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

2. 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
Gomes and Vickery. Separation of storm types
Simiu. First comprehensive analysis of U.S. historical extreme wind speeds. Sampling errors.
Davison and Smith. Excesses over threshold method.
Peterka and Shahid. Re-analysis of U.S. data - ‘superstations’

3. Prediction of design wind speeds
Generalized Extreme Value distribution (G.E.V.) :
c.d.f FU(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 (limit as k 0) : FU(U) = exp {- exp [-(U-u)/a]}
Type I transformation :
If U is plotted versus -loge[-loge(1-FU(U)], we get a straight line

4. Prediction of design wind speeds
Generalized Extreme Value distribution (G.E.V.) : -6 -4 -2 2 4 6 8 -3 -1 1 3
Reduced variate : -ln[-ln(FU(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)
Type I, II : U is unlimited as c.d.f. reduces (reduced variate increases)
Type III: U has an upper limit

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

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

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

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

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

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

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

12. 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

13. Prediction of design wind speeds
Separation by storm type
Probability of annual max. wind being less than Uext due to any storm type =
Probability of annual max. wind from storm type 1 being less than Uext
 Probability of annual max. wind from storm type 2 being less than Uext
 etc… (assuming statistical independence)
In terms of return period,
R1 is the return period for a given wind speed from type 1 storms etc.

14. 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 Uext =
Probability of annual max. wind speed (response)from direction 1 being less than Uext Probability of annual max. wind speed (response)from direction 2 being less than Uext  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

15. 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

16. Prediction of design wind speeds
Compositing data (‘superstations’)
Example of a superstation (Peterka and Shahid ASCE 1978) :
3931 FORT POLK, LA
3937 LAKE CHARLES, LA
12884 BOOTHVILLE, LA
12916 NEW ORLEANS, LA
12958 NEW ORLEANS, LA
13934 ENGLAND, LA
13970 BATON ROUGE, LA
93906 NEW ORLEANS, LA
193 station-years of combined data

17. 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 :u0, u1, u2, 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-u0) between each storm wind and the threshold level u0 are calculated and averaged (only positive excesses are counted)
previous step is repeated for each level, u1, u2 etc, in turn
mean excess for each threshold level is plotted against the level
straight line is fitted

18. 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 uo exceeding x, G(x) =
= [1-(G) k]/k
Value of x exceeded with a probability, G

19. Prediction of design wind speeds
Excesses (peaks) over threshold approach
Average number of excesses above lowest threshold, uo per annum = 
Average number of excesses above uo in R years = R
R-year return period wind speed, UR = u0 +
value of x with average rate of exceedence of 1 in R years
≈ u0 + value of x exceeded with a probability, (1/ R)
= u0 + [1-(R)-k]/k
Upper limit to UR as R  for positive k
UR= u0 +( /k)

20. Prediction of design wind speeds
Excesses (peaks) over threshold approach
Example of plot of mean excess versus threshold level :
MOREE Downburst Gusts 0. 1 2 3 4 5.
y = x 5 10 15
Threshold (m/s)
Average excess (m/s)
Negative slope indicates positive k
(extreme wind speed has upper limit )

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

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

23. Prediction of design wind speeds
Example :
L = 50 years
R = 50 years
Risk of exceedence of a 50-year return period wind speed in 50 years,
There is a 64% chance that U50 will be exceeded in the next 50 years
Wind load factor must be applied
e.g. 1.6 W for strength design in ASCE-7 (Section )

24. End of Lecture 4 John Holmes 225-405-3789 JHolmes@bigpond.com