Chung C. Fu, Ph.D., P.E.
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Transcript of Chung C. Fu, Ph.D., P.E.
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Dynamic Response of Pedestrian Bridges/Floor Vibration and Various Methods of Vibration Remediation
Chung C. Fu, Ph.D., P.E.
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Presentation
• Brief overview of structural vibration• Understanding how people perceive and
react to unwanted vibration• General response of pedestrian bridges to
vibration• Various design guidelines• Damping• Bridge case study
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Structural Vibration
• Stiffness Force: FS = -kx• Damping Force: FD = -cx’• External Force: FE(t)• Inertial Force
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Structural Vibration
• General equation of motion
tFtkxtxctxm e
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Structural Vibration• Free Vibration
• Solution
0 tkxtxctxm 00 x 00 x
t
xxtxetx d
n
oondo
tn
sin
1cos
2
t
xxtxetx d
oondo
tn
sin
1cos
2
mk
n 2 mc
n 2 21 nd
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Structural Vibration• Forced Vibration
• Solution
tFtkxtxctxm e
te
xxtextx d
t
n
oond
to
nn
sin
1cos
2
te
xxtextx d
t
n
ppnd
tpp
nn
sin
1
00cos0
2
te
xxtextx d
toond
to
nn
sin
1cos
2
te
xxtextx d
tppnd
tpp
nn
sin
1
00cos0
2
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Structural Vibration• Steady State Forcing Function
tFtF ooe sin
• Solution
trtr
rrk
Ftx oo
o
ss
sin1cos221
2222
trtrrr
kF
tx oo
oo
ss
sin2cos1
212
222
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Human Perception
• Human Response– Present: Not perceived– Perceived: Does not annoy– Perceived: Annoys and disturbs– Perceived: Severe enough to cause illness
• Peak acceleration limits
Situation Building inStrong Wind
PublicTransportation
Building inEarthquake
AmusementPark Ride
Peak Acceleration (% g) 0.5 – 10 51 – 102 204 – 458 <458
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Peak Acceleration for HumanComfort for Vibrations
Design Guide 11 Fig. 2.1 Recommended peak acceleration for humancomfort for vibrations due to human activities
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Pedestrian Bridge Response
• Vertical Vibration• Lateral Vibration
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Pedestrian Bridge Response
• Vertical Vibration (also apply to floor vibration)
istepi tifPtF 2cos1
P = Person’s weight
i = Dynamic coefficient for the harmonic force
i = Harmonic multiple (1, 2, 3…)
fstep = Step frequency of activity
t = time
i = Phase angle for the harmonic
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Pedestrian Bridge Response
• Lateral Vibration
Synchronous Lateral Excitation
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Design Guidelines
• Serviceability (i.e. functional, usable)– Stiffness– Resonance
• Resonance– Frequency matching– Uncomfortable/damaging vibration– Unfavorable perception
AVOID RESONACE!
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Design Guidelines• Natural Frequency
gmassstiffnessf
22
gfn 18.0
Ex.) Uniformly loaded simple beam:
EIwL
3845 4
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Design Guidelines• Natural Frequency (Vertical Vibration)
– Limiting values (Bridge)• AASHTO
– f > 3.0 Hz– f > 2.85ln(180/W)– W > 180e-0.35f
– Special cases: f > 5.0 Hz
• British Code (1978 BS 5400)/Ontario Bridge Code (1983)– fo > 5.0 Hz
– amax < 0.5(fo)1/2 m/s2
– amax = 4fo2ysK
– F = 180sin(2foT) N
– vt = 0.9fo m/s (> 2.5 m/s per Ontario Code)
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Bridge Design Guidelines
Kyfa so22
max 4
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British Design Guidelines
Kyfa so22
max 4
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Design Guidelines
• Natural Frequency (Vertical Vibration)– Limiting values– AASHTO– British Code (1978 BS 5400)– AISC/CISC Steel Design Guide Series 11
WeP
ga of
op
35.0
< 1.5% (Indoor walkways)
< 5.0% (Outdoor bridges)
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Response to Sinusoidal ForceResonance response function
a/g, a0/g= ratio of the floor acceleration to the acceleration of gravity; acceleration limitfn = natural frequency of floor structurePo = constant force equal to 0.29 kN (65 lb.) for floors and 0.41 kN (92 lb.) for footbridges
Simplified design criterion
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Steel Framed Floor System• The combined Beam or joist and girder panel system
– Spring in parallel (a & b) or in series (c & d)
System frequency
Equivalent panel weight
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Design Guidelines
• Natural Frequency (Lateral Vibration)– Step frequency ½ vertical– 1996 British Standard BS 6399
• 10% vertical load– Per ARUP research
• f > 1.3 Hz– Rule of thumb
• Lateral limits ½ vertical limits
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Design Guidelines
• Stiffening– Uneconomical– Unsightly
• Damping– Inherent damping < 1%– Mechanical damping devices
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Damping• Coulomb Damping
kxxmFd
kF
tkF
xx ddo
cos
kF
xx dot
2
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Damping• Viscous Damping
textx dt sinmax
1ln2
1
1 2 n
1ln2
1n
Welded steel, prestressed concrete, well detailed reinforced concrete.
0.02 < < 0.03
Reinforced concrete with considerable cracking.
0.03 < < 0.05
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Damping
• Mechanical dampers– Active dampers (not discussed here)
• Expensive• Complicated• No proven examples for bridges
(prototypes currently being tested for seismic damping)
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Damping
• Mechanical dampers– Passive dampers
• Viscous Dampers• Tuned Mass Dampers (TMDs)• Viscoelastic Dampers• Tuned Liquid Dampers (TLDs)
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Damping
Viscous Dampers
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Damping
Viscous Dampers
0
5
10
15
20
25
30
35
40
45
0 0.5 1 1.5 2 2.5
Velocity
Dam
ping
For
ce
Linear
Fast RiseSlow Rise
xcFD
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Dampers
Tuned mass damper
Mm
s 21
Ex) Consider mass ratio = 0.01
s = 0.05 (5% damping)
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Dampers
Viscoelastic Dampers
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Dampers
Tuned Liquid Dampers
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Case Study: Millennium Bridge• Crosses River Thames, London, England• 474’ main span, 266’ north span, 350’
south span
• Superstructure supported by lateral supporting cables (7’ sag)
• Bridge opened June 2000, closed 2 days later
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Millennium Bridge• Severe lateral resonance was noted
(0.25g)• Predominantly noted during 1st mode of
south span (0.8 Hz) and 1st and 2nd modes of main span (0.5 Hz and 0.9 Hz)
• Occurred only when heavily congested• Phenomenon called “Synchronous
Lateral Excitation”
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Millennium Bridge• Possible solutions
– Stiffen the bridge• Too costly• Affected aesthetic vision of the bridge
– Limit pedestrian traffic• Not feasible
– Active damping• Complicated• Costly• Unproven
– Passive damping
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Millennium Bridge
• Passive Dampers– 37 viscous dampers installed– 19 TMDs installed
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Millennium Bridge
• Results– Provided 20% critical damping.– Bridge was reopened February, 2002.– Extensive research leads to eventual
updating of design code.