By analysis of statistical characteristics and probability density distribution of extreme values of wind pressures on the surfaces of a typical low-rise building model and a typical high-rise building model, characteristics of the commonly used methods for estimating the extreme-values of wind pressure are discussed. The relationship between the parameters of the extreme value distribution of wind pressure and its observation length is then deduced based on the generalized extreme value theory and the independence of the observed extreme values. A new method for estimating the extreme values is developed by dividing the time history sample of the wind pressure into several subsamples. The extreme values of the wind pressure coefficients calculated with the present method and those with the commonly used methods are compared and the results indicate that the present method can estimate the extreme values of non-Gaussian wind pressure more accurately than the commonly used ones.

Wind pressure on building surfaces is a random process, and its probabilistic and statistical characteristics are some of the key points to wind engineering. In the 1960s, Davenport [

Peterka and Cermak [

Estimating the extreme value of non-Gaussian wind pressure is another important problem in wind engineering. Davenport [

To solve this problem, Kareem and Zhao [

Kasperski [

In this work, the surface wind pressure data on a typical low-rise building model and a typical high-rise building model, which were obtained from wind tunnel tests in simulated wind field, were analyzed. The statistical characteristics of the data were examined. The relationship between the parameters of the extreme value distribution of the wind pressure and its observation length was deduced based on the GEV theory. A new method of estimating extreme values was developed by dividing a sample of the wind pressure time history into several subsamples.

A large number of wind tunnel tests on low-rise and high-rise buildings have been performed on this subject. The surface wind pressure data from a wind tunnel test on one of the low-rise building models in a simulated suburban wind field was chosen. The test model, the definition of the wind angle, and the arrangement of the wind pressure taps are shown in Figure

The test model, the definition of the wind angle, and the arrangement of the test taps of the low-rise building model.

Figure

The test model, the definition of the wind angle, and the arrangement of the test taps of the high-rise building model.

Methods for estimating the extreme values of wind pressures are always based on a certain probability or statistics assumption. In this section, the theoretical basis of those commonly used methods for estimating the extreme values of wind pressure is discussed through making an assay of the probability and statistics characteristics of the wind pressures on the surfaces of the low-rise building as well as the high-rise building.

Kumar and Stathopoulos [

Skewness contours of the wind pressure coefficients at the model surface at different wind angles.

0° wind angle

45° wind angle

90° wind angle

Kurtosis contours of the wind pressure coefficients at the low-rise building model surface at different wind angles.

0° wind angle

45° wind angle

90° wind angle

The skewness of the wind pressure coefficients on the windward wall (Figure

The non-Gaussian characteristics of the surface wind pressure are also ubiquitous on the high-rise building model. The skewness and the kurtosis in test cases of 0, 45, and 90° wind angles were calculated and their contour figures are described in Figures

Skewness contours of the wind pressure coefficients at the model surface at different wind angles.

0° wind angle

45° wind angle

90° wind angle

Kurtosis contours of the wind pressure coefficients at the low-rise building model surface at different wind angles.

0° wind angle

45° wind angle

90° wind angle

Since the wind pressure coefficients of the building cannot satisfy the assumption of a Gaussian process, the Davenport method cannot reliably estimate extreme values of the wind pressure.

Kareem and Zhao transformed the Gaussian random variable to a Hermite polynomial of a non-Gaussian random variable with the high-order moments by using the following expression:

Kwon and Kareem improved the calculating method about the coefficients

However, in theory, it is unavoidable to encounter the truncation error while using the first four-order statistics to establish the polynomial to fit the actual probability density distribution of wind pressure. In order to describe the error of this fitting process, the standardization of the non-Gaussian wind pressure on surfaces of the building was done in the first place. And then, an inverse transformation was taken by using (

The comparative results of representative test taps on surfaces of the high-rise building model in test cases of 0 and 45° wind angles were considered (Figure

The comparative results of representative test taps on surfaces of the high-rise building model.

The probability density distribution functions of the wind pressure coefficients of representative test taps were analyzed (Figures

Comparison of results of fitting with the probability distribution for test taps on the surfaces of low-rise building model at 0 and 45° wind angle tests.

Comparison of results of fitting with the probability distribution for test taps on the surfaces of high-rise building model at 0 and 45° wind angle tests.

As seen from the figures, the wind pressure coefficients of typical areas in the test taps seriously deviate from the Gaussian distribution. The lognormal, gamma, and GEV distributions provide a better fit than the Gaussian distribution does. However, the fitting quality of the distribution patterns varies. The patterns generally do not show a good fit with the tail of the probability distribution function for the wind pressure, which is the key to estimating extreme values. The surface wind pressure at various surfaces of the building has poor fit with any pattern of the probability distribution function, and the wind pressure coefficients of the different test taps are not consistent, which are similar to previous observations [

In sum, the probability distribution of the non-Gaussian wind pressure time history on the surfaces of the buildings cannot be fitted by one certain existing mathematical model well. Both the Sadek-Simiu method based on the commonly used probability distribution functions and Kwon-Kareem method relied on the polynomial established by the first four-order statistics still have some problems while they are used to describe the probability distribution of wind pressure on the surfaces of buildings. Moreover, these two estimating methods are based on the level-crossing rate theory. And the raw non-Gaussian time history was used to replace mapping result while calculating the level-crossing rate because the mapping process approach cannot obtain the standard Gaussian process. It might bring great errors, especially for the Kwon-Kareem method. However, if a method based on extreme-value theory was taken to estimate the extreme values of wind pressures, the problems faced by the methods mentioned above could be avoided effectively.

According to the theory raised by Fisher and Tippett [

Comparison of fitting results of large number of samples for test taps on the surfaces of high-rise building model.

The probability distributions of the positive and negative extreme values of a wind pressure time history for different wind angles at several test taps of low-rise building and high-rise building are shown in Figures

Comparison of fitting results of single sample for test taps on the surfaces of low-rise building model.

Comparison of fitting results of single sample for test taps on the surfaces of high-rise building model.

What is more, the wind pressure extreme values on surfaces of buildings still comply with the extreme value types I and III while the extreme value type II seldom appears except those which are likely to comply with the extreme value type II distribution due to the error in the individual values. Statistical analyses show that these errors have a significant effect on the final estimate of the extreme values. Thus, results for fit with extreme value type I distribution were used to replace those for extreme value type II.

From the above analysis, the current methods of estimating wind pressure extreme values still have some inadequacies in their assumption of the wind pressure distribution. Hence, a new method of estimating extreme values is developed in the present study; it is based on the GEV theory, which is highly versatile.

The classical extreme value theory considers that a large number of independently observed extreme values fit one of the three classes of distribution regardless of the probability distribution of the maternal sample [

If

Provided that a test data sample with an appropriate length of time,

After rearranging, the following expression is obtained:

According to the above conversion, a sample with a duration of

In the present estimating method, larger values of

Relationship between the root-mean-square error of the estimated extreme values and the observation period with the mean value of the correlation coefficients of the wind pressure history.

Root-mean-square error

Autocorrelation coefficients

To facilitate the comprehension and application of the present method, the procedure is listed below.

The auto-correlation coefficients of the sample with a standard observation period

The parent sample was divided into

The maximum values of

The parameters for the observation period of

The expected extreme values of

For the calculation of negative values, the sign of the extreme values could be changed to transform the problem into a calculation of positive values; afterward, the problem could be solved by following the aforementioned procedure.

The BLUE, Kwon-Kareem, and Sadek-Simiu methods are presently the most commonly used methods to estimate extreme values. These methods, as well as the present method, were applied to estimate the extreme values of wind pressure in a standard sample. The accuracy of the different methods was determined by comparing the aforementioned methods. However, before the step of comparison of the four methods, the first problem is to determine the standard values. For the case of low-rise building, the mean value of 15 independent observed extreme values of 15 samples was considered as the standard extreme value while the mean value of 2800 independent observed extreme values of the 2800 samples was taken as the standard extreme value for the high-rise building case. According to the instructions above, the MATLAB program is written. During the estimation by the Sadek-Simiu method, the level-crossing rate of the translated standard Gaussian time period

In this part, the estimating results of the four methods will be illustrated. In order to make it more convenient to explain the match condition between the estimating results (

Results obtained for the test cases of 0, 45, and 90° wind angles and the estimated negative wind pressure and positive wind pressure of the low-rise building are presented in Figures

Comparison of estimated positive extreme values on low-rise building.

0° wind angle

45° wind angle

90° wind angle

Comparison of estimated negative extreme values on low-rise building.

0° wind angle

45° wind angle

90° wind angle

Figure

Accuracy of the different methods under various working conditions of lowing (note that P represents the positive extreme values of calculated cases and N represents the negative extreme values of calculated cases).

Root-mean-square values of errors

Mean values of errors

Results obtained for the test cases of 0, 45, and 90° wind angles and the estimated positive wind pressure and negative wind pressure of the high-rise building are presented in Figures

Comparison of estimated positive extreme values on high-rise building.

0° wind angle

45° wind angle

90° wind angle

Comparation of estimated negative extreme values on high-rise building.

0° wind angle

45° wind angle

90° wind angle

In order to show the accuracy of the estimating results quantificationally, Figure

Accuracy of the different methods under various working conditions of high-rise building (note that P represents the positive extreme values of calculated cases and N represents the negative extreme values of calculated cases).

Root-mean-square values of errors

Mean values of errors

The probabilistic and statistical characteristics of the wind pressure coefficients on a building surface and the disadvantages of the commonly used methods for their extreme values were discussed. Based on the GEV theory and the independence of the observed extreme values, the relationship of the extreme value distribution between the short-term sub-sample and the long-term maternal sample was established. A more accurate method of estimating extreme value was proposed. The following conclusions were obtained.

Wind pressures on the surfaces of buildings, especially those on low-rise buildings exposed to high-turbulence air flow, do not follow a Gaussian distribution assumption and even seriously deviate from the Gaussian distribution in some cases.

Whole wind pressures are difficult to fit to a single probability distribution function. Moreover, there is no consistency in the wind pressure behavior on the different locations.

The extreme values of wind pressure coefficients on the high-rise building adhere to the extreme value types I and III. And the fitting results are not satisfactory enough in the rear part even if 2800 independent time series were taken. Although the number of independent extreme values from a single time history obtained from the sub-samples is far away from the requirement proposed by Holmes and Cochran [

The Kwon-Kareem method attempts to reduce the error by introducing the non-Gaussian correction term. However, if the strong non-Gaussian process is encountered, the mapping result obtained by the polynomial established by the first four statistics would show a significant difference on both tails of the probability distribution. And when it was used to estimate the extreme values, the divergence of the results was significantly large.

The conclusion drawn by Sadek and Simiu that the gamma and Gaussian distributions can describe the probability distribution of the wind pressure well is not as good as one thought. For those taps with strong non-Gaussianity, both tails of the probability distribution cannot be fitted satisfactorily by these two patterns, with great errors existing. According to the maximum domain of attraction theory, the gamma and Gaussian distribution both belong to the maximum attraction domain of extreme value type I distribution. And yet, the probability density distribution of the extreme values of wind pressure on the surfaces of buildings does not always comply with the extreme value type I. Consequently, it still leads to a relatively large error, especially when it was used to estimate the extreme values of wind pressures on the surfaces of low-rise building.

The extreme values predicted by the BLUE method have significant errors sometimes because of the limitation of the extreme value type I distribution assumption.

Compared with the commonly used methods, the present method can more accurately predict extreme values of positive and negative wind pressure coefficients.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors gratefully acknowledge the support from the National Natural Science Foundation of China (51278367, 90715040), the State Key Laboratory of Disaster Reduction in Civil Engineering in China (Grant no. SLDRCE10-B-03), and the Fundamental Research Funds for the Central Universities.