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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: F S = -kx Damping Force: F D = -cx’ External Force: F E (t) Inertial Force

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Structural Vibration General equation of motion

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Structural Vibration Free Vibration Solution

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Structural Vibration Forced Vibration Solution

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Structural Vibration Steady State Forcing Function Solution

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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 SituationBuilding in Strong Wind Public Transportation Building in Earthquake Amusement Park Ride Peak Acceleration (% g)0.5 – 1051 – – 458<458

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Peak Acceleration for Human Comfort for Vibrations 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 Lateral Vibration

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Pedestrian Bridge Response Vertical Vibration (also apply to floor vibration) P = Person’s weight i = Dynamic coefficient for the harmonic force i = Harmonic multiple (1, 2, 3…) f step = 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 Ex.) Uniformly loaded simple beam:

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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) –f o > 5.0 Hz –a max < 0.5(f o ) 1/2 m/s 2 –a max = 4 f o 2 y s K –F = 180sin(2 f o T) N –v t = 0.9f o m/s (> 2.5 m/s per Ontario Code)

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Bridge Design Guidelines

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British Design Guidelines

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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 < 1.5% (Indoor walkways) < 5.0% (Outdoor bridges)

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Response to Sinusoidal Force Resonance response function a/g, a 0 /g= ratio of the floor acceleration to the acceleration of gravity; acceleration limit f n = natural frequency of floor structure P o = 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 % 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

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Damping Viscous Damping Welded steel, prestressed concrete, well detailed reinforced concrete < < 0.03 Reinforced concrete with considerable cracking < < 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

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Dampers Tuned mass damper 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 1 st mode of south span (0.8 Hz) and 1 st and 2 nd 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, –Extensive research leads to eventual updating of design code.

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