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Along-wind dynamic response
Wind loading and structural response
Lecture 12 Dr. J.D. Holmes
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
x(t) High n1
• Time history of response :
• Structure with high natural frequency
Dynamic response
• Time history of fluctuating wind force
D(t)
time• Time history of response :
• Structure with low natural frequency
time
x(t)Low n1
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
• Stable vibration amplitudes : damping forces = applied loads
inertial forces (mass acceleration) balance elastic forces in structure
effective static loads : ( 1 times) inertial forces
• Additional forces resist loading : inertial forces, damping forces
Dynamic response
• Comparison with dynamic response to earthquakes :
• Earthquakes are shorter duration than most wind storms
• Earthquake forces appear as fully-correlated equivalent lateral forces
wind forces (along-wind and cross wind) are partially-correlated fluctuating forces
• Dominant frequencies of excitation in earthquakes are 10-50 times higher than wind loading
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
D(t)
k
c
m
mk2
cη
m
k
2π
1n1
representative of large mass supported by a low-mass column
D(t)kxxcxm • equation of motion :
Dynamic response
• Along-wind response of single-degree-of freedom structure :
• by quasi-steady assumption (Lecture 9) :
• in terms of spectral density :
22
22222
a2
D2222
a2
Do2 u'
U
D4Au'UρCAu'UρCD'
AUρ21
DC
20a
D since :
0
u2
2
0
D (n).dnSU
D4(n).dnS
• hence :)(nS
U
D4(n)S u2
2
D 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)
spectral density :
mean deflection :
where the mechanical admittance is given by :
this is relation between spectral density of deflection and approach velocity
k
DX k = spring stiffness
(n)SH(n)k
1(n)S D
2
2x
2
1
2
22
1
2
nn
4ηnn
1
1H(n)
(n)SU
D4H(n)
k
1(n)S u2
22
2x
Dynamic response
• Aerodynamic admittance:
• Larger structures - velocity fluctuations approaching windward face cannot be assumed to be uniform
where 2(n) is the ‘aerodynamic admittance’
then :
)(nSU
D4(n).(n)S u2
22
D Χ
Dynamic response
• Aerodynamic admittance:
based on experiments :
3
4
U
A2n1
1nχ
Low frequency gusts - well correlated
High frequency gusts - poorly correlated
0.01 0.1 1.0 10
1.0
0.1
0.01
nχ
U
An
Dynamic response
• Aerodynamic admittance:
hence :
substituting D = kX :
(n)(n).S.U
D4H(n)
k
1(n)S u
22
22
2x Χ
(n)(n).S.H(n)U
X4(n)S u
22
2
2
x Χ
Dynamic response
• Mean square deflection :
where :
0
u22
2
2
0
x2
x (n).dn(n).S.H(n)U
X4(n).dnSσ Χ
RBU
σX4.dn
σ
(n)S(n)..H(n)
U
σX4σ
2
2u
2
02
u
u22
2
2u
22
x
Χ
0
2u
u2 .dnσ
(n)S(n).B Χ
0
2
2u
1u1
2 .dnH(n)σ
)(nS).(nR Χ
assumes X2(n) and Su(n) are constant at X2(n1) and Su(n1), near the resonant peak
independent of frequency
Dynamic response
• Mean square deflection :
4η
πn.dnH(n) 1
0
2
(integration by method of poles)
η4σ
)(nSπn).(nR 2
u
1u11
2Χ
Dynamic response
• Gust response factor (G) :
Expected maximum response in defined time period / mean response in same time period
g = peak factor
xgσXX̂
RBU
σ2g1
X
σg1
X
X̂G ux
)υT(log2
577.0)υT(log2g
e
e
= ‘cycling’ rate (average frequency)
Dynamic response
• Dynamic response factor (Cdyn):
Maximum response including correlation and resonant effects / maximum response excluding correlation and resonant effects
This is a factor defined as follows :
U
σ2g1
RBU
σ2g1
Cu
u
dyn
B = 1 (reduction due to correlation ignored)
R = 0 (resonant effects ignored)
Used in codes and standards based on peak gust (e.g. ASCE-7)
Dynamic response
• Gust effect factor (ASCE-7) :
This is a ‘dynamic response factor’ not a ‘gust response factor’
For flexible and dynamically sensitive structures (Section 6.5.8.2)
zv
2R
22Qz
f Ig7.11
RgQgI7.11925.0G
0.925(instead of 1) is ‘calibration factor’
Separate peak factors (gQ and gR) for background and resonant response :
gQ = gv= 3.4 )n3600(log2
577.0)n3600(log2g
1
1R
e
e
1.7 (instead of 2) to adjust for 3-second gust instead of true peak gust
Dynamic response
• Gust effect factor (ASCE-7) :
Previously :
Resonant response factor (Equation 6-8) :
)0.47R(0.53RRRβ
1R LBhn
is critical damping ratio ()
RhRB(0.53 + 0.47RL) is the aerodynamic admittance 2(n1)
η4σ
)(nSπn).(nR 2
u
1u11
2Χ
decomposed into components for vertical separations (Rh), lateral separations (RB) and along-wind (windward/ leeward wall) (RL)
Dynamic response
• Gust effect factor (ASCE-7) :
In fact it is :
where :
Rn should be :2
u
1u1
4σ
)(nSπn
2u
1u1
2
4σ
)(nSπn.
1.7
2
3/51
12
u
1u1
10.3N1
N9.6
σ
)(nSn
Note that : 6.9=(2/3)10.3 so that
1dnσ
(n)S2
u
u
0
z
z11 V
LnN
But Su(0) should = 4u2u /Uz (Lecture 7) Hence Lz = (4/6.9) u = 0.58 u
Note that Su(0) is equal to 6.9u2Lz/Vz
Dynamic response
• Along-wind response of structure with distributed mass :
Use : generalized (modal) mass, stiffness, damping, applied force for each mode
The calculation of along-wind response with distributed masses (many modes of vibration) is more complex (Section 5.3.6 in the book)
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
Based on modal analysis (Lecture 11) :
x(z,t) = j aj (t) j (z) j (z) is mode shape in jth mode
