# Snow Load Shape Coefficients and Snow Prevention Method for Stepped Flat Roofs
## 4 Results and Discussion
### Snow Load Shape Coefficients  
The snow depth \(d\) on the roof is normalized by the variation in snow depth \(\Delta d\) on the ground during the measurement. The dimensionless values obtained are regarded as snow load shape coefficients (\(\mu\)), as shown in Figure 12.  
Figure 11: The snow distribution on the central axis of the roofs: (**a**) lower roof on the windward side; (**b**) upper roof on the windward side. Figure 10: The snow deposition on the roofs. The windward snow load shape coefficients on the lower roof gradually decreased with the increase in \(x/\mathrm{H}\). The snow load on the roof at \(x/\mathrm{H}=1.6\) was the same as the ground snow load. The maximum snow load shape coefficient occurred at \(x/\mathrm{H}=0\), \(\mu_{\mathrm{max}}=3.44\). As indicated in Figure 12a, the snow load shape coefficients demonstrated a strong regularity, which was fitted with a quadratic polynomial of \(x/\mathrm{H}\) to obtain Equation (18) (\(\mathrm{R}^{2}=0.9991\)):  
\[\mu=0.3346(x/\mathrm{H})^{2}-2.0547(x/\mathrm{H})+3.4426. \tag{18}\]  
The snow load shape coefficient of the lower roof when the upper roof was on the windward side (Figure 12b) was about 1 at \(x/\mathrm{H}=0.2\)-2.4. The snow load shape coefficient gradually decreased to 0.25 at \(x/\mathrm{H}\geq 2.4\). However, this area was relatively small compared with the overall low roof; so, the snow load shape coefficient was still unified as 1. The maximum value of the shape coefficient on the lower roof occurred at the junction of the roofs (\(x/\mathrm{H}=0\)), \(\mu_{\mathrm{max}}=1.25\). The snow load shape coefficient on the upper roof was smaller than that on the lower roof, and the maximum value was about 0.5. The snow load on stepped flat roofs is specified in detail in many countries. However, the definitions of the snow load shape coefficient are different. Comparing the codes of different countries, the formula of the snow load values of roofs can be expressed in Equation (19) (the ASCE can also be derived to obtain this form), in which \(S\) is the snow load on the roof, \(\mu\) is the snow load shape coefficient, and \(S_{0}\) is the ground snow load. \[S=\mu S_{0} \tag{19}\]  
In this paper, the different expressions of snow load in different national load codes are unified into Equation (19). As listed in Table 3, the set of dimensionless coefficients that are multiplied by the \(S_{0}\) in each code is defined as the snow load shape coefficient. For easier comparison, the parameters with the same meanings in the formulas of different countries have been standardized. The greater values of snow load shape coefficients at the same dimensionless distance are selected in two measurements, which are used to obtain the expressions for the snow load shape coefficients in this work as Equation (20):  
\[\mu=\left\{ \begin{matrix}0.3346(x/\mathrm{H})^{2}-2.0547(x/\mathrm{H})+3.4426& 0\leq x/\mathrm{H}\leq 1.6\\ 1&x/\mathrm{H}>1.6\end{matrix} \right.. \tag{20}\]  
The fitting equation is compared with the snow load shape coefficients in different codes in Figure 13. Each code has fewer definitions of snow distribution on the upper roof, and the measured snow depth of the lower roof is much greater than that of the upper roof. Moreover, the snow load shape coefficients on the upper roof are far less than 1. Therefore, only the snow load shape coefficients for the lower roof are compared. For stepped flat roofs, different national load codes hold that the maximum snow load shape coefficient emerges at the roof junction (\(x=0\)), which is consistent with the measured results in this paper. However, the codes of different countries assume that the nonuniform distribution of snow load is a linear regularity. The measurement results justify that the law of quadratic polynomials is more suitable for expressing the snow distribution  
Figure 12: Snow load shape coefficients for the stepped flat roofs: (a) lower roof on the windward side; (b) upper roof on the windward side. coefficients.