin

1

cos2

( ) ( ) ( ) ( ) ( )

( )

−

′++′−′

−−t e

x x

t e xt x d t p pn

d

t

p p nn ω ζ

ζ ω

ω ζω ζω

sin1

00

cos0 2


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&tructural Vibration

! &teady &tate Forcing Function

( ) ( )t F t F ooe ω sin=

! &olution

( )( ) ( )

( ) ( ) ( )[ ]t r t r r r

k

F

t xoo

o

ss ω ω ζ

ζ sin1cos2

21

2

222−+−

+−=

( )( ) ( )

( ) ( ) ( )[ ]t r t r r r

k F

t x oo

oo

ss ω ζ ω ζ

ω

sin2cos1

21

2

222

+−+−

=′


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Pea*

4ccelerationfor 1uman

Comfort forVibrations

Design Guide 11 Fig. 2.1 Recommended peak acceleration for human

comfort for vibrations due to human activities


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Pedestrian Bridge Response

! Vertical Vibration

! 5ateral Vibration


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! Vertical Vibration -also apply to floor "ibration

( ) ( )[ ]∑ ∑ ++= i stepi t if P t F φ π α 2cos1

P ( Persons #eight

αi ( Dynamic coefficient for theharmonic force

i ( 1armonic multiple -6, 7, 89

f step ( &tep fre0uency of acti"ity

t ( time

φ ( Phase angle for the harmonic


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! 5ateral Vibration

&ynchronous 5ateral +citation


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Design %uidelines

! &er"iceability -i.e. functional, usable

2 &tiffness

2 Resonance! Resonance

2 Fre0uency matching

2 $ncomfortable/damaging "ibration

2 $nfa"orable perception

4V:D R&:34C;


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Design %uidelines

! 3atural Fre0uency

∆

== g

mass

stiffness f

22

π π

∆= g

f n 18.0

+. $niformly loaded simple beam'

EI

wL

384

5 4

=∆


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Design %uidelines

! 3atural Fre0uency -Vertical Vibration 2 5imiting "alues -Bridge

! 44&1 1?

2 f = 7.@Aln-6@>/

2 = 6@>e)>.8Af

2 &pecial cases' f = A.> 1?

! British Code -1978 ! "#$$ /:ntario Bridge Code -6@8

2 f o = A.> 1?

2 ama+ >.A-f o6/7 m/s7

2 ama+ ( Eπ2f o7ysΨ 2 F ( 6@>sin-7πf o


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Bridge Design %uidelines

Ψ= K y f a so22

max 4π


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British Design %uidelines

Ψ= K y f a so22

max 4π


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Design %uidelines

! 3atural Fre0uency -Vertical Vibration

2 5imiting "alues

2 44&1


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Response to &inusoidal Force

Resonance response function

a%g& a$ %g ( ratio of the floor

acceleration to the accelerationof gra"ityH acceleration limit

f n ( natural fre0uency of floor

structure

' o ( constant force e0ual to >.7

*3 -IA lb. for floors and >.E6 *3

&implified design criterion


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&teel Framed Floor &ystem

!


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Design %uidelines

! 3atural Fre0uency -5ateral Vibration

2 &tep fre0uency L "ertical

2 6I British &tandard B& I8

! 6>G "ertical load

2 Per 4R$P research

! f = 6.8 1?

2 Rule of thumb! 5ateral limits L "ertical limits


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Design %uidelines

! &tiffening

2 $neconomical

2 $nsightly

! Damping

2 nherent damping 6G

2 Mechanical damping de"ices


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Damping

! Coulomb Damping

kx xm F d +′′=

k

F t

k

F x x

d d

o +

−= ω cos

k

F x x d ot 2+−==

ω π


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Damping

! Viscous Damping

( ) ( )φ ω ζω += − t e xt x d t sin

max

=− δ π ζ

ζ 1ln

2

1

1 2 n

= δ π ζ

1ln

2

1

n

Wele steel! "restresse concrete! #elletaile rein$orce concrete.

0.02 < ζ < 0.03

ein$orce concrete #it& consiera'lecracking.

0.03 < ζ < 0.05


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Damping

! Mechanical dampers

2 4cti"e dampers -not discussed here

! +pensi"e

! Complicated

! 3o pro"en e+amples for bridges

-prototypes currently being tested for

seismic damping


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Damping

! Mechanical dampers

2 Passi"e dampers

! Viscous Dampers

!


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Damping

Viscous Dampers


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Damping

Viscous Dampers

( )η xc F D ′=


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Dampers

.>6

βs ( >.>A -AG damping


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Dampers

Viscoelastic Dampers


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Dampers


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Case &tudy' Millennium Bridge

! Crosses Ri"er

south span

! &uperstructure supported by lateralsupporting cables - sag

! Bridge opened Nune 7>>>, closed 7 days

later


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Millennium Bridge

! &e"ere lateral resonance #as noted

->.7Ag

! Predominantly noted during 6st mode of

south span ->.@ 1? and 6st and 7nd modes of main span ->.A 1? and >. 1?

! :ccurred only #hen hea"ily congested

! Phenomenon called O&ynchronous

5ateral +citation


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Millennium Bridge

! Possible solutions 2 &tiffen the bridge

!


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Millennium Bridge

! Passi"e Dampers

2 8 "iscous dampers installed

2 6


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Millennium Bridge

! Results

2 Pro"ided 7>G critical damping.

2 Bridge #as reopened February, 7>>7.

2 +tensi"e research leads to e"entual

updating of design code.

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Ped Bridge Vibration

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