# Calculation of Safety Factors of the Eurocodes
## 2 Materials and Methods  
The assumptions, terms, and symbols of the Eurocodes [1] apply. The target one-year reliability index is \(\beta_{0}=4.7\), meaning that the 50-year reliability index is \(\beta_{50}\approx 3.83\) and the 50-year failure probability is \(\mathrm{P}_{E0}\approx 1/15,400\).  
In the calculation algorithm explained here one value must be selected to associate all distributions, i.e., the permanent load, the variable load, and the material property, with the same value. In the Eurocodes, the characteristic values of the distributions, the mean of the permanent load, the 0.98 fractile of the one-year variable load, and the 0.05 fractile of the material property are the same. This value is here set to unity. Such selection is possible as all distributions can be multiplied by an arbitrary number and the calculation result remains unchanged. This value is called here a design point. This selection means that all materials have the same value i.e., unity, at the 0.05 fractile. Other material parameters, like means, are different as explained later. In this calculation, the design point is unity in the ULS, too, which means that the load distributions must be divided by the load factors and the material distribution must be multiplied by the material factors.  
The permanent load distribution, cumulative distribution \(\mathrm{G}(\mathrm{x},\mathrm{\mu}_{\mathrm{G}},\mathrm{\sigma}_{\mathrm{G}})\), and density distribution \(\mathrm{g}(\mathrm{x},\mathrm{\mu}_{\mathrm{G}},\mathrm{\sigma}_{\mathrm{G}})\) are assumed to be normal, \(\mathrm{\mu}_{\mathrm{G}}=1\), \(\mathrm{\sigma}_{\mathrm{G}}=0.1\), and \(\mathrm{V}_{\mathrm{G}}=0.1\)[1, 2, 3, 4, 5]. The permanent load safety factor is \(\mathrm{\gamma}_{G}=1.35\).  
The variable load distribution, cumulative distribution \(\mathrm{G}(\mathrm{x},\mathrm{\mu}_{\mathrm{Q}},\mathrm{\sigma}_{\mathrm{Q}})\), and density distribution \(\mathrm{g}(\mathrm{x},\mathrm{\mu}_{\mathrm{G}},\mathrm{\sigma}_{\mathrm{G}})\) are assumed to be Gumbel distributions. The characteristic load is the 50-year return load, i.e., the one-year 0.98-fractile is set to the design point, \(\mathrm{\mu}_{\mathrm{Q}}=0.4909\), \(\mathrm{\sigma}_{\mathrm{Q}}=0.1964\); \(\mathrm{V}_{\mathrm{Q}}=0.4\). This distribution applies to the one-year loads. In the current Eurocodes, the variable load distribution corresponds to 5-year loads and distribution is  
\[\mathrm{Q}=(\mathrm{x};\frac{\mathrm{\mu}_{\mathrm{Q}}}{\mathrm{\gamma}_{ \mathrm{G}}},\frac{\mathrm{\sigma}_{\mathrm{Q}}}{\mathrm{\gamma}_{\mathrm{G}}}) ^{5}\]  
which is due to the reliability reduction by the sensitivity factor \(\mathrm{\alpha}_{\mathrm{E}}=0.7\). The variable load safety factor is \(\mathrm{\gamma}_{\mathrm{Q}}=1.5\).  
The distribution of the material property is assumed to be log-normal. The 0.05 fractile is set at the design point. The safety factors are calculated for three materials with coefficients of variation: \(\mathrm{V}_{\mathrm{M}}=0.1\), 0.2, 0.3, assumed to apply to steel, timber, and concrete. The parameters are given in Table 1.  
The load ratio \(\alpha\) and the variable load proportion in the total load is  
\[\alpha=\frac{\mathrm{\mu}_{\mathrm{Q}}}{(\mathrm{\mu}_{\mathrm{G}}+\mathrm{ \mu}_{\mathrm{Q}})} \tag{1}\]  
\begin{table}
\begin{tabular}{c c c} \hline \(\mathrm{V}_{\mathrm{M}}\) & \(\mathrm{\mu}_{\mathrm{M}}\) & \(\mathrm{\sigma}_{\mathrm{M}}\) \\ \hline
0.1 & 1.1841 & 0.1184 \\
0.2 & 1.4125 & 0.2825 \\
0.3 & 1.6921 & 0.5076 \\ \hline \end{tabular}
\end{table}
Table 1: Parameters of the material properties.