prediction of design wind speeds wind loading and structural response lecture 4 dr. j.d. holmes
TRANSCRIPT
Prediction of design wind speeds
Wind loading and structural response
Lecture 4 Dr. J.D. Holmes
Prediction of design wind speeds
• Historical :
1928. Fisher and Tippett. Three asymptotic extreme value distributions
1954. Gumbel method of fitting extremes. Still widely used for windspeeds.
1955. Jenkinson. Generalized extreme value distribution
1982. Simiu. First comprehensive analysis of U.S. historical extreme wind speeds. Sampling errors.
1977. Gomes and Vickery. Separation of storm types
1990. Davison and Smith. Excesses over threshold method.
1998. Peterka and Shahid. Re-analysis of U.S. data - ‘superstations’
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 (k0) Gumbel Type II (k<0) Frechet
Type III (k>0) ‘Reverse Weibull’
Type I transformation :
Type I (limit as k 0) : FU(U) = exp {- exp [-(U-u)/a]}
k
a
uUk/1
)(1exp
(U))(FloglogauU Uee
If U is plotted versus -loge[-loge(1-FU(U)], we get a straight line
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
-6
-4
-2
0
2
4
6
8
-3 -2 -1 0 1 2 3 4
Reduced variate : -ln[-ln(FU(U)]
(U-u)/a
Type I k = 0Type III k = +0.2Type II k = -0.2
(In this way of plotting, Type I appears as a straight line)
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 = exceedenceofyProbabilit
1
(U)F1
1
U
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
Prediction of design wind speeds
• Type I Extreme value distribution
Large values of R :
(U))(FloglogauU Uee
)
R
1-(1loglogauU ee
RaloguU e
In terms of return period :
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
1N
mp
• 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
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
0.88-1N
0.44-mp
• Otherwise the method is the same as the Gumbel method
12.0N
0.44-m
Prediction of design wind speeds
• Gumbel/ Gringorten methods - example
• Baton Rouge Annual maximum gust speeds 1970-1989
BATON ROUGE LA
Year Gust speed (mph) Gumbel Gringorten Gumbel Gringorten(corrected to 33 ft) ordered rank p p y y
1970 67.58 40.97 1 0.048 0.028 -1.113 -1.2761971 48.57 45.4 2 0.095 0.078 -0.855 -0.9391972 54.91 46.46 3 0.143 0.127 -0.666 -0.7241973 52.8 47.97 4 0.190 0.177 -0.506 -0.5491974 76.03 47.97 5 0.238 0.227 -0.361 -0.3951975 51.74 48.57 6 0.286 0.276 -0.225 -0.2521976 46.46 48.57 7 0.333 0.326 -0.094 -0.1141977 53.85 49.97 8 0.381 0.376 0.036 0.0211978 48.57 50.68 9 0.429 0.425 0.166 0.1571979 62.3 51.74 10 0.476 0.475 0.298 0.2961980 53.85 51.74 11 0.524 0.525 0.436 0.4391981 50.68 52.8 12 0.571 0.575 0.581 0.5901982 51.74 52.8 13 0.619 0.624 0.735 0.7521983 45.4 53.85 14 0.667 0.674 0.903 0.9301984 52.8 53.85 15 0.714 0.724 1.089 1.1291985 40.97 53.96 16 0.762 0.773 1.302 1.3591986 47.97 54.91 17 0.810 0.823 1.554 1.6361987 53.96 62.3 18 0.857 0.873 1.870 1.9941988 49.97 67.58 19 0.905 0.922 2.302 2.5171989 47.97 76.03 20 0.952 0.972 3.020 3.567
y = - loge (-loge p)
Prediction of design wind speeds
• Gringorten method -example
• Baton Rouge Annual maximum gust speeds 1970-1989
BATON ROUGE ANNUAL MAXIMA 1970-89
y = 6.24x + 49.4
0
20
40
60
80
-2 -1 0 1 2 3 4
reduced variate (Gringorten) -ln(-ln(p))
Gu
st w
ind
sp
ee
d (
mp
h)
Prediction of design wind speeds
• Gringorten method -example
• Baton Rouge Annual maximum gust speeds 1970-1989
Mode = 49.40Predicted values Slope = 6.24
Return Period UR(mph)10 63.420 67.950 73.7
100 78.1200 82.4500 88.2
1000 92.5
)
R
1-(1loglogauU ee
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
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,
21
11
11
11
RRRc
R1 is the return period for a given wind speed from type 1 storms etc.
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,
Ri is the return period for a given wind speed from direction sector i
N
ia iRR 1
11
11
Prediction of design wind speeds
• Compositing data (‘superstations’)
Most places have insufficient history of recorded data (e.g. 20-50 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
Prediction of design wind speeds
• Compositing data (‘superstations’)
Example of a superstation (Peterka and Shahid ASCE 1978) :
3931 FORT POLK, LA 1958 -1990 3937 LAKE CHARLES, LA 1970 - 1990 12884 BOOTHVILLE, LA 1972 - 1981 12916 NEW ORLEANS, LA 1950 - 1990 12958 NEW ORLEANS, LA 1958 - 1990 13934 ENGLAND, LA 1956 - 1990 13970 BATON ROUGE, LA 1971 - 1990 93906 NEW ORLEANS, LA 1948 - 1957
193 station-years of combined data
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
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) =
k
1
σ
kx1
• Value of x exceeded with a probability, G = [1-(G) k]/k
Prediction of design wind speeds• Excesses (peaks) over threshold approach
• Average number of excesses above lowest threshold, uo per annum =
= u0 + [1-(R)-k]/k
• Upper limit to UR as R for positive k
• UR= u0 +( /k)
≈ u0 + value of x exceeded with a probability, (1/ R)
• 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
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
0.
1
2
3
4
5.y = -0.139x + 4.36
0 5 10 15
Threshold (m/s)
Average excess (m/s)
Prediction of design wind speeds• Excesses (peaks) over threshold approach
• Prediction of extremes :
upper limit (R) = 51.7 m/s
MOREE Downburst Gusts
Return Period UR (m/s)
scale = 5.067 m/s 10 32.820 34.8
shape = 0.161 50 37.1100 38.7
rate = 2.32 per annum 200 40.1500 41.7
1000 42.7
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
L
R
)
1(1
L
Rr
)
1(11
)1
(1R
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,
Prediction of design wind speeds• Example :
L = 50 years
636.0)50
1(11
50
r
R = 50 years
There is a 64% chance that U50 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. 2.3.2)
