Wind loading and structural response Lecture 15 Dr

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 = 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 : since N(s) = K/sm

Fatigue under wind loading • Narrow band random loading : By Miner’s Rule : Probability distribution of peaks is Rayleigh : (Lecture 3) substituting, damage (x) is the Gamma Function EXCEL gives loge (x) : ( n! = (n+1) ) GAMMALN()

Fatigue under wind loading • Narrow band random loading : Fatigue life : set D =1, rearrange as expression for T 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. 033 m (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 Probability distribution of U : (Weibull) Probability of exceedence

Fatigue under wind loading • Effect of varying wind speed : Probability density of U (Weibull) : The fraction of the time T during which the mean wind speed falls between U and U+ U is f. U(U). U. Amount of damage generated during this time :

Fatigue under wind loading • Effect of varying wind speed : Total damage for all mean wind speeds :

Fatigue under wind loading • Fatigue life : Lower limit (based on narrow band vibrations) : 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

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 from EXCEL : GAMMALN() function = 0. 926 - 0. 033 m =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 JHolmes@lsu. edu