Fatigue Under Wind Loading: Failure Model, Random Loading, and Varying Wind Speed, Slides of Environmental Law and Policy

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• Occurs on slender chimneys, masts under vortex shedding - narrow

(frequency) band

• Occurs on steel roofing under wide band loading
• May occur in along-wind dynamic response - background - wide band
• resonant - narrow band

• Failure model - based on sinusoidal test results

Typical s-N graph

:

• Failure model

Miner’s Rule :  1 

 i

i

N

n

Assumes fractional damage at different stress amplitudes adds

linearly to give total damage

ni = number of stress cycles at given amplitude

Ni = number of stress cycles for failure at that amplitude

No restriction on order of

loading

‘High-cycle’ fatigue (stresses below yield

stress)

• Narrow band random loading :

since N(s) = K/s

m

total number of cycles with amplitudes in the range s to s,

n(s) = o

T fp(s). s

fractional damage at stress level, s

K

υ Tf (s) s δs

N(s)

n(s)

m o p



• Narrow band random loading :

By Miner’s Rule :

Probability distribution of peaks is

Rayleigh : (Lecture 3)

K

υ T f (s) s ds

N(s)

n(s) D

m o 0 p

0

 

    

2

2

p (^2) 2 σ

s exp σ

s f (s)

substituting, damage ds

2 σ

s s exp Kσ

υ T D 2

2 m 1 (^20)

o 

 





m ( 2 σ) Γ( K

υ (^) o T m  



(x) is the Gamma Function ( n! = (n+1) )

EXCEL gives loge (x) : GAMMALN()

• Wide band loading :

More typical of wind loading

Fatigue damage under wide band loading : D wb =

D nb

 = empirical factor

Lower limit for  = 0.926 - 0.033m (m = exponent of s-N

curve)

• Effect of varying wind speed :

Standard deviation of stress is a function of mean wind

speed :  = AU

n

Probability distribution of U :

(Weibull)

k

U c

U

F (U) 1 exp

Loxton 1984-2000 (all directions)

0 5 10 15 20 25 Wind speed (m/s)

data Weibull fit (k=1.36, c=3.40)

Probability of exceedence

• Effect of varying wind speed :

Total damage for all mean wind speeds :

1. U f (U) dU 2

m Γ( K

υ T( 2 A) D (^) U

mn

0

m o 





 

dU c

U

exp c

k

1. U 2

m Γ( K

υ T( 2 A)

k

k

mn k 1

0

m o

   



k

mn k

1. Γ( 2

m Γ( K

υ T( 2 A) c D

m mn o   



• Fatigue life :

Lower limit (based on narrow band vibrations) :

) k

mn k

1. Γ( 2

m υ ( 2 A) c Γ(

K T m mn o

lower  

 

) k

mn k

1. Γ( 2

m λυ ( 2 A) c Γ(

K T m mn o

upper  

 

Upper limit (based on wide band vibrations) ( < 1) :

o

(cycling rate or ‘effective’ frequency)

Can be taken as natural frequency for lower limit;

0.5 x natural frequency for upper limit

Sensitivity :

Fatigue life is inversely proportional to

A

m

• sensitive to stress concentrations

Fatigue life is inversely proportional to

c

mn

• sensitive to wind climate