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]

## Recently Viewed Presentations

• The functional organizational structure is typically used in businesses that primarily sell and produce standard products. The advantages of a functional organizational structure are no duplication of activities and functional excellence. Disadvantages include insularity, slow response time, and lack of...
• Miodrag Bolic Advisor: Prof. Petar M. Djuric Introduction - Motivations and Goals Implementation of PFs - VLSI Signal Processing Architectures Implementation of PFs - Methodology Introduction - Motivations and Goals Implementation of PFs - VLSI Signal Processing Architectures ...
• By Robert J. Carbaugh 9th Edition Chapter 2: Foundations of Modern Trade Theory Historical development of trade theory Mercantilism Regulation to ensure a positive trade balance Critics: possible only for short term; assumes static world economy Absolute advantage (Adam Smith)...
• GENIUS HOUR is designed to provide an outlet for students to develop ownership of knowledge related to topics of interest and to present what they learn through project development to an audience. We will work on project presentation skills, citing...
• Introduction to the newest anticonvulsants. Barbara Lynne Phillips, M.D. Assistant Professor of Neurology. WSU BSOM. ... Other concerns: enzyme inducers reduce its effectiveness, may reduce BCP efficacy, possible euphoria, not rec in severe hep/renal.
• Calibri Arial Symbol Office Theme Akatsuki Mission Update: New Images and Status of the Mission New Image from 2 micron Camera 2 micron Image of Venus' Day Side UV Image of Venus in Reflected Sunlight UV Image of Venus in...
• Built-in web content and document management capabilities allow you to create, store, edit, and distribute web pages and store and distribute most office documents such as PDF, spreadsheets, and text documents. Workflow tools allow you to automate business processes such...
• The overall fin efficiency is COMMENTS: (1) The gas-side resistance is substantially decreased by using the fins and q is increased. KNOWN: Geometry of finned, annular heat exchanger.