Wind loading and structural response
Lecture 12 Dr. J.D. Holmes
Along-wind dynamic response
Dynamic response
• Significant resonant dynamic response can occur under wind actions
for structures with n1 < 1 Hertz (approximate)
• All structures will experience fluctuating loads below resonant
frequencies (background response)
• Significant resonant response may not occur if damping is high enough
• e.g. electrical transmission lines - ‘pendulum’ modes - high aerodynamic
damping
Dynamic response
• Spectral density of a response to wind :
background
component
resonant
contributions
Dynamic response
• Time history of fluctuating wind force
D(t)
time
Dynamic response
• Time history of fluctuating wind force
D(t)
time
• Time history of response :
x(t)
High
n1
time
• Structure with high natural frequency
Dynamic response
• Time history of fluctuating wind force
D(t)
time
• Time history of response :
x(t)
Low n1
time
• Structure with low natural frequency
Dynamic response
• Features of resonant dynamic response :
• Time-history effect : when vibrations build up structure response at
any given time depends on history of loading
• Additional forces resist loading : inertial forces, damping forces
• Stable vibration amplitudes : damping forces = applied loads
inertial forces (mass  acceleration) balance elastic forces in structure
effective static loads : ( 1 times) inertial forces
Dynamic response
• Comparison with dynamic response to earthquakes :
• Earthquakes are shorter duration than most wind storms
• Dominant frequencies of excitation in earthquakes are 10-50 times higher
than wind loading
• Earthquake forces appear as fully-correlated equivalent lateral forces
wind forces (along-wind and cross wind) are partially-correlated fluctuating
forces
Dynamic response
• Comparison with dynamic response to earthquakes :
Dynamic response
• Random vibration approach :
• Uses spectral densities (frequency domain) for calculation :
Dynamic response
• Along-wind response of single-degree-of freedom structure :
• mass-spring-damper
system, mass small w.r.t.
length scale of turbulence
representative of large mass
supported by a low-mass column
c
D(t)
m
k
n1 
• equation of motion :
1 k
2π m
η
c
2 mk
mx  cx  kx  D(t)
Dynamic response
• Along-wind response of single-degree-of freedom structure :
• by quasi-steady assumption (Lecture 9) :
D'  C Do
2
2
4D 2 2
ρ a U u' A  CD ρ a U u' A  2 u'
U
2
2
2
2
2
2
2
since :
• in terms of spectral density :
• hence :
4D 2
SD (n)  2 Su (n )
U

2
2
CD 
D
1
ρ a U 02 A
2

4D 2
0 SD (n).dn  U 2 0 Su (n).dn
this is relation between spectral density
of force and velocity
Dynamic response
• Along-wind response of single-degree-of freedom structure :
• deflection :
X(t) = X + x'(t)
mean deflection :
X
D
k
k = spring stiffness
spectral density :
Sx (n) 
1
2
H(n)
SD (n)
k2
H(n) 
2
where the mechanical admittance is given by :
2
1
2 4D
Sx (n)  2 H(n)
Su (n)
2
k
U
1
2
2
  n 2 


n
1      4η2  
  n1  
 n1 
this is relation between spectral density
of deflection and approach velocity
Dynamic response
• Aerodynamic admittance:
• Larger structures - velocity fluctuations approaching
windward face cannot be assumed to be uniform
then :
4D 2
SD (n)  Χ (n). 2 Su (n )
U
2
where 2(n) is the ‘aerodynamic admittance’
Dynamic response
• Aerodynamic admittance:
Low frequency gusts well correlated
1.0
χ n 
0.1
High frequency gusts poorly correlated
0.01
0.01
0.1
1.0
10
A
U
n
based on experiments :
χ n  
1
 2n A 
1 

U


4
3
Dynamic response
• Aerodynamic admittance:
hence :
2
1
2 4D
2
Sx (n)  2 H(n)
.
Χ
(n).S u (n)
2
k
U
substituting D = kX :
4X 2
2
Sx (n)  2 H(n) . Χ 2 (n).S u (n)
U
Dynamic response
• Mean square deflection :

σx
σx
2

4X 2
2
  Sx (n).dn   2 H(n) . Χ 2 (n).S u (n).dn
U
0
0
4X 2σ u

U2

where :
2
2 
2
Su (n)
4X 2 σ u
2
0 H(n) .Χ (n). σ u 2 .dn  U 2 B  R 
2
S (n)
B   Χ (n). u 2 .dn
σu
0
2
independent of
frequency

S (n )
2
R  Χ (n1 ). u 21  H(n) .dn
σu 0
2
assumes X2(n) and Su(n) are constant at X2(n1) and
Su(n1), near the resonant peak
Dynamic response
• Mean square deflection :

 H(n) .dn 
2
0
πn1
4η
πn1Su (n1 )
R  Χ (n1 ).
2
4σ u η
2
(integration by method of poles)
Dynamic response
• Gust response factor (G) :
Expected maximum response in defined time period /
mean response in same time period
X̂  X  gσ x
G
σ
X̂
σ
 1  g x  1  2g u
X
X
U
BR
g = peak factor
g  2 log e ( υT) 
0.577
2 log e ( υT)
 = ‘cycling’ rate (average frequency)
Dynamic response
• Dynamic response factor (Cdyn):
This is a factor defined as follows :
Maximum response including correlation and resonant effects /
maximum response excluding correlation and resonant effects
B = 1 (reduction due to correlation ignored)
R = 0 (resonant effects ignored)
Cdyn 
1  2g
σu
U
1  2g
BR
σu
U
Used in codes and standards based on peak gust (e.g. ASCE-7)
Dynamic response
• Gust effect factor (ASCE-7) :
For flexible and dynamically sensitive structures (Section 6.5.8.2)
 1  1.7I g 2Q 2  g 2 R
z
Q
R

G f  0.925
1  1.7g v I z






This is a ‘dynamic response factor’ not a ‘gust response factor’
0.925(instead of 1) is ‘calibration factor’
1.7 (instead of 2) to adjust for 3-second gust instead of true peak gust
Separate peak factors (gQ and gR) for background and resonant response :
gQ = gv= 3.4
g R  2 log e (3600n1 ) 
0.577
2 log e (3600n1 )
Dynamic response
• Gust effect factor (ASCE-7) :
Resonant response factor (Equation 6-8) :
R
Previously :
1
R n R h R B (0.53  0.47R L )
β
πn1Su (n1 )
R  Χ (n1 ).
2
4σ u η
2
 is critical damping ratio ()
RhRB(0.53 + 0.47RL) is the aerodynamic admittance 2(n1)
decomposed into components for vertical separations (Rh), lateral
separations (RB) and along-wind (windward/ leeward wall) (RL)
Dynamic response
• Gust effect factor (ASCE-7) :
Rn should be :
In fact it is :
where :

πn1Su (n1 )
2
4σ u
2
 2  πn1Su (n 1 )

.
2
4σ u
 1.7 
n1Su (n1 )
6.9 N1

2
1  10.3N1 5 / 3
σu

Note that : 6.9=(2/3)10.3 so that

0
N1 
n 1L z
Vz
Su (n)
dn  1
2
σu
Note that Su(0) is equal to 6.9u2Lz/Vz
But Su(0) should = 4u2lu /Uz (Lecture 7)
Hence Lz = (4/6.9) lu = 0.58 lu
Dynamic response
• Along-wind response of structure with distributed mass :
The calculation of along-wind response with distributed masses (many
modes of vibration) is more complex (Section 5.3.6 in the book)
Based on modal analysis (Lecture 11) :
x(z,t) = j aj (t) j (z)
j (z) is mode shape in jth mode
Use : generalized (modal) mass, stiffness, damping, applied force for each mode
Two approaches :
i) use modal analysis for background and resonant parts (inefficient needs many modes) - Section 5.3.6
ii) calculate background component separately; use modal analysis
only for resonant parts - Section 5.3.7
Easier to use (ii) in the context of effective static load distributions
End of Lecture 12
John Holmes
225-405-3789 JHolmes@lsu.edu