Transcript Dynamic Response of Pedestrian Bridges and Various Methods

Dynamic Response of Pedestrian
Bridges/Floor Vibration and Various
Methods of Vibration Remediation
Chung C. Fu, Ph.D., P.E.
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
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t   Fe t 
Structural Vibration
• Free Vibration
mxt   cxt   kxt   0
• Solution
xt   e
 n t
xt   e
k
 
m
2
n
x0  0 x0  0


 n xo  xo


sin  d t 
 xo cos d t  
2
n 1  




 nt


 n xo  xo







x
cos

t

sin

t
 o
d
d 
2
1




c
2 n 
m
d  n 1   2
Structural Vibration
• Forced Vibration
mxt   cxt   kxt   Fe t 
• Solution


 n xo  xo  nt
  nt

xt    xo e
cos d t  
e
sin  d t  
2

1




n




 n x p 0  xp 0  nt


 nt








x
t

x
0
e
cos

t

e
sin

t
 p
p
d
d 
2

1




n




 n xo  xo  nt
  nt

xt    xo e
cos d t  
e
sin  d t  
2
1








 n x p 0  xp 0  nt


 nt
cos d t  
e
sin  d t 
 xp t   xp 0e
1 2




Structural Vibration
• Steady State Forcing Function
Fe t   Fo sin o t 
• Solution
xss t  
xss t  
Fo




2
k



2

r
cos

t

1

r
sin ot 
o
2
2 2
1  r  2r 


Fo o
1  r 
2 2


2
k
1  r  cos o t   2r sin o t 
2
 2r 
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 in
Strong Wind
Public
Transportation
Building in
Earthquake
Amusement
Park Ride
Peak Acceleration (% g)
0.5 – 10
51 – 102
204 – 458
<458
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
Pedestrian Bridge Response
• Vertical Vibration
• Lateral Vibration
Pedestrian Bridge Response
• Vertical Vibration (also apply to floor vibration)
 F t   P1  
i

cos2if stept  i 
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
Pedestrian Bridge Response
• Lateral Vibration
Synchronous Lateral Excitation
Design Guidelines
• Serviceability (i.e. functional, usable)
– Stiffness
– Resonance
• Resonance
– Frequency matching
– Uncomfortable/damaging vibration
– Unfavorable perception
AVOID RESONACE!
Design Guidelines
• Natural Frequency
f 

2
stiffness 

m ass
2
g

Ex.) Uniformly loaded simple beam:
g
f n  0.18

4
5wL

384EI
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 = 42fo2ysKY
F = 180sin(2foT) N
vt = 0.9fo m/s (> 2.5 m/s per Ontario Code)
Bridge Design Guidelines
amax  4 f ys KY
2
2
o
British Design Guidelines
amax  4 f ys KY
2
2
o
Design Guidelines
• Natural Frequency (Vertical Vibration)
– Limiting values
– AASHTO
– British Code (1978 BS 5400)
– AISC/CISC Steel Design Guide Series 11
Po e 0.35 f o

g
W
ap
< 1.5% (Indoor walkways)
< 5.0% (Outdoor bridges)
Response to Sinusoidal Force
Resonance response function
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
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
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
Design Guidelines
• Stiffening
– Uneconomical
– Unsightly
• Damping
– Inherent damping < 1%
– Mechanical damping devices
Damping
• Coulomb Damping
Fd  mx  kx
Fd 
Fd

x   xo 
 cost 
k 
k

xt 

Fd
  xo  2
k
Damping
• Viscous Damping
xt   xmaxe
t

1
2
sind t   
1
1

ln 
2 n   
1
1
 
ln 
2n   
Welded steel, prestressed concrete, well
detailed reinforced concrete.
0.02 <  < 0.03
Reinforced concrete with considerable
cracking.
0.03 <  < 0.05
Damping
• Mechanical dampers
– Active dampers (not discussed here)
• Expensive
• Complicated
• No proven examples for bridges
(prototypes currently being tested for
seismic damping)
Damping
• Mechanical dampers
– Passive dampers
• Viscous Dampers
• Tuned Mass Dampers (TMDs)
• Viscoelastic Dampers
• Tuned Liquid Dampers (TLDs)
Damping
Viscous Dampers
Damping
Viscous Dampers
FD  cx

45
40
35
Damping Force
30
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
1 m
s 
2 M
Ex) Consider mass ratio = 0.01
s = 0.05 (5% damping)
Dampers
Viscoelastic Dampers
Dampers
Tuned Liquid Dampers
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
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”
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
Millennium Bridge
• Passive Dampers
– 37 viscous dampers installed
– 19 TMDs installed
Millennium Bridge
• Results
– Provided 20% critical damping.
– Bridge was reopened February, 2002.
– Extensive research leads to eventual
updating of design code.