This paper summarizes a comprehensive study for non-Gaussian properties of wind pressure. The field measurements are implemented on the structure surface of a rigid structure, while a large-span membrane structure is selected as the flexible structure wind pressure contrast group. The non-Gaussian characteristics of measured pressure data were analyzed and discussed through probability density distribution, characteristic statistical parameters, power spectral density function, and the correlations, respectively. In general, the non-Gaussian characteristics of wind pressure present depending on tap location, wind direction, and structure geometry. In this study, the fluctuating wind pressures on the windward side and leeward side of the structures show obvious different degrees of non-Gaussian properties; that is, the surface of rigid structure shows strong non-Gaussian property in leeward, while the roof of flexible structure shows obvious non-Gaussian property in windward. Finally, this paper utilizes the present autospectrum empirical formula to fit the wind pressure power spectrum density of the measured data and obtains the conclusion that the existing empirical formula is not ideal for fitting of the flexible structure.
Wind loads possess stochastic characteristics, especially the fluctuating component, which may force the engineering structure to flutter, torsional vibration, and other forms of wind-induced random coupling vibration. In the 1970s and 1980s, research on characteristics of wind load on structural has begun; Dalgliesh [
Previous studies on the non-Gaussian characteristics of wind pressures are mainly conducted through wind-tunnel test data, while the field measurement data of non-Gaussian wind pressure is very limited. Li et al. [
Large-span membrane structure, as a new appearance structure, has superior mechanical properties and light transmission. Some researchers [
The power spectrum of pulsating wind pressure is based on a limited data collection to describe the power (on the frequency) distribution of fluctuating wind pressure. Many scholars [
This paper summarizes a comprehensive study for non-Gaussian properties, in which wind pressure time series is measured at different locations on two types of structure: the structure surface of a rigid structure and the roof of a flexible structure. Probability density distribution, characteristic statistical parameters, and the correlations of the measured pressure data were analyzed and discussed. Finally, this paper utilizes the present autospectrum empirical formula to fit the wind pressure power spectrum density of the measured data.
According to the theory of random process, the first four-order statistical parameters, the mean, variance, skewness, and kurtosis, are the mathematical characteristics of random variables. For the random Gaussian process, the probability density function can be determined by only using the first two-order statistics parameters (i.e., the mean and variance), while it is very difficult for the non-Gaussian stochastic process. It needs high-order statistics parameters such as the third-order, fourth-order, or higher-order ones [
The mean
Treated as a random process, the discrete random fluctuating wind pressure
Yueqing City Sports Center is located in Xu Yang Road, Yueqing City, which is composed of the stadium, swimming pool, and gymnasium. The stadium building is about 229 m from north to south, 211 m from east to west, and about 42 m from the top of the column. The roof is covered with meniscus nonenclosed space cable-truss system. The maximum cantilever span is about 57 m. The wave structure of the membrane structure is supported by the 273 × 10 steel pipe arch of the cable-truss system. Under the structure of the two waves, the rope structure is arranged under the cable. The effect of the stadium is shown in Figure
Effect drawing of Yueqing Sports Center.
Zhi-hong et al. [
Surface wind pressure detection system.
Equipment name | Model/specification | Quantity | Unit |
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Lightning-proof load cell CYG1721 (top surface) and mine-type differential pressure wind load sensor CYG1722 (lower surface) | CYG1721/1722-±3 KPa/accuracy class: 0.5% FS; output: 4–20 mA; power supply: 24 VDC; installation interface: Φ54 × 16 mm | 124 | Couple |
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256-channel sampler | SQCSCP-256: the system main instrument box, the linear power source, the 256-channel signal lightning protection module (must have the mine proof authentication certificate), the signal sampling conversion board, the A/D board, the connector, the industry control computer, and the observation and control software and so on; composes 256 test systems | 1 | / |
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Shielding gas-shielded waterproof cable | Shielded air-guided waterproof cable, 2-core conductor core color red, white; single-core conductor resistance is less than 94.6 ohms | 50000 | Meter |
Flexible structure wind pressure measuring points layout.
Wind pressure equipment on-site layout is shown in Figure
Layout of field wind pressure instrument. (a) Arrangement of side span of wind pressure instrument. (b) Arrangement of middle span of wind pressure instrument. (c) Wind pressure instrument. (d) Sampling channels and cables.
3D scanning of the middle three pieces of film.
Planar layout of the flexible structure.
The measured wind pressure data includes 21 points’ upper surface wind pressure
The measured value of the wind pressure on the 21 measuring points.
According to the records of the wind speed and wind direction measuring instruments, the horizontal wind speed and wind direction are obtained in Figures
Time series of horizontal wind speed and wind direction. (a) Time series of horizontal wind speed. (b) Time series of wind direction.
Select
In general,
Mean wind speed and mean wind direction. (a) Mean wind speed (3 min). (b) Mean wind direction (3 min).
In order to analyze the non-Gaussian characteristics of wind pressure acting on the rigid structural surface, a rectangular masonry structure (length: 5.5 m; width: 3.6 m; height: 1.85 m) on the roof of an office building located in East China Jiaotong University was chosen as the test object, and the field measurement of wind pressure was implemented, respectively, on 23 November 2012 and 1 March 2013.
The first scheme of rigid structure (R1) as shown in Figure
The first scheme of rigid structure field measurement. (a) Site layout. (b) Plane layout of the rigid structure.
R1 was arranged to test the wind pressure on the corner of the masonry. In this scheme, the pressure sensors from #1 to #5 were fixed on the metope AB of the rectangular structure as shown in Figure
R2 was designed to measure the wind pressure around the structure. In the same horizontal plane, as indicated in Figure
The second scheme of rigid structure field measurement. (a) Site layout. (b) Plane layout of the rigid structure.
A kind of pressure sensor, CYG1513T, has been developed to measure wind pressures by the way of attaching to the building surface, and the field measurement adopting CYG1513T to test the wind pressure on a building structure has been carried out in the literature [
In the process of R1, it is northeaster on that day. Therefore, DA metope was located in the windward side; AB metope belonged to the nonwindward side. On the day of the R2, it is norther. So, AB and DA walls were situated in the windward side, while BC and CD walls were seated in the nonwindward side. The measured wind pressures of two tests are shown in Figures
Time history of the measured wind pressures (R1).
Time history of the measured wind pressures (R2).
However, in order to further research the features of the non-Gaussian wind pressures for rigid structure and flexible structure, it is necessary to analyze the probability distribution function (PDF), high-order statistical characteristic, correlation, and power spectral density (PSD).
The probability density function (PDF) describes the numerical distribution characteristics of random variables, and it is an important basis for studying the non-Gaussian properties of wind pressure. The PDF is calculated for the wind pressure values of field measurement programs and compared with the Gaussian-type distribution, respectively. The MATLAB code can be used to calculate PDF for the data of each measuring pressure point (e.g., point 10): [mu, sigma] = normfit(point 10) [f,xi]=ksdensity(point 10);plot(xi,f,'.'); hold on; y=-50:0.1:50;dd=normpdf(y,mu,sigma); plot(y,dd,'r-').
As shown in Figures
Corresponding measuring points of the windward side and leeward side in field measurement programs.
Field measurement program | Windward side | Leeward side |
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Flexible structure | BC side: #1, #13, #8, #17, #21 | AD side: #10, #5, #14, #18 |
The first scheme of rigid structure (R1) | AD side: #6, #7, #8, #9, #10 | AB side: #1, #2, #3, #4, #5 |
The second scheme of rigid structure (R2) | AB side: #1, #2, #3 |
BC side: #4, #5, #6 |
The PDF of the measured wind pressures (the flexible structure). (a) AD side measuring points. (b) Selected measuring points inside the membrane. (c) BC side measuring points.
The PDF of the measured wind pressures (R1). (a) AB side measuring points. (b) AD side measuring points.
The PDF of the measured wind pressures (R2). (a) AB side measuring points. (b) AD side measuring points. (c) BC side measuring points. (d) CD side measuring points.
For the non-Gaussian wind pressures, it is significant to master the high-order statistic characteristics, such as skewness and kurtosis. In order to analyze the high-order statistic characteristics, the measured wind pressures of rigid structure are divided into many segments in ten-minute interval, while the measured wind pressures of flexible structure are divided into many segments in three-minute interval on account of the short sampling time. Based on these segments of the measured wind pressures, the relationships between skewness and kurtosis are displayed in Figures
The relationship between skewness and kurtosis of the fluctuating wind pressures (the flexible structure). (a) AD side; (b) CD side; (c) AB side; and (d) BC side.
The relationship between skewness and kurtosis of the fluctuating wind pressures (R1).
The relationship between skewness and kurtosis of the fluctuating wind pressures (R2).
By observing Figure
A correlation function is a function that gives the statistical correlation between random variables; it can be classified into autocorrelation function and cross-correlation functions. The correlation of the non-Gaussian wind pressures measured in R1 is displayed in Figure
The correlation function of the measured non-Gaussian wind pressures (R1).
In R2, between these horizontal measuring points, the correlation of non-Gaussian fluctuating wind pressure also conforms to the rule that the correlation decreases with the increase of spacing as shown in Figure
The correlation function of the measured non-Gaussian wind pressures (R2).
The analysis of Figure
The correlation function of the measured non-Gaussian wind pressures (the flexible structure). (a) AD side (leeward side); (b) AB side; and (c) BC side (windward side).
In order to further study the special relationship between the wind pressures of the flexible structure, the correlation analysis of the measured wind pressure data of the upper and lower surface of the membrane structure is carried out.
For large-span membrane structures, the direction of the wind pressure coincides with the direction of the upper surface wind pressure. The wind pressure coefficient
The wind pressure root mean square coefficient
According to the nature of the multidimensional random variable [
In order to facilitate the calculation, (
If the upper and lower surface wind pressure signals are not related to each other, that is,
Comparing (
When
Figure
The correlation of the wind pressures on upper and lower surface of the flexible structure. (a) The correlation coefficient of measurement point #1. (b) The correlation coefficient histogram of 21 measurement points.
By observing Figure
The measuring points with negative correlation coefficients.
Power spectral density (PSD) estimation methods mainly include classical spectral estimation and modern power spectral estimation, namely, the parameter spectral estimation method. The process is estimating parameter model through the observation of data, and then the power spectrum of fluctuating wind pressure can be estimated by the output power of parameter model method. This method is proposed for improving the bad resolution and variance performance in the classical spectral estimation. This paper adopts the minimizing prediction error autoregression (AR) spectral estimation in modern power spectral estimation, and an analysis of power spectrum is made in measuring non-Gaussian pulsating wind pressure for both the rigid and flexible structures with the fitted wind pressure autospectral formula in (
As shown in Figures
The related parameters of measured fluctuating non-Gaussian wind pressures (R1).
Leeward side | Windward side | ||||||||||
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Measuring points |
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Measuring points |
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Wall |
#1 |
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12.39 | 254.3 | 1.022 | Wall |
#6 |
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31.02 | 414.6 | 1.044 |
#2 | 16.65 | 304.9 | 1.033 | #7 | 19.94 | 270.7 | 1.065 | ||||
#3 | 26.42 | 612.4 | 1.015 | #8 | 84.26 | 1467 | 1.035 | ||||
#4 | 50.48 | 1469 | 1.002 | #9 | 61.76 | 1356 | 1.024 | ||||
#5 | 17.59 | 528.7 | 1.028 | #10 | 126.7 | 3672 | 1.007 |
The related parameters of measured fluctuating non-Gaussian wind pressures (R2).
Leeward side | Windward side | ||||||||||
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Measuring points |
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Measuring points |
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Wall |
#4 |
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845.8 | 99800 | 1.025 | Wall |
#1 |
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430.4 | 21790 | 1.019 |
#5 | 1616 | 236300 | 1.031 | #2 | 698 | 38390 | 1.034 | ||||
#6 | 1352 | 196000 | 1.038 | #3 | 985 | 42850 | 1.014 | ||||
Wall |
#7 | 500.1 | 64950 | 1.034 | Wall |
#10 | 251.5 | 11630 | 1.03 | ||
#8 | 389.4 | 46910 | 1.032 | #11 | 280.4 | 13430 | 1.022 | ||||
#9 | 41.43 | 3292 | 1.044 | #12 | 92.32 | 4021 | 1.016 |
The related parameters of measured fluctuating non-Gaussian wind pressures (the flexible structure).
Windward side | Leeward side | ||||||||||
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Measuring points |
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Measuring points |
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Wall |
#1 | 9.23 | 1453 | 2634 | 0.932 | Wall |
#10 | 9.23 | 5087 | 4581 | 0.9096 |
#13 | 7158 | 4117 | 0.9595 | #5 | 15670 | 6670 | 0.9436 | ||||
#8 | 3636 | 3773 | 0.9264 | #14 | 41670 | 9192 | 0.9458 | ||||
#17 | 1978 | 2698 | 0.956 | #18 | 0.2976 | 0.2118 | 0.8034 | ||||
#21 | 1789 | 2615 | 0.9554 |
Power spectrum of measured fluctuating non-Gaussian wind pressures (R1).
Power spectrum of measured fluctuating non-Gaussian wind pressures (R2).
Power spectrum of measured fluctuating non-Gaussian wind pressures fitted with (
However, the result of Pan’s fitting autospectral formula for the flexible structure is even worse as shown in Figure
Power spectrum of measured fluctuating non-Gaussian wind pressures fitted with (
In this paper, the non-Gaussian characteristics of measured pressure data were analyzed and discussed by comparing the measuring wind pressures on rigid structure and flexible structure, and the main conclusions are summarized as follows.
The authors declare that they have no conflicts of interest.
The authors would like to acknowledge the financial contributions received from the National Natural Science Foundation of China (Grant no. 51378304) and the Science Foundation of Jiangxi Province (Grant no. 2017BAB206051). Also, the authors wish to thank Professor Zhi-hong Zhang for help in providing some figures presented in this paper and the precious wind pressure data for the flexible structure.