03 prediction of design wind speeds

8/13/2019 03 Prediction of Design Wind Speeds

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Prediction of design wind speeds

Wind loading and structural response

Lecture 4 Dr. J.D. Holmes


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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 windspeeds. 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


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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 (k0) 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


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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 ofplotting, Type I

appears as a straight

line)


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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


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Type I Extreme value distribution

Large values of R :

(U))(FloglogauU Uee

)

R

1-(1loglogauU ee

RaloguU e

In terms of

return period :


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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

1Nm

p

Form reduced variate : y = - loge(-logep)

Plot U versus y, and draw straight line of best fit, using least squares

method (linear regression) for example


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Gringorten method

Gringorten formula is unbiased :

same as Gumbel but uses different formula for p

Gumbel formula is biased at top and bottom ends

0.88-1N0.44-m

p

Otherwise the method is the same as the Gumbel method

12.0N0.44-m


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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 y1970 67.58 40.97 1 0.048 0.028 -1.113 -1.276

1971 48.57 45.4 2 0.095 0.078 -0.855 -0.939

1972 54.91 46.46 3 0.143 0.127 -0.666 -0.724

1973 52.8 47.97 4 0.190 0.177 -0.506 -0.549

1974 76.03 47.97 5 0.238 0.227 -0.361 -0.395

1975 51.74 48.57 6 0.286 0.276 -0.225 -0.252

1976 46.46 48.57 7 0.333 0.326 -0.094 -0.114

1977 53.85 49.97 8 0.381 0.376 0.036 0.021

1978 48.57 50.68 9 0.429 0.425 0.166 0.157

1979 62.3 51.74 10 0.476 0.475 0.298 0.2961980 53.85 51.74 11 0.524 0.525 0.436 0.439

1981 50.68 52.8 12 0.571 0.575 0.581 0.590

1982 51.74 52.8 13 0.619 0.624 0.735 0.752

1983 45.4 53.85 14 0.667 0.674 0.903 0.930

1984 52.8 53.85 15 0.714 0.724 1.089 1.129

1985 40.97 53.96 16 0.762 0.773 1.302 1.359

1986 47.97 54.91 17 0.810 0.823 1.554 1.636

1987 53.96 62.3 18 0.857 0.873 1.870 1.994

1988 49.97 67.58 19 0.905 0.922 2.302 2.517

1989 47.97 76.03 20 0.952 0.972 3.020 3.567

y = - loge(-logep) .. note: loge =ln


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Gringorten method -example


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))

Gustw

indspeed(mph)


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Gringorten method -example


Mode = 49.40

Predicted values Slope = 6.24

Return Period UR(mph)

10 63.4

20 67.9

50 73.7

100 78.1

200 82.4

500 88.2

1000 92.5

)

R

1-(1loglogauU ee


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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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Separation by storm type

Probability of annual max. wind being less than Uextdue 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

R1is the return period for a given wind speed from type 1 storms etc.


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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 UextProbability 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


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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


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Compositing data (superstations)

Example of a superstation (Peterka and Shahid ASCE 1978) :

3931 FORT POLK, LA 1958 -1990

3937 LAKE CHARLES, LA 1970 - 199012884 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 - 199093906 NEW ORLEANS, LA 1948 - 1957

193 station-years of combined data


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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, u2etc, in turn

mean excess for each threshold level is plotted against the level

straight line is fitted


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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 uoexceeding x, G(x) =k

1

kx1

Value of x exceeded with a probability, G = [1-(G)k]/k


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Average number of excesses above lowest threshold, uoper annum =

= u0

+ [1-(R)-k]/k

Upper limit to URas R for positive k

UR= u0+(/k)

u0 + value of x exceeded with a probability, (1/ R)

Average number of excesses above uoin R years = R

R-year return period wind speed, UR= u0 +

value of x with average rate of exceedence of 1 in R years


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Prediction of extremes :

upper limit (R) = 51.7 m/s

MOREE Downburst Gusts

Return Period UR(m/s)

scale = 5.067 m/s 10 32.8

20 34.8

shape = 0.161 50 37.1

100 38.7

rate = 2.32 per annum 200 40.1

500 41.7

1000 42.7


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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,


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End of Lecture 4

John Holmes225-405-3789 [email protected]

03 prediction of design wind speeds

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