## 4 Non-Gaussian Property of Wind Pressures
### Skewness and Kurtosis of Wind Pressure  
Skewness \(m_{3}\) as well as kurtosis \(m_{4}\) of wind pressure are calculated to identify whether the process is Gaussian. The definition of \(m_{3}\) and \(m_{4}\) for wind pressure coefficient \(C_{pi}\) is expressed as:  
\[m_{3}=\frac{1}{N}\sum_{m=1}^{N}\left[\frac{C_{pi}(t_{m})-C_{pi,\text{mean}}}{C _{pi,\text{rms}}}\right]^{3},\text{ and} \tag{27}\]  
\[m_{4}=\frac{1}{N}\sum_{m=1}^{N}\left[\frac{C_{pi}(t_{m})-C_{pi,\text{mean}}}{C _{pi,\text{rms}}}\right]^{4} \tag{28}\]  
The statistical and spatial distributions for wind pressure on the entire roof are examined. Statistical analysis is operated on wind pressure coefficient histories measured from 110 taps, including 24 wind azimuths. The histogram of skewness \(m_{3}\) is presented in Figure 8a, in which the purple curve is the CDF of \(m_{3}\). Most of the values for \(m_{3}\) are distributed in the range \([-1,0]\), and 80% of them are smaller than \(-0.1\). This indicates that most of the wind pressures are left-skewed regardless of azimuths. Moreover, roughly 18% of \(m_{3}\) are smaller than \(-0.6\), showing strong non-Gaussian properties.  
The minimal value of \(m_{3}\) for each tap among all tested wind azimuths is calculated. The spatial distribution of \(m_{3,\text{ min}}\) is presented in Figure 8b. As can be seen, all \(m_{3,\text{ min}}\) values are smaller than \(-0.3\). The values of \(m_{3,\text{ min}}\) for edge \(ab\) and edge \(cd\) of the BCR, as well as edge _ef_ of the SCR, are less than \(-1.2\). Moreover, values of \(m_{3,\text{ min}}\) for corners \(a\) and \(d\) are even smaller than \(-2.0\). The wind pressures for such regions are highly left-skewed. This is because flow separation and vortex shedding at curved edges and corners are distinct, resulting in evident non-Gaussian wind flow for those regions. Thus, wind pressures here exhibit a more predominant wind suction than that of a Gaussian distribution.  
Figure 9a presents the histogram of excess kurtosis (\(m_{4}-3\)), in which the purple line is the exceedance probability of excess kurtosis, i.e., 1-CDF. Over 90% of (\(m_{4}-3\)) values are larger than 0. Roughly 17% of (\(m_{4}-3\)) values are larger than 1.0.  
Figure 8: Statistical analysis for skewness \(m_{3}\) of wind pressure coefficient. (**a**) Histogram of skewness \(m_{3}\), (**b**) spatial distribution contour of \(m_{3,\text{ min}}\).  
Furthermore, the maximal kurtosis \(m_{4,\,\max}\) for every tap among all azimuths is calculated, whose distribution on the entire roof is shown in Figure 9. All \(m_{4,\,\max}\) values are larger than 3.5. This indicating the wind pressure history for each tap is the softening process. Moreover, the values of \(m_{4,\,\max}\) around edge \(ab\) and edge \(cd\) of the BCR, as well as edge \(ef\) of the SCR, are more than 8.0, while corners \(a\) and \(d\) are even more than 13.0. All this suggests that the wind pressures on the long-span roof have more outliers than the Gaussian distribution and are leptokurtic. Therefore, extreme wind pressures are deemed to be considerable. Still, using the Gaussian assumption will underestimate extreme suction in the sensitive regions.