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Prediction of design wind speeds Wind loading and structural response Lecture 4 Dr. J.D. Holmes

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Prediction of design wind speeds Historical : Fisher and Tippett. Three asymptotic extreme value distributions Gumbel method of fitting extremes. Still widely used for windspeeds Jenkinson. Generalized extreme value distribution Simiu. First comprehensive analysis of U.S. historical extreme wind speeds. Sampling errors Gomes and Vickery. Separation of storm types Davison and Smith. Excesses over threshold method Peterka and Shahid. Re-analysis of U.S. data - superstations

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Prediction of design wind speeds Generalized Extreme Value distribution (G.E.V.) : c.d.f. F U (U) = k is the shape factor; a is the scale factor; u is the location parameter Special cases : Type I (k 0) Gumbel Type II (k<0) Frechet Type III (k>0) Reverse Weibull Type I transformation : Type I (limit as k 0) : F U (U) = exp {- exp [-(U-u)/a]} If U is plotted versus -log e [-log e (1-F U (U)], we get a straight line

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Prediction of design wind speeds Generalized Extreme Value distribution (G.E.V.) : Type I, II : U is unlimited as c.d.f. reduces (reduced variate increases) Type III: U has an upper limit Reduced variate : -ln[-ln(F U (U)] (U-u)/a Type I k = 0 Type III k = +0.2 Type II k = -0.2 (In this way of plotting, Type I appears as a straight line)

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Prediction of design wind speeds Return Period (mean recurrence interval) : Unit : depends on population from which extreme value is selected A 50-year return-period wind speed has an probability of exceedence of 0.02 in any one year Return Period, R = e.g. for annual maximum wind speeds, R is in years it should not be interpreted as occurring regularly every 50 years or average rate of exceedence of 1 in 50 years

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Prediction of design wind speeds Type I Extreme value distribution Large values of R : In terms of return period :

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Prediction of design wind speeds Gumbel method - for fitting Type I E.V.D. to recorded extremes - procedure Assign probability of non-exceedence Extract largest wind speed in each year Rank series from smallest to largest m=1,2…..to N Form reduced variate : y = - log e (-log e p) Plot U versus y, and draw straight line of best fit, using least squares method (linear regression) for example

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Prediction of design wind speeds Gringorten method Gringorten formula is unbiased : same as Gumbel but uses different formula for p Gumbel formula is biased at top and bottom ends Otherwise the method is the same as the Gumbel method

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Prediction of design wind speeds Gumbel/ Gringorten methods - example Baton Rouge Annual maximum gust speeds y = - log e (-log e p)

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Prediction of design wind speeds Gringorten method -example Baton Rouge Annual maximum gust speeds

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Prediction of design wind speeds Gringorten method -example Baton Rouge Annual maximum gust speeds

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Prediction of design wind speeds Separation by storm type Baton Rouge data (and that from many other places) indicate a mixed wind climate Some annual maxima are caused by hurricanes, some by thunderstorms, some by winter gales Effect : often an upward curvature in Gumbel/Gringorten plot Should try to separate storm types by, for example, inspection of detailed anemometer charts, or by published hurricane tracks

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Prediction of design wind speeds Separation by storm type Probability of annual max. wind being less than U ext due to any storm type = Probability of annual max. wind from storm type 1 being less than U ext Probability of annual max. wind from storm type 2 being less than U ext etc…. (assuming statistical independence) In terms of return period, R 1 is the return period for a given wind speed from type 1 storms etc.

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Prediction of design wind speeds Wind direction effects If wind speed data is available as a function of direction, it is very useful to analyse it this way, as structural responses are usually quite sensitive to wind direction Probability of annual max. wind speed (response) from any direction being less than U ext = Probability of annual max. wind speed (response)from direction 1 being less than U ext Probability of annual max. wind speed (response)from direction 2 being less than U ext etc…. (assuming statistical independence of directions) In terms of return periods, R i is the return period for a given wind speed from direction sector i

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Prediction of design wind speeds Compositing data (superstations) Most places have insufficient history of recorded data (e.g years) to be confident in making predictions of long term design wind speeds from a single recording station Sampling errors : typically 4-10% (standard deviation) for design wind speeds Compositing data from stations with similar climates : reduces sampling errors by generating longer station-years Disadvantages : disguises genuine climatological variations assumes independence of data

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Prediction of design wind speeds Compositing data (superstations) Example of a superstation (Peterka and Shahid ASCE 1978) : 3931 FORT POLK, LA LAKE CHARLES, LA BOOTHVILLE, LA NEW ORLEANS, LA NEW ORLEANS, LA ENGLAND, LA BATON ROUGE, LA NEW ORLEANS, LA station-years of combined data

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Prediction of design wind speeds Excesses (peaks) over threshold approach Uses all values from independent storms above a minimum defined threshold Example : all thunderstorm winds above 20 m/s at a station Procedure : several threshold levels of wind speed are set :u 0, u 1, u 2, etc. (e.g. 20, 21, 22 …m/s) the exceedences of the lowest level by the maximum wind speed in each storm are identified and the average number of crossings per year,, are calculated the differences (U-u 0 ) between each storm wind and the threshold level u 0 are calculated and averaged (only positive excesses are counted) previous step is repeated for each level, u 1, u 2 etc, in turn mean excess for each threshold level is plotted against the level straight line is fitted

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Prediction of design wind speeds Excesses (peaks) over threshold approach Procedure contd.: a scale factor,, and shape factor, k, can be determined from the slope and intercept : Shape factor, k = -slope/(slope +1) - (same shape factor as in GEV) Scale factor, = intercept / (slope +1) These are the parameters of the Generalized Pareto distribution Probability of excess above u o exceeding x, G(x) = Value of x exceeded with a probability, G = [1-(G) k ]/k

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Prediction of design wind speeds Excesses (peaks) over threshold approach Average number of excesses above lowest threshold, u o per annum = = u 0 + [1-( R) -k ]/k Upper limit to U R as R for positive k U R = u 0 +( /k) u 0 + value of x exceeded with a probability, (1/ R) Average number of excesses above u o in R years = R R-year return period wind speed, U R = u 0 + value of x with average rate of exceedence of 1 in R years

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Prediction of design wind speeds Excesses (peaks) over threshold approach Example of plot of mean excess versus threshold level : Negative slope indicates positive k (extreme wind speed has upper limit ) MOREE Downburst Gusts y = x Threshold (m/s) Average excess (m/s)

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Prediction of design wind speeds Excesses (peaks) over threshold approach Prediction of extremes : upper limit (R ) = 51.7 m/s MOREE Downburst Gusts

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Prediction of design wind speeds Lifetime of structure, L Appropriate return period, R, for a given risk of exceedence, r, during a lifetime ? Assume each year is independent Probability of non exceedence of a given wind speed in any one year = Probability of non exceedence of a given wind speed in L years = Risk of exceedence of a given wind speed in L years,

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Prediction of design wind speeds Example : L = 50 years R = 50 years There is a 64% chance that U 50 will be exceeded in the next 50 years Risk of exceedence of a 50-year return period wind speed in 50 years, Wind load factor must be applied e.g. 1.6 W for strength design in ASCE-7 (Section )

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End of Lecture 4 John Holmes

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