7/27/2019 03 Lect 15 Fatigue

1/17

Fatigue under wind loading

Wind loading and structural response

Lecture 15 Dr. J.D. Holmes

7/27/2019 03 Lect 15 Fatigue

2/17

Fatigue under wind loading

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

7/27/2019 03 Lect 15 Fatigue

3/17

Fatigue under wind loading

Failure model - based on sinusoidal test results

Nsm = K

N = cycles to failure

s = stress amplitude

K = a constant depending on material

m = exponent between 5 and 20

7/27/2019 03 Lect 15 Fatigue

4/17

Fatigue under wind loading

Failure model - based on sinusoidal test results

Typical s-N graph

:

7/27/2019 03 Lect 15 Fatigue

5/17

Fatigue under wind loading

Failure model

Miners 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)

7/27/2019 03 Lect 15 Fatigue

6/17

Fatigue under wind loading

Narrow band random loading :

total number of cycles in a time period, T, is o+T

for narrow-band random stress s(t), the proportion of

cycles with amplitudes in the range from s to s + s,= fp(s).

sfp(s) is the probability density of the peaks

o+ is the rate of crossing of the mean stress ( natural

frequency)

s(t)

time

7/27/2019 03 Lect 15 Fatigue

7/17

Fatigue under wind loading

Narrow band random loading :

since N(s) = K/sm

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

ss(s)Tf

N(s)

n(s)m

po

7/27/2019 03 Lect 15 Fatigue

8/17

Fatigue under wind loading

Narrow band random loading :

By Miners Rule :

Probability distribution of peaks is

Rayleigh : (Lecture 3)

K

dss(s)fT

N(s)

n(s)D

m

p0

o

0

2

2

2p 2

sexp

s(s)f

substituting, damage ds2

s

expsK

T

D 2

21m

02

o

1)

2

m()2(

K

T mo

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

EXCEL gives loge (x) : GAMMALN()

7/27/2019 03 Lect 15 Fatigue

9/17

Fatigue under wind loading

Narrow band random loading :

Fatigue life : set D =1, rearrange

as expression for T

1)2

m()2(

KTm

o

Only applies for one mean wind speed,U, since

standard deviation of stress, , varies with wind speed

need to incorporate probability distribution of

U

7/27/2019 03 Lect 15 Fatigue

10/17

Fatigue under wind loading

Wide band loading :

More typical of wind loading

Fatigue damage under wide band loading : Dwb=

Dnb = empirical factor

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

curve)

7/27/2019 03 Lect 15 Fatigue

11/17

Fatigue under wind loading

Effect of varying wind speed :

Standard deviation of stress is a function of mean wind

speed : = AUn

Probability distribution ofU :

(Weibull)

k

Uc

Uexp1)U(F

Loxton 1984-2000 (all directions)

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20 25

ind speed (m/s)

dataWeibull fit (k=1.36, c=3.40)

Probability of

exceedence

7/27/2019 03 Lect 15 Fatigue

12/17

Fatigue under wind loading

Effect of varying wind speed :

Amount of damage generated during this time :

The fraction of the time T during which the mean wind speed fallsbetween U and U+U is fU(U).U.

Probability density ofU

(Weibull) :

k

k

1k

Uc

Uexp

c

Uk)U(f

1)2

m()AU2(

K

U(U)TfD mn

Uo

U

7/27/2019 03 Lect 15 Fatigue

13/17

Fatigue under wind loading

Effect of varying wind speed :

Total damage for all mean wind speeds :

dU(U)fU1)2

m(

K

A)2T(D U

mn

0

m

o

dUc

Uexp

c

kU1)

2

m(

K

A)2T(k

k

1kmn

0

m

o

)k

kmn(1)

2

m(

K

cA)2T(D

mnm

o

7/27/2019 03 Lect 15 Fatigue

14/17

Fatigue under wind loading

Fatigue life :

Lower limit (based on narrow band vibrations) :

)k

kmn(1)2

m(cA)2(

KT

mnm

o

lower

)k

kmn(1)

2

m(cA)2(

KT

mnm

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

7/27/2019 03 Lect 15 Fatigue

15/17

Fatigue under wind loading

Example :

m = 5 ; n = 2 ; k = 2; 0+ = 0.5 Hertz

from EXCEL : GAMMALN() function

= 0.926 - 0.033m =0.761

K = 2 x 1015 [MPa]1/5; c = 8 m/s ; A = 0.1 2(m/s)

MPa

1205!(6))2

2mn(

3.323e(3.5)1)2

m( 1.201

secs101.65120.03.32380.1)2(0.5

102T 8105

15

lower

years13.8years0.7615.242

TT lowerupper

years5.2years360024365

101.65 8

7/27/2019 03 Lect 15 Fatigue

16/17

Fatigue under wind loading

Sensitivity :

Fatigue life is inversely proportional to

Am

- sensitive to stress concentrations

Fatigue life is inversely proportional to

cmn

- sensitive to wind climate

7/27/2019 03 Lect 15 Fatigue

17/17

End of Lecture 15

John Holmes225-405-3789 JHolmes@lsu.edu