AN EXTREME-VALUE ESTIMATING METHOD OF NON- GAUSSIAN WIND PRESSURE

Transcription

1 The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan AN EXTREME-VALUE ESTIMATING METHOD OF NON- GAUSSIAN WIND PRESSURE Yong Quan 1 Ming Gu 2 Yukio Tamura 3 Bin Chen 4 1 Associate Professor, State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai , China, quanyong@tongji.edu.cn 2 Professor, State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai , China, minggu@tongji.edu.cn 3 Professor, Department of Building Engineering, Tokyo Polytechnic University, Atsugi, ,Japan, Yukio@arch.t-kougei.ac.jp 4 Engineer, Hangzhou Municipal Installation Supervision and Administration Center; Hangzhou ; China, jeetchen_123@hotmail.com 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 non- Gaussian 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 non- Gaussian 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

2 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( 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 Test(1/100 Category III(AIJ2004 Mean wind speed Turbulence intensity Height Z(m Figure1 Arrangement of the wind field U(z or I(z Figure.2 Simulated wind field of suburban terrain

3 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 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, 2 Cp(, i t = p(, i t/(0.5 ρvh (1 where Cp (, i t and p(, it are the time series of pressure coefficients and wind pressures, 2 respectively, and /(0.5 ρv H 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 21.8 o, 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

4 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. Wind pressure coefficient (C p Time series of C p on Tap Time(sec Probability distribution E-3 Gaussian process C p on Tap 1(E=-1.2;s=0.6;C s =-1.7;C k =8.1 1E Wind pressure coefficient (C p Wind pressure coefficient (C p Figure 5 Time series and Probability distribution of Cp on Tap 1 Time series of C p on Tap Time(sec Probability distribution E-3 1E-4 Gaussian process Cp on Tap 34(E=-0.2;s=0.2;C s =-0.4;C k = Wind pressure coefficient (C p 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.

5 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 = ax ( e 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( x e (3 x e = u + y /α (4 The expected value and standard deviation of the extreme-values is, x e = u + γ /α (5 π 1 σ xe = (6 6 α where γ = 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 (7 α = 0.78/σ xe (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 t 1 into n ( n= t1 t2 non-overlapping sub-sections with a length of t 2.The probability of the extreme-value smaller than or equal to x is expressed as F(T= t 1 in the parent section and F(T= t 2 in the sub-section, then the following relation can be obtained assuming the observed extreme-values are independent, FT ( = t1 = F n ( T= t2 (9 Based on Eqs.(1,2,9, the following expressions can be obtained: 1/ at ( = t1 = 1/ at ( = 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 t 2, 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?

6 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. MeanSE Observing period of sub-section(sec Figure 7 Mean standard errors of the calculated values for varied observe periods average of auto-correlation coefficient 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, t 1, on each tap and find the shortest time delay, t 2, for auto-correlation coefficient decaying from 1 to the level of 0; 2 Divide the parent sample into n(n=t 1 /t 2 sub-sections with a observing period of t 2 ; 3 Observe the maximum (or minimum peak values of each sub-section, x e _ sub, then their expected value and standard, x e _ sub and σ xe _ sub, can be calculated. 4 Calculate the parameters of u sub and x e _ sub and σ xe _ sub with Eqs.(7 and (8; α sub for the extreme distribution type I with 5 Transform the parameters for observing period of t 2, usub 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 t 1, 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

7 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 C p _ est p _ std γ err = (12 C p _ std where C p _, std C p _ and est γ are standard extreme values and estimated extreme values of the err 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. γ The mean 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 γ method, respectively. The RMS values of err are 8%, 21%, 23% and 35% for the four methods, respectively. Estimated extreme values "Standard" extreme values Proposed method Sadek-Simiu method Improved peak factor method Peak factor method g err Proposed method Sadek-Simiu method Improved peak factor method Peak factor method "Standard" extreme values Figure 9 Comparison of estimated extreme values with "standard" ones Mean values of g err Proposed method Sadek-Simiu method Improved peak factor method Peak factor method No. of model cases Figure 11 Mean values of γ err for 96 taps for 18 samples of 16 model cases RMS values of g err No. of taps Figure 10 Error ratios of the methods Proposed method Sadek-Simiu method Improved peak factor method Peak factor method 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

8 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, 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, Holmes, J.D, Cochran, L.S, (2003. Probability distributions of extreme pressure coefficients, Journal of Wind Engineering and Industrial Aerodynamics, Vol 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; 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, 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 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.