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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 n 1 < 1 Hertz (approximate).

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along wind dynamic response

Along-wind dynamic response

Wind loading and structural response

Lecture 12 Dr. J.D. Holmes

dynamic response
Dynamic response
  • Significant resonant dynamic response can occur under wind actions for structures with n1 < 1 Hertz (approximate)
  • All structures will experience fluctuatingloads 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 response3

background component

resonant contributions

Dynamic response
  • Spectral density of a response to wind :
dynamic response4

D(t)

time

Dynamic response
  • Time history of fluctuating wind force
dynamic response5

High n1

x(t)

D(t)

time

time

Dynamic response
  • Time history of fluctuating wind force
  • Time history of response :
  • Structure with high natural frequency
dynamic response6

Low n1

D(t)

x(t)

time

time

Dynamic response
  • Time history of fluctuating wind force
  • Time history of response :
  • Structure with low natural frequency
dynamic response7
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 response8
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 response9
Dynamic response
  • Comparison with dynamic response to earthquakes :
dynamic response10
Dynamic response
  • Random vibration approach :
  • Uses spectral densities (frequency domain) for calculation :
dynamic response11

c

D(t)

m

k

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

  • equation of motion :
dynamic response12

since :

Dynamic response
  • Along-wind response of single-degree-of freedom structure :
  • by quasi-steady assumption (Lecture 9) :
  • in terms of spectral density :
  • hence :

this is relation between spectral density of force and velocity

dynamic response13
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

dynamic response14
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’

dynamic response15

1.0

0.1

0.01

0.01 0.1 1.0 10

Dynamic response
  • Aerodynamic admittance:

Low frequency gusts - well correlated

High frequency gusts - poorly correlated

based on experiments :

dynamic response16
Dynamic response
  • Aerodynamic admittance:

hence :

substituting D = kX :

dynamic response17

independent of frequency

assumes X2(n) and Su(n) are constant at X2(n1) and Su(n1), near the resonant peak

Dynamic response
  • Mean square deflection :

where :

dynamic response18
Dynamic response
  • Mean square deflection :

(integration by method of poles)

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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)

dynamic response20
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)

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

dynamic response21
Dynamic response
  • Gust effect factor (ASCE-7) :

For flexible and dynamically sensitive structures (Section 6.5.8.2)

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

dynamic response22
Dynamic response
  • Gust effect factor (ASCE-7) :

Resonant response factor (Equation 6-8) :

Previously :

 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 response23

Rn should be :

Note that : 6.9=(2/3)10.3 so that

Dynamic response
  • Gust effect factor (ASCE-7) :

In fact it is :

where :

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 response24
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