1. Along-wind dynamic response
Wind loading and structural response
Lecture 12 Dr. J.D. Holmes
Along-wind dynamic response

2. 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

3. Dynamic response Spectral density of a response to wind :
background component
resonant contributions

4. Dynamic response
Time history of fluctuating wind force
D(t)
time

5. Dynamic response Time history of fluctuating wind force
D(t)
time
Time history of response :
time
x(t)
High n1
Structure with high natural frequency

6. Dynamic response Time history of fluctuating wind force
D(t)
time
Time history of response :
time
x(t)
Low n1
Structure with low natural frequency

7. 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

8. Dynamic response Comparison with dynamic response to earthquakes :
Earthquakes are shorter duration than most wind storms
Dominant frequencies of excitation in earthquakes are 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

9. Dynamic response
Comparison with dynamic response to earthquakes :

10. Dynamic response Random vibration approach :
Uses spectral densities (frequency domain) for calculation :

11. 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
D(t) k c m
equation of motion :

12. Dynamic response
Along-wind response of single-degree-of freedom structure :
by quasi-steady assumption (Lecture 9) :
since :
in terms of spectral density :
hence :
this is relation between spectral density of force and velocity

13. Dynamic response
Along-wind response of single-degree-of freedom structure :
deflection : X(t) = X + x'(t)
mean deflection :
k = spring stiffness
spectral density :
where the mechanical admittance is given by :
this is relation between spectral density of deflection and approach velocity

14. Aerodynamic admittance:
Dynamic response
Aerodynamic admittance:
Larger structures - velocity fluctuations approaching windward face cannot be assumed to be uniform
then :
where 2(n) is the ‘aerodynamic admittance’

15. Aerodynamic admittance:
Dynamic response
Aerodynamic admittance:
Low frequency gusts - well correlated
1.0
0.1
0.01
High frequency gusts - poorly correlated
based on experiments :

16. Aerodynamic admittance:
Dynamic response
Aerodynamic admittance:
hence :
substituting D = kX :

17. Mean square deflection :
Dynamic response
Mean square deflection :
where :
assumes X2(n) and Su(n) are constant at X2(n1) and Su(n1), near the resonant peak
independent of frequency

18. Mean square deflection :
Dynamic response
Mean square deflection :
(integration by method of poles)

19. Gust response factor (G) :
Dynamic response
Gust response factor (G) :
Expected maximum response in defined time period / mean response in same time period
g = peak factor
 = ‘cycling’ rate (average frequency)

20. 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)
Used in codes and standards based on peak gust (e.g. ASCE-7)

21. Gust effect factor (ASCE-7) :
Dynamic response
Gust effect factor (ASCE-7) :
For flexible and dynamically sensitive structures (Section )
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

22. Gust effect factor (ASCE-7) :
Dynamic response
Gust effect factor (ASCE-7) :
Resonant response factor (Equation 6-8) :
Previously :
 is critical damping ratio ()
RhRB( RL) is the aerodynamic admittance 2(n1)
decomposed into components for vertical separations (Rh), lateral separations (RB) and along-wind (windward/ leeward wall) (RL)

23. Gust effect factor (ASCE-7) :
Dynamic response
Gust effect factor (ASCE-7) :
Rn should be :
In fact it is :
where :
Note that : 6.9=(2/3)10.3 so that
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

24. Along-wind response of structure with distributed mass :
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 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

25. End of Lecture 12 John Holmes 225-405-3789 JHolmes@lsu.edu