## 4 Design Values of Shape Coefficients
### Flat Roofs  
Design values of shape coefficients of flat roofs (\(\mu_{1,d}\)) are illustrated in Figure 4 as a function of the coefficient of variation of the annual maxima of ground snow load (\(COV_{g}\)). In the diagram, where each dashed curve corresponds to a specific exposure condition (the green curve to the sheltered condition, the blue curve to normal condition, and the red curve to the exposed condition), the analogous, green, blue, and red, horizontal solid lines corresponding to the constant values of the \(\mu_{1,d}\) provided in EN1991-1-3:2003 have also been reported. These have been kept unchanged in the prEN1991-1-3:2020. The considered range of \(COV_{g}\), \(COV_{g}=0.2-2.0\), has been chosen to cover all the values that can be realistically expected in Europe (see Section 2.1). Figure 4 shows the point values derived, in the three exposure conditions, from the data available from 25 Italian weather stations, suitably covering lowland (\(a\leq 100\) m a.s.l. ), medium altitude (\(100<a\leq 600\) m a.s.l. ), and mountain (\(a>600\) m a.s.l.) sites. To facilitate the interpretation of the results, Figure 4 shows the annual probabilities of exceedance correlated with the design value of the snow load on roofs derived using the constant value shape coefficients \(\mu_{1,d}\) provided in the Eurocode, in the case the coefficient of variation of the extreme value distribution of the snow load on the ground equals its lower or upper bound, i.e., \(COV_{g}=0.2\) or \(COV_{g}=2.0\). Eurocode provisions lead to roof snow loads characterized by an annual probability of exceedance varying between \(p=0.082\), if \(COV_{g}=0.2\), and \(p=0.022\), if \(COV_{g}=2.0\), in case of normal exposure conditions. These vary between \(p=0.057\), if \(COV_{g}=0.2\), and \(p=0.019\), if the \(COV_{g}=2.0\), in exposed conditions. It is worth noting that, in sheltered conditions, owing to the fact that the variations of the measured conversion factors are small, the Eurocode leads to roof snow loads characterized by nearly constant annual probabilities of exceedance (\(p\approx 0.015\)). Looking at the results reported in Figure 4, it appears that the Eurocode approach (adopted in EN1991-1-3:2003 as well as in prEN1991-1-3:2020) is slightly unconservative for normal conditions, while it fits the target probability for exposed conditions, and is slightly conservative for sheltered conditions. In any case, the differences are generally limited and comparable with those linked with the usual model uncertainties. Moreover, considering the Norwegian database [16; 24; 25], where snow load on roofs is usually characterized by low values of the coefficient of variation (see Table 2), different outcomes are obtained for flat roofs (see Figure 5a). In fact, in that case, the design shape coefficients determined using the Eurocode approach are conservative both for sheltered (\(p=0.005-0.008\)) and normal exposure (\(p=0.009-0.010\)), while no data are available for exposed conditions. Figure 4: Design shape coefficient for flat roofs (\(\mu_{1,d}\)) as a function of the coefficient of variation (\(COV_{g}\)) of ground snow load, parameterized in terms of exposure conditions (sheltered: green lines, normal: blue lines, exposed: red lines) for 25 Italian weather stations. Considering that in other climatic European regions higher variability can be expected for snow load on roofs, the applicability of the Norwegian dataset to different climates has been also checked, adopting a safe-sided approach using an artificial increase in the coefficient of variation of the shape factor (\(COV_{\mu_{1}}\)), thus assuming \(COV_{\mu_{1}}=0.4\). The obtained results are summarized in Figure 5b. It must be underlined that in that case the Eurocode obviously leads to increased annual probabilities of exceedance, with \(p=0.04-0.013\) for sheltered exposure and \(p=0.052-0.0136\) for normal exposure. Clearly, the design shape coefficient curves depend on the assumed distribution for ground snow load extremes.