Edificios altos

1. Tall buildings
Wind loading and structural response
Lecture 19 Dr. J.D. Holmes

2. Tall buildings
• Very wind-sensitive in synoptic winds (including hurricanes)
• Stimulated development of boundary-layer wind tunnel
• Usually governed by serviceability response (peak accelerations and
deflections in top floors)
• Cladding pressures can be v. high especially at unusual corners and change
of cross section
• Resonant dynamic response for along- and cross-wind very significant (> 100
metres)
(‘Rule-of-thumb’ first mode frequency : 46/h Hertz (h in metres) )
• Sometimes torsional response is significant depending on geometry and
structural system

3. Tall buildings
• Empire State Building - full-scale and wind-tunnel studies in 1930’s
Much stiffer in east-west direction
Y
(N-S)
X
(E-W)
a
wind
D - Mean deflection (inches)
Uh - Mean wind speed at 1250 feet in MPH (uncorrected)
1.0
0.5
0 10 20 30 40 50 60 70 80 90
Angle of attack - degrees
x
x
x
N-S
E-
W x
10
U
3
2
x
h
D

4. Tall buildings
• Commerce Court building, Toronto, Canada - 1970’s
Full-scale and wind-tunnel measurements of local cladding pressures and
overall building response (accelerations)
Studies of local pressure peaks and implications for glass design :
Acceleration measurements showed significance of torsional component (twist)
1/200 scale aeroelastic model showed good agreement with full scale
0 1 2 3 4 5 6
Time (minutes)
Wind
pressure

5. • World Trade Center – New York 1973-2001
Tall buildings
• First buildings to be tested in a turbulent
boundary-layer flow wind tunnel (mid 1960’s)

6. Tall buildings
• Flow around a tall building :

7. Tall buildings
• Pressure fluctuations on a tall building :
(movie by Shimizu Corporation, Tokyo, Japan)

8. Tall buildings
• Pressure fluctuations on a tall building :
(movie by Shimizu Corporation, Tokyo, Japan)

9. Tall buildings
• Cladding pressures :
Four values of pressure coefficients :
2
h
a
0
p
U
ρ
2
1
p
p
C


2
h
a
0
p
U
ρ
2
1
p
p̂
Ĉ

 2
h
a
0
p
U
ρ
2
1
p
p
C




2
h
a
2
Cp
p
U
ρ
2
1
p
σ
C




Time
Cp (t)
Cp
ˆ
Cp
C p

Cp


10. Tall buildings
• Square cross section - height/width =2.1
0.8
0.6
0.4
0.2 0.2
0.0
-0.2 -0.2
-0.4 -0.4
1.8
1.6
1.4
1.2
1.0 1.0
p
C p
Ĉ
p
C

stagnation
point  0.8h
minimum maximum
Windward wall :

11. Tall buildings
• Square cross section - height/width =2.1
mean Cp’s :
-0.6 to -0.8
largest minimum Cp : -3.8
Side wall (wind from left) :
-0.9
-0.9
-0.5
-0.6
-0.8
-0.8
-0.7
-0.6
-0.5
-2.2
-2.4
-2.0
-2.0
-1.8 -2.2
-2.4
-2.6
-2.8
-3.2
-3.8
-3.4
-3.0
-2.8
-2.6
-2.4
0.6
0.4
0.2
0.0
p
C p
Ĉ
p
C


12. Tall buildings
• Square cross section - height/width =2.1
mean Cp’s :
-0.35 to -0.45
largest minimum Cp : -1.6
Leeward wall :
-0.45 -0.45
-0.4
-0.35
-1.6
-1.6
-1.4 -1.4
-1.2
-1.6 -1.6
-0.1
p
C p
C

p
Ĉ

13. Tall buildings
• Glass strength under wind loading
Glass strength is dependent on duration of loading :
Microscopic flaws on tension side grow at a rate dependent on local stress
  dt
t
s
D
n
T


0
)
(
Accumulated damage at constant temperature and humidity
(Brown’s integral) :
s(t) is stress; T is total time over which it acts; n is a high power (15 to 20)

14. Tall buildings
• Glass strength under wind loading
Under wind loading p(t) : assume s(t) = K[p(t)]m/n (nonlinear)
i.e. mth moment of probability density function of Cp
  dt
t
p
E
K
D
E
m
T
}
)
(
{
}
{
0


p
p
Cp
m
p dC
C
f
C
U
KT
D
E )
(
)
(
}
{
0
2
2
1


 

15. Tall buildings
• Glass strength under wind loading
Glass testing is usually carried out with a linearly increasing ‘ramp’ load :
damage produced by 1-minute ramp load :
m)
(1
60.p
K
60
t
.
K
D
m
max
m
60
0
max








  dt
p
time
load failure
pmax
pmax is specified load in glass design charts

16. Tall buildings
• Glass strength under wind loading
Ck is approximately equal to the peak pressure coefficient during the hour
of storm winds
Ck = equivalent glass design pressure coefficient - gives pressure which
produces same damage in 1 hour of wind loading as that produced by a 1-
minute ramp load
























p
p
Cp
m
p
m
m
k
dC
C
f
C
m
C
)
(
U
ρ
2
1
)
3600
(
K
)
1
(
U
ρ
2
1
.
.
60
K.
0
2
a
2
a
writing pmax as Ck. (1/2)aU2 , where Ck is an equivalent glass design
pressure coefficient, and equating damage in ramp load test to that in 1
hour (3600 sec.) of wind :
m
p
p
Cp
m
p
k dC
C
f
C
m
C
/
1
0
)
(
)
1
(
60





 
 


17. Tall buildings
• Glass strength under debris impact
Glazing is vulnerable to damage and failure by roof gravel in the U.S.
ASCE-7 (6.5.9.3) requires glazing above 18.3 m above ground level, and
over 9.2m above gravel source, to be protected
Gravel acts like a sphere or cube – will only go up if there is a vertical
wind velocity component

18. Cross-wind vibrations are usually greater than
along-wind vibrations for buildings of heights greater than
100m (330 feet)
along wind
cross wind
Tall buildings
• Overall loading and dynamic response

19. Tall buildings
• Overall loading and dynamic response
along wind
Standard deviation of deflections at top of a tall building :
η
1
b
n
U
ρ
ρ
A
h
σ
kx
1
h
b
a
x
x

















η
1
b
n
U
ρ
ρ
A
h
σ
ky
1
h
b
a
y
y
















 cross wind
Ax and Ay - depend on building shape
kx - 2 to 2.5 ky - 2.5 to 3.5 (cross-wind)
b - average building density
n1 - first mode frequency  - critical damping ratio

20. Tall buildings
• Overall loading and dynamic response
Standard deviation of deflections at top of a tall building :
Circular cross section :
10
1
5
2
100
5
2
10
- 1
5
2 3 5 7 10 15
wind
X
Y
x
cross wind
1000 x deflection
height
sy
h
sx
h
1

21. Tall buildings
• Overall loading and dynamic response
Deflections at top of a tall building :
Effect of cross section :
Peak
deflection
height
0
.001
.002
.003
.004
30 50 100 500 1000
Return period/years
Direction
of
motion
Modification of corners are effective in reducing response

22. Tall buildings
• Torsional loading and response
Two mechanisms :
• applied moments from aerodynamic forces produced by non-uniform
pressure distributions or non-symmetric cross-sections
• structural eccentricity between elastic center and geometric center
(a 10% eccentricity on a square building: doubled mean twist and increased
dynamic twist by 40-50%)

23. Tall buildings
• Torsional loading and response
Mean torque coefficient :
depends on ratio between minimum and maximum projected widths of
the cross section
0.2
0.1
0 0.2 0.4 0.6 0.8 1.0
f = 2
max
min








b
b

24. Tall buildings
• Interference effects
Surrounding buildings can produce increases or decreases in peak wind
loads :
shows percentage change in peak cross-wind response of building B, due to
a similar building A at position (X,Y)
10b 8b 6b 4b 2b -2b
b
Building B
Wind direction
(X,Y)
Building A
V
b
2b
3b
4b
0%
+30%
+20% +10%
-10%
+10%
+20%
X
Y
0%
-20%
increases
increases
decreases

25. Tall buildings
• Damping
Damping is the mechanism for dissipation of vibration energy
Structural damping (Japanese buildings) :
0018
.
0
470
014
.
0 1
1 








h
x
n t

0029
.
0
400
013
.
0 1
1 








h
x
n t

reinforced concrete
steel frame
n1 = first mode natural frequency xt = amplitude of vibration

26. Tall buildings
• Damping
Auxiliary damping :
Viscoelastic damper :
used on World Trade Center buildings, New York
F/2 F/2
Steel flange
V.E. material
Centreplate
F

27. Tall buildings
• Damping
Auxiliary damping :
Tuned mass damper :
used on CityCorp building, New York (M2=400 ton of concrete)
K
C
M
K
C
M
y
y
1
1
1
2
2
2
2
1 (t) (t)

28. Tall buildings
• Damping
Auxiliary damping :
Tuned liquid (sloshing) damper :
used on Shin-Yokohama hotel, Japan
h
2R

29. Tall buildings
• Damping
Auxiliary damping :
Tuned liquid column damper :
to be used on Eureka tower building, Melbourne, Australia (under construction)
X
X
Flow
A
Orifice

30. End of Lecture 19
John Holmes
225-405-3789 JHolmes@lsu.edu