Download an extreme-value estimating method of non

The Seventh Asia-Pacific Conference on
Wind Engineering, November 8-12, 2009,
Taipei, Taiwan
AN EXTREME-VALUE ESTIMATING METHOD OF NONGAUSSIAN WIND PRESSURE
1
Yong Quan1 Ming Gu2 Yukio Tamura3 Bin Chen4
Associate Professor, State Key Laboratory of Disaster Reduction in Civil Engineering,
Tongji University, Shanghai 200092, China, [email protected]
2
Professor, State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji
University, Shanghai 200092, China, [email protected]
3
Professor, Department of Building Engineering, Tokyo Polytechnic University,
Atsugi, 243-0213,Japan, [email protected]
4
Engineer, Hangzhou Municipal Installation Supervision and Administration Center;
Hangzhou 310003; China, [email protected]
ABSTRACT
With non-Gaussian pressure measurement wind tunnel test data of low-rise building models in a simulated
suburban terrain in the boundary layer wind tunnel in Tokyo Polytechnic University in Japan, an estimating
method of expected extreme-value of non-Gaussian wind pressure coefficients which needs just one single
sample is proposed based on the classical extreme-value theory. The extreme-values of wind pressure
coefficients on roofs of tested low-rise building models calculated with the proposed method and methods used
widely at present, the peak factor method, the improved peak factor method and the Sadek-Simiu method, are
compared carefully. The results indicate that the proposed method can estimate extreme-values of non-Gaussian
wind pressure coefficients on the low-rise building roofs more accurately than the other methods used widely.
KEYWORDS: NON-GAUSSIAN WIND PRESSURE; EXTREME-VALUE; ESTIMATING METHOD
Introduction
As the wind pressures on buildings are varied temporally and spatially, their extremevalues are useful for researchers and engineers when designing the claddings or components
of buildings. Great attention is paid to methods for extreme-values in wind engineering and
they have been developing rapidly for decades.
In early stage, researchers represented by Davenport(1964) estimated the extremevalues of fluctuating wind pressures by summing up their mean value and standard deviation
multiplied by a peak factor obtained with the level-cross method, in which the fluctuating
wind pressures was assumed to be the Gaussian process (peak factor method by short).
However, Holmes(1985) and Massimiliano & Vittorio(2002) pointed out that the assumption
of Gaussian processes is not applicable to the design of claddings or the connections because
distribution of wind pressures are quite different from it, especially on the condition of high
turbulence intensity and at the separation area. The extreme-values calculated with the peak
factor method are always smaller than the real ones, which makes the design based on these
results unsafe. Holmes(1985) and Massimiliano & Vittorio(2002) found that the nonGaussian property of wind pressure at the separation area has great effect on the extremevalues. By comparing the models, Holmes(1985) found the structure damage induced by nonGaussian wind loads is 15%~30% more severe than damage induced by Gaussian ones. Due
to this problem of the peak factor method, many kinds of improvement are introduced, in
which the method proposed by Kareem & Zhao (1994) is the most representative (improved
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan
peak factor method by short). Based on the work of Davenport, they transform the Gaussian
random variable to a Hermite polynomial of a non-Gaussian random variable with their high
order moments, skewness coefficients and kurtosis coefficients. This improvement makes the
peak factor method not only applicable to Gaussian process but also to non-Gaussian process.
Based on the translation process approach proposed by Grigoriu(1995), Sadek&Simiu(2002)
proposed a method (Sadek-Simiu method by short) for extreme-value of non-Gaussian
process. Ge(2004) checked this method with wind tunnel test data and found it quite reliable.
Kasperski (2000) studied the probability distribution of the extreme-values of wind
pressure coefficients on 4 points on a low-rise building model with more than 3000 samples.
The result shows that the probability distribution of extreme-value matches well with Gumbel
distribution (extreme distribution type I). Holmes&Cochran(2003) fitted the extreme-values
of thousands of tested wind pressure with extreme distribution type I and GEV, which are
classical and ideal methods for estimating the extreme-values because the extreme-values of
these samples are stated directly. These methods are widely used in the estimating of extreme
wind velocity and the extreme water level of flood. However these methods are not suitable to
estimate the extreme wind pressure on the surface of structure in practical engineering,
because large numbers of samples are needed. It is not economic to get so much data in wind
tunnel test for practical engineering. A method to estimate extreme-values with just one single
sample is very practical in engineering.
Based on the classical extreme-value theory, a method to estimate the expected value
of extreme-values with just one single sample is proposed under the assumption of stationary
and ergodic stochastic process in the present study. Comparison of the proposed method with
other methods used widely at present is done with the pressure measurement wind tunnel test
data on low-rise building models in a simulated suburban terrain in the boundary layer wind
tunnel in the Tokyo Polytechnic University in Japan. The results indicate the veracity of the
proposed method.
Outline of the test
Data of wind pressure coefficients used in this paper to check the methods for
extreme-values are gotten from the database for low-rise building（http://www.wind.arch.tkougei.ac.jp/info_ center /windpressure /lowrise/mainpage.html）, which are established by
the first author when he was working in Tokyo Polytechnic University in Japan.
The data of the database came from a pressure measurement wind tunnel test on a
series of low-rise building models in a simulated suburban terrain in the boundary layer wind
tunnel on the Tokyo Polytechnic University in Japan. The wind tunnel has a test section of
2.2m in width and 1.8m in height. Considered on the block ratio and the test wind velocity,
the length ratio, wind velocity ratio and time ratio are selected to be 1/100,1/3 and 3/100,
respectively.
70
60
Test(1/100) Category III(AIJ2004)
Mean wind speed
Turbulence intensity
Height Z(m)
50
40
30
20
10
0
0.0
Figure1 Arrangement of the wind field
0.2
0.4
0.6
U(z) or I(z)
0.8
1.0
Figure.2 Simulated wind field of suburban terrain
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan
The terrain category III defined in the Recommendations for Loads on Buildings (AIJ,
2004) was simulated, with the mean velocity profile exponent of α=0.20 and the turbulence
intensity at the height of 10m about 0.25. The simulated wind field and its parameters are
showed in Figure 1 and Figure 2, respectively. Wind velocity in the wind tunnel test at the
height of 10cm is about 7.3m/s, which is 22m/s at the height of 10m in full scale.
Wind
Figure.3 The definition of geometric parameters and the axes of the model
Figure 4 Arrangement of pressure taps
116 gable- and hip-roofed models with varied roof pitches and ratios of height/breadth
and depth/breadth were tested for their surface pressures, which are made of synthetic glass.
The definition of geometric parameters and the axes for gable-roofed models are showed in
the Figure 3.
The data were sampled 10 or 18 times for each test case with a sampling period of 18
seconds (10 minutes in full scale) and a sampling frequency of 500Hz (15Hz in full scale).
Based on the time series of the sampled wind pressure, the time series of pressure coefficients
of each tap can be calculated with follows,
C p (i, t ) = p (i, t ) /(0.5 ρVH2 )
（1）
where C p (i, t ) and p (i, t ) are the time series of pressure coefficients and wind pressures,
respectively, and/(0.5 ρVH2 ) is the wind pressure of the approach flow at the mid-height of the
roofs.
The analyzing of test data on just one of these 116 models is shown in this paper in
detail. The arrangement of pressure taps on this model is shown in Figure 4. The roof pitch,
ratio of height/breadth and ratio of depth/breadth of this model are 21.8o, 4/4 and 3/2,
respectively.
Probability distribution of wind pressure coefficients
In order to assess the non-Gaussian property of wind pressure coefficients of low-rise
buildings, wind pressure coefficients on two taps, Tap 1 near the windward eave and Tap 34
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan
1
10
5
Time series of Cp on Tap 1
0
Probability distribution
Wind pressure coefficient (Cp)
near the leeward eave (Figure 4), on the roof of the selected tested model for wind direction of
0o are analyzed with their time series and probability distributions as shown in Figure 5 and
Figure 6, in which E, ,Cs and Ck are mean value, standard deviation, skewness coefficient
and kurtosis coefficient of wind pressure coefficients, respectively.
-1
-2
-3
-4
-5
0
100
200
300
Time(sec)
400
500
Gaussian process
Cpon Tap 1(E=-1.2;s=0.6;Cs=-1.7;Ck=8.1)
2
1
0.5
0.1
0.01
1E-3
1E-4
-7
600
-6
-5
-4
-3
-2
-1
0
1
Wind pressure coefficient (Cp)
Figure 5 Time series and Probability distribution of Cp on Tap 1
0.5
10
Probability distribution
Wind pressure coefficient (Cp)
20
Time series of Cp on Tap 34
0.0
-0.5
-1.0
0
100
200
300
400
500
600
Time(sec)
Gaussian process
Cp on Tap 34(E=-0.2;s=0.2;Cs=-0.4;Ck=4.7)
5
2
1
0.5
0.1
0.01
1E-3
1E-4
-1.5
-1.0
-0.5
0.0
0.5
Wind pressure coefficient (Cp)
1.0
Figure 6 Time series and Probability distribution of Cp on Tap 34
In Figure 5, there are some high peaks as "spikes" over very short time periods in the
time series of wind pressure coefficient on Tap 1. All of those peaks point downward and
none points upward, which make the time series like a rake. The skeness coefficient and
kurtosis coefficient are -1.7 and 8.1, respectively, far from those for Gaussian process, 0 and
3.0, because of those high peaks. Compared with Gaussian process, there is a longer tail on
the left side of the probability distribution graph of the wind pressure coefficient, which will
make it has a larger negative peak than the Gaussian process.
For the downwind Tap 34, the high peaks of the time series of Cp in Figure 6 point
into two sides, which make the skeness coefficient low. In this case, the skeness coefficient is
-0.4, near the value for Gaussian process. However, the kurtosis coefficient is 4.7, far from the
value for Gaussian process, 3.0, else. As one can see from the probability distribution, there
are long tails at both sides of the graph, which will make the both amplitudes of positive and
negative peak values of the wind pressure coefficients larger than those of the Gaussian
process.
The analysis of the probability distributions of Cp on more taps indicates that the wind
pressures on roofs on low-rise buildings in high turbulence wind field are non-Gaussian
processes.
Proposed extreme-value estimating method
In order to estimate the extreme values of the non-Gaussian wind pressure from just
one single sample, a new method are proposed based on classical Gumbel method.
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan
Classical Gumbel method
Gumbel (1954) found that if the parent distribution of a sample follows exponential
distribution, such as Gaussian distribution, Gamma distribution and Weibull distribution, etc,
its asymptotic distribution of the extreme-values will obey the extreme value type I
distribution (the Gumbel distribution) as follows,
F ( xe ) = exp(− exp(− y ))
（1）
where y is a reduced variable, which is,
y = a( xe − u )
（2）
where fitted variables, u and 1 / α , are mode and dispersion, respectively.
In order to estimate the extreme-value, the two expresses above can be transformed to
the follows:
y = − ln(− ln( F ( xe )))
（3）
xe = u + y / α
The expected value and standard deviation of the extreme-values is,
xe = u + γ / α
π 1
σ xe =
6α
where γ = 0.5772 is the Euler number.
From Eq(5) and Eq(6), the parameters of u and α can be expressed as,
u = x e − 0.45
α = 0.78 / σ xe
（4）
（5）
（6）
（7）
（8）
Many samples are needed in the Gumbel method. The function of the probability
distribution is obtained by extreme-values extracted from the samples.
Estimation of extreme-value with sub-sections
Divide a sample with a length of t1 into n ( n = t1 t2 ) non-overlapping sub-sections
with a length of t2 .The probability of the extreme-value smaller than or equal to x is
expressed as F(T= t1 ) in the parent section and F(T= t2 ) in the sub-section, then the following
relation can be obtained assuming the observed extreme-values are independent,
F (T = t1 ) = F n (T = t2 )
（9）
Based on Eqs.(1,2,9), the following expressions can be obtained:
1/ a(T = t1 ) = 1/ a(T = t2 )
（10）
U (T = t1 ) = U (T = t2 ) + ln(t1 / t2 ) / α (T = t2 )
（11）
Above equations introduce us that the parameters of extreme distribution of a sample
can be calculated with its own sub-sections and then its expected extreme-value can be
calculated.
Optimal observing period of sub-sections
The amount and length of the sub-sections in Eq(9-11), n and t2, can be varied. The
more sub-sections the original data is divided into, the more details are considered and more
inherent characteristics are included in the results. However, there might be not so
independent between the observed extreme-values of the sub-sections if the subsections are
too short, which makes Eq.(9) not applicable. How long is the most optimal observing period
for the steadiest and the most reliable extreme-values?
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan
18 wind pressure samples, with an observing period of 10 min in full scale, on 96
measured taps on the roof of the selected wind tunnel test model are used to check the
methods here. The averages of the observed extreme values of the 18 samples are calculated
and considered as the "standard" extreme values to compare here. The extreme values
estimated by proposed method and other methods are calculated with just one of the 18
samples.
In order to find the best observing period of the sub-sections, the extreme values are
calculated with varied observing periods, 120s, 60s, 40s, 33.3s, 30s, 25s, 16.7s, 8.3s, 6s, 4.8s,
3.0s, 2s, 1.7s, 1.2s, 1s, 0.8s, 0.7s and 0.3s. The average of standard errors (meanSE) of the
estimated extreme values on each taps for varied observing periods of sub-sections from the
"standard" extreme values are calculated as shown in Figure 7. The average of autocorrelation coefficients of the wind pressures at each tap are shown in Figure 8. Comparing
Figure 7 with Figure 8, one can find that the shortest time delay for auto-correlation
coefficient to decay from 1 to the level of 0 is near to the best observing period for least
standard errors. The reliability of this conclusion is proved by more wind tunnel test data
cases. So, the optimal observing period of sub-sections for each tap can be determined with
the auto-correlation analysis of the sample on each tap respectively.
average of auto-correlation coefficient
0.22
0.20
0.18
MeanSE
0.16
0.14
0.12
0.10
0.08
0.06
0
20
40
60
80
100
Observing period of sub-section(sec)
120
Figure 7 Mean standard errors of the
calculated values for varied observe periods
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
0
20
40
60
80
100
120
Time(sec)
Figure 8 Average of auto-correlation
coefficients of wind pressures on 96 taps
Calculating Procedure
In order to facilitate the comprehension and application of the proposed method, the
procedure is listed below:
1) Analyze the auto-correlation coefficients of the sample with a standard observing
period, t1 , on each tap and find the shortest time delay, t2, for auto-correlation
coefficient decaying from 1 to the level of 0;
2) Divide the parent sample into n(n=t1/t2) sub-sections with a observing period of t2;
3) Observe the maximum (or minimum) peak values of each sub-section, xe _ sub , then
their expected value and standard, xe _ sub and σ xe _ sub , can be calculated.
4) Calculate the parameters of u sub and α sub for the extreme distribution type I with
xe _ sub and σ xe _ sub with Eqs.(7) and (8);
5) Transform the parameters for observing period of t2 , u sub and α sub , into ones for t1 ,
u and α ,by Eqs.(10) and (11);
6) Calculate the expected extreme-value for observing period of t1 , x e ,with Eq.(5).
Comparisons
Extreme values of wind pressure coefficients on 96 taps on the selected model roof are
calculated with one of the 18 samples for 10min in full scale for wind direction angle of 0o by
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan
the proposed method, the peak factor method by Davenport (1964), the improved peak factor
method by Kareem & Zhao(1994) and the method by Sadek & Simiu(2002), respectively. The
extreme values by the proposed method and others are compared with the "standard" extreme
values in Figure 9. As one can see, the results of the proposed method are located uniformly
at both sides of the straight line for "standard" values, while the values of other methods are
located at the under side of the straight line, which means underestimation for the "standard"
values.
Figure 10 shows their error ratios on each taps, which are calculated as follows,
C p _ est − C p _ std
γ err =
(12)
C p _ std
where C p _ std , C p _ est and γ err are standard extreme values and estimated extreme values of the
wind pressure coefficients on each taps and their error ratios, respectively.
As shown in Figure 10, the points for the proposed method are distributed uniformly
at both sides of the x-axis while the points for other methods are distributed under the x axis.
The mean γ err for all of 96 taps on the roof are -0.9%, -17%, -22% and -34% for the proposed
method, the Sadek-Simiu method, the improved peak factor method and the peak factor
method, respectively. The RMS values of γ err are 8%, 21%, 23% and 35% for the four
methods, respectively.
"Standard" extreme values
Proposed method
Sadek-Simiu method
Improved peak factor method
Peak factor method
-5
0.2
-4
err
0.0
-3
-0.2
-2
-0.4
-1
-0.6
-1
-2
-3
-4
-5
"Standard" extreme values
0
-6
0.25
Proposed method
Sadek-Simiu method
Improved peak factor method
Peak factor method
0.45
gerr
0.10
0.05
0.00
-0.05
-0.10
-0.15
-0.20
-0.25
60
80
100
Proposed method
Sadek-Simiu method
Improved peak factor method
Peak factor method
0.50
RMS values of
gerr
0.15
40
Figure 10 Error ratios of the methods
0.55
0.20
20
No. of taps
Figure 9 Comparison of estimated extreme
values with "standard" ones
Mean values of
Proposed method
Sadek-Simiu method
Improved peak factor method
Peak factor method
0.4
g
Estimated extreme values
-6
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
-0.30
-0.35
0.00
0
2
4
6
8
10
12
14
16
No. of model cases
Figure 11 Mean values of γ err for
96 taps for 18 samples of 16 model cases
0
2
4
6
8
10
12
14
16
No. of model cases
Figure 12 RMS values of γ err for
96 taps for 18 samples of 16 model cases
The most unfavorable negative wind pressure coefficients on the roofs of 16 tested
model cases which were sampled 18 times are calculated with these four methods. The total
mean and RMS values of the error ratios for all of the 96 taps for 18 samples are shown for
these four methods in Figure 11 and Figure 12. The mean error ratios for proposed method
are less than 0.5%~6% and the positive errors means conservative. Except the proposed one,
the Sadek-Simiu method is better than other two methods. Its mean error ratios are -4%~-11%
while those for other two methods are -21%~-31%. In Figure 12, one can find that the RMS
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan
error ratios for proposed method are the smallest ones for the four methods, which means that
the proposed method is the most stable one.
Conclusions
Based on the classical extreme-value theory, an estimating method for expected
extreme-value of non-Gaussian wind pressure coefficients is proposed, which need just one
single sample. Comparisons between the proposed method and other methods used widely are
performed and the advantage of the proposed method is shown. The following conclusions
can be obtained:
1) The wind tunnel test results indicate that the wind pressure coefficients on roofs on
low-rise buildings in high turbulence wind field are non-Gaussian processes.
2) Mean error ratios of the extreme-values of non-Gaussian wind pressure coefficients
estimated with the Sadek-Simiu method, the improved peak factor method and the
peak factor method are -4~-11%, -21~-26% and -26~-31%, respectively. The
negative error ratios mean dangerous.
3) The extreme-value of a non-Gaussian wind pressure sample can be estimated
accurately by the distribution of the extreme-values of its sub-sections and the
optimal observing period of the sub-sections can be obtained with the autocorrelation analysis.
4) Compared with the other methods, the proposed method can estimate the extremevalues of non-Gaussian wind pressures more accurately, with mean error ratios no
more than 6%.
Acknowledgements
This research is supported by Technology R&D Program of China (Grant No.
2006BAJ06B05 and 2008BAJ08B14) and the data used in this project comes from the database for low-rise building in Tokyo Polytechnic University, Japan, which was established by
the first author supported by GCOE research project, which are gratefully acknowledged.
References
Davenport A. G, (1964). Note on the distribution of the largest value of a random function with application to
gust loading, Proceedings of the Institution of Civil Engineers, paper No.6739, Vol. 28, 187- 196.
Grigoriu, M, (1995). Applied non-Gaussian processes, Prentice Hall, Englewood Cliffs, N.J.
Gumbel, E. J. (1954). Statistical Theory of Extreme-values and Some Practical Applications, National Bureau of
Standards Applied Mathematics Series 33, Washington.
Holmes, J.D, (1985). Wind action on glass and Brown's integral, Eng. Struct, 4, 226-230
Holmes, J.D, Cochran, L.S, (2003). “Probability distributions of extreme pressure coefficients”, Journal of Wind
Engineering and Industrial Aerodynamics, Vol. 91 893–901
Kareem, A. and Zhao, J. (1994). Analysis of Non-Gaussian Surge Response of Tension Leg Platforms under
Wind Loads, Journal of Offshore Mechanics and Arctic Engineering, ASME, Vol. 116; 137-144.
Kasperski, M, (2000). Specification and codification of design wind loads, Habilitation thesis, Department of
Civil Engineering, Ruhr University Bochum.
Massimiliano Gioffre` and Vittorio Gusella (2002). ‘‘Damage Accumulation in Glass Plates.’’ J. Journal of
Engineer Mechanics, ASCE, 7, 801-805.
Sadek, F., and Simiu, E., (2002). "Peak Non-Gaussian Wind Effects for Database-Assisted Low-Rise Building
Design," Journal of Engineering Mechanics, ASCE, Vol. 128(5), pp. 530-539.
Zhongfu Ge, (2004). Analysis of surface pressure and velocity fluctuations in the flow over surface-mounted
prisms, Ph.D. Dissertation, Virginia Polytechnic Institute and State University, the Department of Engineering
Science and Mechanics.