 Typical s-N graph : Fatigue under wind loading Failure model Miners Rule : ni N 1 i 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) o Tf p (s) s ss N(s) K since N(s) = K/sm Fatigue under wind loading Narrow band random loading : By Miners Rule : m T f (s) s ds o

p n(s) 0 D 0 N(s) K Probability distribution of peaks is Rayleigh : (Lecture 3) substituting, damage o T D 2 K 0 s2

s f p (s) 2 exp 2 2 s2 m 1 ds s exp 2 2 (x) is the Gamma Function ( n! = ) is the Gamma Function ( n! = (n+1) )) ) EXCEL gives loge (x) is the Gamma Function ( n! = ) : GAMMALN() o T m ( 2) m ( 1) K

2 Fatigue under wind loading Narrow band random loading : Fatigue life : set D =1) ), rearrange as ex) is the Gamma Function ( n! = pression for T T K o ( 2) m ( 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 = ex) is the Gamma Function ( n! = ponent of s-N curve) Fatigue under wind loading Effect of varying wind speed : Standard deviation of stress is a function of mean wind speed : = AUn Probability distribution of U : (Weibull)

FU (U) 1 exp Loxton 1984-2000 (all directions) data Weibull fit (k=1.36, c=3.40) 1.0 Probability of exceedence 0.8 0.6 0.4 0.2 0.0 0 5 10 15

Wind speed (m/s) 20 25 U c k Fatigue under wind loading Effect of varying wind speed : Probability density of U (Weibull) :

kU k 1 f U (U) k exp c U c k 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 o Tf U (U) sU K ( 2AU n ) m ( m 1) 2

Fatigue under wind loading Effect of varying wind speed : Total damage for all mean wind speeds : oT( 2A)m m D ( 1) U mn f U (U) dU 0 K 2 k o T( 2A)m m k

U mn k 1 ( 1) U exp dU k 0 K 2 c c o T( 2A) m c mn m mn k D ( 1) ( ) K 2 k Fatigue under wind loading

Fatigue life : Lower limit (based on narrow band vibrations) : Tlower K m mn k o ( 2A) m c mn ( 1) ( ) 2 k Upper limit (based on wide band vibrations) ( < 1) )) : K Tupper m mn k o ( 2A)m c mn ( 1) ( ) 2 k + o (cycling rate or effective frequency) Can be taken as natural frequency for lower limit; 0.5 x) is the Gamma Function ( n! = natural frequency for upper limit Fatigue under wind loading Example : m = 5 ; n = 2 ; k = 2; 0+ = 0.5 Hertz MPa (m/s) 2 K = 2 x) is the Gamma Function ( n! = 1) )01) )5 [MPa]1) )/5 ; c = 8 m/s ; A = 0.1) ) m 1) (3.5) e1.201 3.323 2 mn 2 ( ) (6) 5!120 2 (

Tlower from EXCEL : GAMMALN() function 2 1015 8 1.65 10 secs 5 10 0.5 ( 2 0.1) 8 3.323120.0 1.65 108 years 5.2 years 365 24 3600

= 0.926 - 0.033m Tupper Tlower 2 5.24 years 13.8 years 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]