# Research on Snow Load Characteristics on a Complex Long-Span Root Based on Snow-Wind Tunnel Tests
## 4 Analysis of Snow Distribution Characteristics
### Combination of **E**Of and **Pc**  
Since the original field matrix **X** can be decomposed into a spatial matrix **EOF** and a wind direction matrix **PC**, conversely it is possible to generate the snow distribution on this complex membrane roof through the combination of the spatial matrix **EOF** and wind direction matrix **PC**. As the first three spatial modes made the greatest contribution to the snow distribution on this structure and reflected the influences of the local and overall roof shapes, respectively, the three modes were selected for combination. The combination method is shown in Equation (16). Each column of the generated matrix \(\textbf{X}^{{}^{\prime}}\) represents the snow distribution under a specific wind direction condition.  
\[\textbf{X}_{mcn}=\textbf{EOF}_{m\times 3}\times\textbf{PC}_{3\times n} \tag{16}\]  
The generated snowdrifts in different inflow directions are shown in Figure 15. Through comparison with the experimental results as shown in Figure 9, the snowdrift patterns, i.e., the overall fan-shaped deposition occurring on the windward and leeward surfaces along the air stream and the local packing formed in the concave regions along the radial cables and at the center of each membrane piece, show good correspondence with the experimental results. However, it should be noted that a significant underestimation of the shape coefficient values, especially the peak values, is observed. This underestimation was mainly caused by the fact that only the first three modes participated in the combination, and the contribution of the latter modes to the snow distribution was not considered  
Figure 14: Wind direction coefficients for \(EOF_{1}\), \(EOF_{2}\), and \(EOF_{3}\).  
yet. In general, it is preliminarily verified that the snowdrift on a complex roof can be restored by the combination of snow distributions on several corresponding simple roofs.  
In comparison with the conventional methods, the existing national load codes only provide the snow load requirements for simple roofs, as mentioned in Section 1. The applications of the CFD and experimental methods in the prediction of snow loads on complex roofs are also limited, due to the characteristics of time and money consumption. However, this study provides a simplified idea of estimating the snow load on a complex roof. Specifically, in the structural design stage, it is possible for the snow load on a complex roof to be estimated preliminarily by the combination of the  
Figure 15: Generated snow distribution shape coefficient \(\mu\): (**a**) 0\({}^{\circ}\); (**b**) 22.5\({}^{\circ}\); (**c**) 45\({}^{\circ}\); (**d**) 67.5\({}^{\circ}\); and (**e**) 90\({}^{\circ}\).  
snow distributions on several simple roofs, which have been provided in the load code (e.g., dome, curved roof, pitch roof, and obstacle, among others) or can be obtained by numerical studies. This will save a lot of time and money. Unfortunately, as an initial exploration, there are still many problems to be solved. Firstly, the correct selection of simple roofs plays a decisive role in determining the accuracy of combinational results, therefore, the selection method of the corresponding simple roofs should be studied emphatically; secondly, the **PC** coefficients are closely related to roof forms, and a large number of in-depth and large-scale studies should be carried out to clarify the **PC** coefficient; finally, reliable measurement data of snowdrifts on actual complex roofs is indispensable for examining prediction results. However, there is no measurement data available now, due to a large number of works and high risks in the process of measurement. These problems should be considered in future investigations.