Dynamic Response of Pedestrian Bridges/Floor Vibration and Various Methods of Vibration Remediation

Presentation • Brief overview of structural vibration • Understanding how people perceive and react to unwanted vibration • General response of pedestrian bridges to vibration ib ti • Various design guidelines • Damping • Bridge case study

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

Structural Vibration

• • • •

Stiffness Force: FS = -kx Damping Force: FD = -cx’ External Force: FE(t) Inertial Force

Structural Vibration

• General equation of motion

mx ′′(t ) + cx ′(t ) + kx k (t ) = Fe (t )

Structural Vibration

Structural Vibration • Forced Vibration

• Free Vibration

mx ′′(t ) + cx ′(t ) + kx(t ) = 0 • Solution x(t ) = e

−ζω n t

x(0) = 0 x ′(0) = 0

⎧⎪ ⎫⎪ ζω x + xo′ −ζω nt x(t ) = ⎨ xo e −ζω nt cos(ω d t ) + n o e sin (ω d t )⎬ + ωn 1− ζ 2 ⎪⎩ ⎪⎭

⎧⎪ ⎫⎪ ζω n xo + xo′ ( ) ( ) + ω x cos t sin ω t ⎨ o d d ⎬ ωn 1− ζ 2 ⎪⎩ ⎪⎭

⎧⎪ ⎫⎪ ζω n x p (0) + x ′p (0) −ζω nt −ζω t e sin (ω d t )⎬ ⎨ x p (t ) − x p (0)e n cos(ω d t ) − 2 ωn 1− ζ ⎪⎭ ⎪⎩

⎧⎪ ⎫⎪ ω x + ζxo′ x ′(t ) = e −ζω nt ⎨ xo′ cos(ω d t ) − n o sin i (ω d t )⎬ 1−ζ 2 ⎪⎩ ⎪⎭

ω n2 =

k m

c m

2ζω n =

mx ′′(t ) + cx ′(t ) + kx(t ) = Fe (t ) • Solution

⎧⎪ ⎫⎪ ω x + ζxo′ −ζω nt x ′(t ) = ⎨ xo′ e −ζω nt cos(ω d t ) − n o e sin (ω d t )⎬ + 1−ζ 2 ⎩⎪ ⎭⎪

ωd = ωn 1 − ζ 2

⎫⎪ ⎧⎪ ω n x p (0) + ζx ′p (0) −ζω nt −ζω t e sin (ω d t )⎬ ⎨ x ′p (t ) − x ′p (0)e n cos(ω d t ) + 2 1−ζ ⎪⎭ ⎪⎩

Structural Vibration

Human Perception

• Steady State Forcing Function

• Human Response

Fe (t ) = Fo sin (ω o t )

– Present: Not perceived – Perceived: Does not annoy – Perceived: Annoys and disturbs – Perceived: Severe enough to cause illness

• Solution xss (t ) =

x ′ss (t ) =

Fo

(1 − r )

2 2

[

Foω o

(1 − r )

2 2

(

)

]

k − 2ζr cos(ωot ) + 1 − r 2 sin (ωot ) 2 + (2ζr )

[

• Peak acceleration limits Situation

Building in Strong Wind

Public Transportation

Building in Earthquake

Amusement Park Ride

Peak Acceleration (% g)

0.5 – 10

51 – 102

204 – 458

3.0 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 = 4π 4 2fo2ysKΨ F = 180sin(2πfoT) N vt = 0.9fo m/s (> 2.5 m/s per Ontario Code)

Design Guidelines • Natural Frequency f =

π 2

stiffness π = mass 2

g Δ

Ex ) Uniformly loaded simple beam: Ex.) f n = 0.18

g Δ

5wL4 Δ= 384 EI

Bridge Design Guidelines

a max = 4π 2 f o2 y s KΨ

British Design Guidelines

• Natural Frequency (Vertical Vibration)

a max = 4π f y s KΨ 2

Design Guidelines

2 o

– Limiting values – AASHTO – British Code (1978 BS 5400) – AISC/CISC Steel Design Guide Series 11 Po e −0.35 f o = g βW

ap

Response to Sinusoidal Force Resonance response function

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

Steel Framed Floor System • The combined Beam or jjoist and g girder p panel system y – Spring in parallel (a & b) or in series (c & d) System y frequency q y

Simplified design criterion

a/g, a0/g= ratio of the floor acceleration to the acceleration of gravity; acceleration limit fn = natural frequency of floor structure Po = constant force equal to 0.29 kN (65 lb.) for floors and 0.41 kN (92 lb.) for footbridges

Equivalent panel weight

Design Guidelines

Design Guidelines

• Natural Frequency (Lateral Vibration)

• Stiffening

– Step frequency ½ vertical – 1996 British Standard BS 6399

– Uneconomical – Unsightly

• 10% vertical load

• Damping

– Per ARUP research

– Inherent damping < 1% % – Mechanical damping devices

• f > 1.3 Hz

– Rule of thumb • Lateral limits ½ vertical limits

Damping

Damping • Viscous Damping

• Coulomb Damping

ζ t x(t ) = x max e −ζω sin (ω d t + φ )

Fd = mx ′′ + kx

ζ 1−ζ 2

F ⎞ F ⎛ x = ⎜ xo − d ⎟ cos ωt + d k ⎠ k ⎝

ζ =

xt =π = − xo + 2 ω

Fd k

=

1 ⎛1⎞ ln⎜ ⎟ 2nπ ⎝ δ ⎠

1 ⎛1⎞ ln⎜ ⎟ 2 π ⎝δ ⎠ 2n

Welded steel, steel prestressed concrete, concrete well detailed reinforced concrete.

0 02 < ζ < 0.03 0.02 0 03

Reinforced concrete with considerable cracking.

0.03 < ζ < 0.05

Damping • Mechanical dampers – Active dampers (not discussed here)

Damping • Mechanical dampers – Passive dampers

• Expensive E i • Complicated • No proven examples for f bridges (prototypes currently being tested for seismic damping)

• Viscous Dampers • Tuned Mass Dampers (TMDs) • Viscoelastic Dampers • Tuned Liquid Dampers (TLDs)

Damping

Viscous Dampers

FD = c( x ′)

η

45 40 35 30 Damping Force e

Viscous Dampers

Damping

Linear

25

Fast Rise 20

Slow Rise

15 10 5 0 0

0.5

1

1.5 Velocity

2

2.5

Dampers Tuned mass damper

βs =

Dampers Viscoelastic Dampers

1 m 2 M

Ex) Consider mass ratio = 0.01 βs = 0 0.05 05 (5% damping)

Dampers Tuned Liquid Dampers

Case Study: Millennium Bridge • Crosses River Thames, London, England • 474’ main i span, 266’ north th span, 350’ south span

• S Superstructure t t supported t d by b lateral l t l supporting cables (7’ sag) • Bridge opened June 2000 2000, closed 2 days later

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 (0.5 5 Hz and 0 0.9 9 Hz) • Occurred only when heavily congested • Phenomenon Ph called ll d “S “Synchronous h Lateral Excitation”

Millennium Bridge • Possible solutions – Stiffen Stiff the th bridge b id • Too costly • Affected aesthetic vision of the bridge

– Limit pedestrian traffic • Not feasible

– Active damping • Complicated • Costly • Unproven

– Passive damping

Millennium Bridge • Passive Dampers – 37 viscous dampers installed – 19 TMDs installed

Millennium Bridge • Results – Provided 20% critical damping. – Bridge was reopened February February, 2002 2002. – Extensive research leads to eventual updating of design code code.