Transcript Prediction of design wind speeds

Wind loading and structural response
Lecture 4 Dr. J.D. Holmes
Prediction of design wind speeds
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
1977. Gomes and Vickery. Separation of storm types
1982. Simiu. First comprehensive analysis of U.S. historical extreme wind
speeds. Sampling errors.
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.
1/ k

  k (U  u )  

FU(U) = exp 1 


a

 
 

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 :
U  u  a loge  loge (FU (U))
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 k = 0
Type III k = +0.2
Type II k = -0.2
8
(In this way of
plotting, Type I
appears as a straight
line)
6
(U-u)/a
4
2
0
-3
-2
-1
-2
0
1
2
3
4
-4
-6
Reduced variate : -ln[-ln(FU(U)]
Type I, II :
U is unlimited as c.d.f. reduces (reduced variate increases)
Type III:
U has an upper limit
Prediction of design wind speeds
• Return Period (mean recurrence interval) :
1
Return Period, R =
Probability of exceedence

1
1  FU (U)
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
0.02 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
Prediction of design wind speeds
• Type I Extreme value distribution
U  u  a loge  loge (FU (U))
In terms of
return period :
Large values of R :

1 

U  u  a  loge  loge (1- ) 
R 


U  u  aloge R
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 :
p
m
N  1
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
• same as Gumbel but uses different formula for p
• Gumbel formula is ‘biased’ at top and bottom ends
• Gringorten formula is ‘unbiased’ : p 
m - 0.44
m - 0.44

N  1 - 0.88 N  0.12
• Otherwise the method is the same as the Gumbel method
Prediction of design wind speeds
• Gumbel/ Gringorten methods - example
• Baton Rouge Annual maximum gust speeds 1970-1989
y = - loge (-loge p)
BATON ROUGE
Year
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
LA
Gust speed (mph)
(corrected to 33 ft) ordered
67.58
40.97
48.57
45.4
54.91
46.46
52.8
47.97
76.03
47.97
51.74
48.57
46.46
48.57
53.85
49.97
48.57
50.68
62.3
51.74
53.85
51.74
50.68
52.8
51.74
52.8
45.4
53.85
52.8
53.85
40.97
53.96
47.97
54.91
53.96
62.3
49.97
67.58
47.97
76.03
rank
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Gumbel
p
0.048
0.095
0.143
0.190
0.238
0.286
0.333
0.381
0.429
0.476
0.524
0.571
0.619
0.667
0.714
0.762
0.810
0.857
0.905
0.952
Gringorten Gumbel
p
y
0.028
-1.113
0.078
-0.855
0.127
-0.666
0.177
-0.506
0.227
-0.361
0.276
-0.225
0.326
-0.094
0.376
0.036
0.425
0.166
0.475
0.298
0.525
0.436
0.575
0.581
0.624
0.735
0.674
0.903
0.724
1.089
0.773
1.302
0.823
1.554
0.873
1.870
0.922
2.302
0.972
3.020
Gringorten
y
-1.276
-0.939
-0.724
-0.549
-0.395
-0.252
-0.114
0.021
0.157
0.296
0.439
0.590
0.752
0.930
1.129
1.359
1.636
1.994
2.517
3.567
Prediction of design wind speeds
• Gringorten method -example
• Baton Rouge Annual maximum gust speeds 1970-1989
BATON ROUGE ANNUAL MAXIMA 1970-89
Gust wind speed (mph)
y = 6.24x + 49.4
80
60
40
20
0
-2
-1
0
1
2
reduced variate (Gringorten) -ln(-ln(p))
3
4
Prediction of design wind speeds
• Gringorten method -example
• Baton Rouge Annual maximum gust speeds 1970-1989
Predicted values
Mode =
Slope =
Return Period
10
20
50
100
200
500
1000
UR(mph)
63.4
67.9
73.7
78.1
82.4
88.2
92.5
49.40
6.24

1 

U  u  a  loge  loge (1- ) 
R 


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,

1  
1 
1 
1    1  1  
 Rc   R1  R2 
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,

1  N 
1
1     1 
 Ra  i 1  R i




Ri is the return period for a given wind speed from direction sector i
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
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
1958 -1990
1970 - 1990
1972 - 1981
1950 - 1990
1958 - 1990
1956 - 1990
1971 - 1990
1948 - 1957
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) =
• Value of x exceeded with a probability, G
= [1-(G) k]/k
  kx 
1   σ 
  
1
k
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)
Prediction of design wind speeds
• Excesses (peaks) over threshold approach
• Example of plot of mean excess versus threshold level :
MOREE Downburst Gusts
y = -0.139x + 4.36
5.
4
Average
excess
(m/s)
3
2
1
0.
0
5
10
15
Threshold (m/s)
Negative slope indicates positive k
(extreme wind speed has upper limit )
Prediction of design wind speeds
• Excesses (peaks) over threshold approach
• Prediction of extremes :
MOREE Downburst Gusts
Return Period
scale =
5.067 m/s
shape =
0.161
rate =
2.32 per annum
upper limit (R) = 51.7 m/s
10
20
50
100
200
500
1000
UR (m/s)
32.8
34.8
37.1
38.7
40.1
41.7
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 ?
Probability of non exceedence of a given wind speed
1
in any one year = 1  ( )
R
Assume each year is independent
Probability of non exceedence of a given wind speed
in L years =
1 

1

(
)

R 

L
1 

Risk of exceedence of a given wind speed in L years, r  1  1  ( )
R 

L
Prediction of design wind speeds
• Example :
L = 50 years
R = 50 years
50
Risk of exceedence of a 50-year return period wind
1 

r  1  1  ( )  0.636
speed in 50 years,
50 

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. 2.3.2)
End of Lecture 4
John Holmes
225-405-3789 [email protected]