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## Along-wind dynamic response

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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

x(t)

D(t)

time

time

Dynamic response- Time history of fluctuating wind force

- Time history of response :

- Structure with high natural frequency

D(t)

x(t)

time

time

Dynamic response- Time history of fluctuating wind force

- Time history of response :

- 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 :

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 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 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 response

- Aerodynamic admittance:

- Larger structures - velocity fluctuations approaching windward face cannot be assumed to be uniform

then :

where 2(n) is the ‘aerodynamic admittance’

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 :

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 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 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 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 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)

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 6.9u2Lz/Vz

But Su(0) should = 4u2lu /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

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