## 2 Materials and Methods
### The Basic Equations  
The author has revealed the safety factor calculation procedure in detail earlier [13; 18; 19; 20], and thus, it is explained only concisely here. Two equations are needed in the reliability calculation.  
When one load (L) with the cumulative distribution \(\mathrm{F_{L}}(\mathrm{x};\;\mu_{\mathrm{L}},\;\sigma_{\mathrm{L}})\) and the safety factor \(\mathrm{\gamma_{L}}\) strikes a material with the resistance \(\mathrm{f_{M}}(\mathrm{x};\;\mu_{\mathrm{M}},\;\sigma_{\mathrm{M}})\), the equation to calculate the safety factors (\(\mathrm{\gamma_{L}}\) or \(\mathrm{\gamma_{M}}\)) or the failure probability (\(\mathrm{P_{f}}\)) is [13; 18; 19; 20]  
\[\int_{0}^{\infty}\mathrm{F_{L}}\bigg{(}\mathrm{x};\frac{\mu_{\mathrm{L}}}{ \gamma_{\mathrm{L}}},\frac{\sigma_{\mathrm{L}}}{\gamma_{\mathrm{L}}}\bigg{)} \mathrm{f_{M}}(\mathrm{x};\;\mu_{\mathrm{M}}\mathrm{\gamma_{M}},\;\sigma_{ \mathrm{M}}\mathrm{\gamma_{M}})\mathrm{d}\mathrm{x}=1-\mathrm{P_{f}} \tag{8}\]  
Both distributions are set in the design point with equal characteristic values.  
When the load comprises the permanent load and the variable load, and the variable load proportion is \(\alpha\), in relation to the total load, \(\alpha=\mu_{\mathrm{Q}}/\mu_{\mathrm{G}}+\mu_{\mathrm{Q}}\), the cumulative distribution can be obtained from adding up the partial distributions by fractiles. Monte Carlo simulation with one seed number is one calculation option, i.e., a random seed number is selected, the permanent load and the variable load is computed from the distributions and the obtained values are added up; this denotes the combination load value. The combination distribution crosses the crossing point of the partial distributions, and this fact can be used for the combination distribution calculation [19]. These methods are impractical, and the next recursive equation is a simple method to calculate the dependent combination distribution (\(\mathrm{F_{d}}\)) in load proportion (\(\alpha\)) [13]  
\[\mathrm{F_{d}}(\mathrm{x},\alpha)=\Bigg{|}\begin{array}{c}\mathrm{y}\, \leftarrow\,\mathrm{root}\Big{[}\mathrm{F_{G}}\Big{[}\mathrm{x}-\mathrm{r}; \;\frac{\mu_{\mathrm{G}}(1-\alpha)}{\mathrm{\gamma_{G}}},\frac{\sigma_{\mathrm{G }}(1-\alpha)}{\mathrm{\gamma_{G}}}\Big{]}-\mathrm{F_{Q}}\Big{[}\mathrm{r};\; \frac{\mu_{\mathrm{Q}}\;\alpha}{\mathrm{\gamma_{Q}}},\frac{\sigma_{\mathrm{Q }}\;\alpha}{\mathrm{\gamma_{Q}}}\Big{]},\mathrm{r}\Big{]}\\ \mathrm{F_{d}}\,\leftarrow\,\mathrm{F_{Q}}\Big{[}\mathrm{y};\;\frac{\mu_{ \mathrm{Q}}\;\alpha}{\mathrm{\gamma_{Q}}},\frac{\sigma_{\mathrm{Q}}\;\alpha}{ \mathrm{\gamma_{Q}}}\Big{]}\end{array} \tag{9}\]  
The safety factors \(\mathrm{\gamma_{G}}\), \(\mathrm{\gamma_{Q}}\), \(\mathrm{\gamma_{M}}\) or the failure probability \(\mathrm{P_{f}}\) are calculated using Equation (8) when the load distribution \(\mathrm{F_{L}}\) is replaced by \(\mathrm{F_{d}}\).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & \(\mathrm{V_{M}}\) & \(\mu_{\mathrm{M}}\) & \(\sigma_{\mathrm{M}}\) & \(\gamma_{\mathrm{M}}\) \\ \hline Steel & 0.1 & 1.1841 & 0.1184 & 1.0 \\ Timber & 0.2 & 1.4125 & 0.2825 & 1.3 \\ Concrete & 0.3 & 1.6921 & 0.5076 & 1.5 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The three examined materials steel, timber and concrete with parameters: \(\mathrm{V_{M}}=\) coefficient of variation; \(\mu_{\mathrm{M}}=\) mean; \(\sigma_{\mathrm{M}}=\) deviation; \(\mathrm{\gamma_{M}}=\) material safety factor.