## 2 Stochastic Modeling of Roof Snow Load
### Shape Coefficients  
As pointed out before, the shape coefficient, allowing for the conversion of ground snow load into roof snow load, is a function of the roof's geometry, the exposure of the site (i.e., the location of the roof relative to its surroundings [23]), and the wind conditions (e.g., the wind velocity at the building's site during the winter season). Three different conditions are commonly defined: sheltered, semi-sheltered (normal condition), and windswept (exposed condition), depending on how trees, taller buildings, and other obstructions in the surroundings shelter the constructed roof. Although in recent years ground snow load data have been collected and extensively studied all over the Europe [10; 11], information about the shape coefficients is still inadequate [9]. Aiming to derive an appropriate probability density function for roof snow loads, a broad measurement campaign was performed in the United States [9; 23], allowing for a statistical calibration of roof conversion factors for structural design. Assuming a log-normal distribution for both ground and roof snow loads, design values of the shape coefficients (\(\mu_{i,d}\)) can be determined as  
\[\mu_{i,d}=\frac{s}{s_{k}}=\frac{\exp\Bigl{[}\mu_{s}+\sigma_{s}\Phi^{-1}(1-p) \Bigr{]}}{\exp\Bigl{[}\mu_{g}+\sigma_{g}\Phi^{-1}(1-p)\Bigr{]}}=\exp\Bigl{[} \mu_{s}-\mu_{g}+\bigl{(}\sigma_{s}-\sigma_{g}\bigr{)}\Phi^{-1}(1-p)\Bigr{]}. \tag{3}\]  
where \(\mu_{s}\) and \(\sigma_{s}\) are the parameters of the distribution of roof snow load, \(\mu_{g}\) and \(\sigma_{g}\) are the parameters of the distribution of ground snow load, and \(p\) is the annual probability of exceedance (\(p=0.02\) in EN1990 [7]). Snow data on roofs in Switzerland, Italy, the United Kingdom, and Germany were also collected over the course of the European Snow Load Research Project [15], but the dataset covers a limited number of years of records and roof geometries. A significant database of nearly 1300 measurements of snow loads on roofs was collected by Heibo [24; 25] for more than 200 agricultural buildings in Norway, covering the period 1966-1986. This large dataset of roof snow measurements, described in [16], has been used to define a new model for gable roofs, and has been included in the draft proposal of the new Eurocode on snow loads on structures (prEN1991-1-3:2020). The mean values (\(\overline{X}\)) and the standard deviations (\(\sigma\)) of shape factors (\(\mu_{1}\)) for flat roofs in different exposure conditions, as derived from available measurements, are summarized in Table 2 for Europe, the United States, and Norway. Generally, mean values are higher for Europe than for the United States and Norway, while coefficients of variations for Europe and Norway are smaller than for the United States. According to the assumption made in [9] for data from the United States, in the following a log-normal distribution is adopted for the conversion factors in Europe. However, since the ground snow load for Europe is generally not described by a log-normal distribution, Equation (3) cannot be directly applied to derive design shape coefficients, and a specific procedure to combine the _pdf_s of ground snow loads and conversion factors is proposed in view of the reliability assessment of the reliability of roof structures and the calibration of partial factors for snow loads. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Site**} & \multicolumn{2}{c}{**Europe [15]**} & \multicolumn{2}{c}{**United States [9]**} & \multicolumn{2}{c}{**Norway [16; 24; 25]**} \\ \cline{2-7}  & \(\overline{X}\) & \(\sigma\) & \(\overline{X}\) & \(\sigma\) & \(\overline{X}\) & \(\sigma\) \\ \hline Sheltered & 0.90 & 0.08 & 0.83 & 0.36 & 0.75 & 0.12 \\ Normal & 0.74 & 0.29 & 0.63 & 0.27 & 0.66 & 0.11 \\ Exposed & 0.58 & 0.19 & 0.60 & 0.26 & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Shape factors (\(\mu_{1}\)) for flat roofs in sheltered, normal, and exposed conditions.