Transcript Fatigue under wind loading - LSU Hurricane Engineering

Wind loading and structural response
Lecture 15 Dr. J.D. Holmes
Fatigue under wind loading
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
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
Fatigue under wind loading
• Failure model - based on sinusoidal test results
Typical s-N graph
:
Fatigue under wind loading
• Failure model
Miner’s Rule :
 ni
  N
 i

  1

ni = number of stress cycles at given amplitude
Ni = number of stress cycles for failure at that amplitude
Assumes fractional damage at different stress amplitudes adds
linearly to give total damage
No restriction on order of
loading
‘High-cycle’ fatigue (stresses below yield
stress)
Fatigue under wind loading
• Narrow band random loading :
s(t)
time
for narrow-band random stress s(t), the proportion of
cycles with amplitudes in the range from s to s + s,= fp(s).
s
fp(s) is the probability density of the peaks
total number of cycles in a time period, T, is o+T
o+ is the rate of crossing of the mean stress ( natural
frequency)
Fatigue under wind loading
• Narrow band random loading :
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
:

m
n(s) υo T fp (s) s δs

N(s)
K
since N(s) = K/sm
Fatigue under wind loading
• Narrow band random loading :
By Miner’s Rule :

D  0



0
K
 s2 
m 1
s exp  2 ds
 2σ 
(x) is the Gamma Function
EXCEL gives loge (x) :
0
 s2 
s
f p (s)  2 exp   2 
σ
 2σ 
Probability distribution of peaks is
Rayleigh : (Lecture 3)
υo T
substituting, damageD 
Kσ 2
n(s)

N(s)
υo T  f p (s) s m ds
υo T
m

( 2σ)m Γ(  1)
K
2
( n! = (n+1) )
GAMMALN()
Fatigue under wind loading
• Narrow band random loading :
Fatigue life : set D =1, rearrange
as expression for T
T
K
υo ( 2σ)m Γ(
m
 1)
2
Only applies for one mean wind speed,U, since
standard deviation of stress, , varies with wind speed
need to incorporate probability distribution of
U
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)
Fatigue under wind loading
• Effect of varying wind speed :
Standard deviation of stress is a function of mean wind
speed :  = AUn
  U k 
FU (U)  1  exp   
  c  
Probability distribution of U :
(Weibull)
Loxton 1984-2000 (all directions)
data
Weibull fit (k=1.36, c=3.40)
1.0
0.8
Probability of
exceedence
0.6
0.4
0.2
0.0
0
5
10
15
Wind speed (m/s)
20
25
Fatigue under wind loading
• Effect of varying wind speed :
  U k 
kU k 1
f U (U)  k exp   
c
  c  
Probability density of U
(Weibull) :
The fraction of the time T during which the mean wind speed falls
between U and U+U is fU(U).U.
Amount of damage generated during this time :
DU 
υo T fU (U) δU
K
( 2AUn ) m Γ(
m
 1)
2
Fatigue under wind loading
• Effect of varying wind speed :
Total damage for all mean wind speeds :

υo T ( 2A)m m
D
Γ(  1) U mnf U (U) dU
0
K
2
k



υo T ( 2A)m m
k
U


mn k 1

Γ(  1) U
exp    dU
k
0
K
2
c
  c  
υo T ( 2A)m c mn m
mn  k
D
Γ(  1) Γ(
)
K
2
k
Fatigue under wind loading
• Fatigue life :
Lower limit (based on narrow band vibrations) :
Tlower 
K
m
mn  k
υo ( 2A)m c mn Γ(  1) Γ(
)
2
k
Upper limit (based on wide band vibrations) ( < 1) :
Tupper 
K
m
mn  k
λυo ( 2A)m cmn Γ(  1) Γ(
)
2
k
o+ (cycling rate or ‘effective’ frequency)
Can be taken as natural frequency for lower limit;
0.5 x natural frequency for upper limit
Fatigue under wind loading
• Example :
m = 5 ; n = 2 ; k = 2; 0+ = 0.5 Hertz
K = 2 x 1015 [MPa]1/5 ; c = 8 m/s ; A = 0.1
m
 1)  Γ(3.5) e1.201  3.323
2
mn  2
Γ(
)  Γ(6)  5! 120
2
Γ(
Tlower
MP a
(m/s)2
from EXCEL : GAMMALN() function
2 1015
8


1.65

10
secs
5
10
0.5 ( 2  0.1)  8  3.323120.0
1.65108

years 5.2 years
365 24 3600
 = 0.926 - 0.033m =0.761
Tupper 
Tlower 2  5.24

years  13.8 years
λ
0.761
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
End of Lecture 15
John Holmes
225-405-3789 [email protected]