## 2 Materials and Methods
In a fire design situation, this value can be computed according to the following formula [6]:  
\[M_{fi,\theta,Rd}=k_{y,\theta}M_{Rd} \tag{1}\]  
where \(k_{y,\theta}\)--effective yield strength; \(M_{Rd}\)--bending moment resistance in persistent design situation [31] according to Equation (2). In Equation (1), the partial safety factor is intentionally uncredited, because the further computation takes into account randomness. \[M_{Rd}=W_{pl,y}f_{y} \tag{2}\]  
where \(f_{y}\)--yield strength of steel; \(W_{pl,y}\)--plastic section modulus of cross-section according to the \(y\) (horizontal) axis. Formulation of the probabilistic model makes it necessary to approximate the coefficient of variation for the bending moment resistance. According to Ref. [32], this value can be computed as follows:  
\[v_{MRd,i}=\sqrt{v_{Wpl,y}^{2}+v_{fy}^{2}}=\sqrt{0.1^{2}+0.08^{2}}=0.13 \tag{3}\]  
The first step of reliability analysis is the assessment of the reliability of a single node. This value in the following minutes of fire duration was computed according to the formulas presented below. It should be noted that at nodes when different profiles were connected (the corners), the characteristics of the \"weaker\" element were taken into account. The computation of reliability for the single \(i\)-th element was computed according to the following equations: \begin{table}
\begin{tabular}{c c c} \hline \hline
**Value** & **Symbol** & **Coefficient of Variation (c.o.v.)** \\ \hline Effect of action & \(\text{M}_{\text{Ed}}\) & \(10\%\) \\ \hline Yield strength & \(\text{f}_{y}\) & \(8\%\) \\ \hline Cross-sectional area & \(\text{W}_{\text{pl,y}}\) & \(10\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Random variables. * Coefficient of variation for effect of action and bearing capacity: \[\
u_{MEd,i}=0.1\cdot M_{EId,i}\] (4) \[\
u_{Mfi,\theta,Rd,i}=0.13\cdot M_{fi,\beta,Rd,i}\] (5)
* Safety margin (\(SM_{i}\)): \[SM_{i}=M_{fi,\theta,Rd,i}-M_{EId,i}\] (6)
* Coefficient of variation for safety margin: \[\
u_{SM,i}=\sqrt{\
u_{M_{fi,\theta,Rd,i}}^{2}+\
u_{M_{EId,i}}^{2}}\] (7)
* Reliability index of a single element (\(\beta_{\rm i}\)): \[\beta_{\rm i}=\frac{SM_{i}}{\
u_{SMi}}\] (8)
* Probability of failure for a single element (\(\rm p_{fi}\)): \[\rm p_{fi}=\Phi(-\beta_{i})\] (9) where \(\Phi\) is the inverse cumulative standard normal distribution. * Reliability of element (\(R_{i}\)): \[R_{i}=1-p_{fi}\] (10)  
Knowing the reliability of a single element (node) makes it possible to assess the reliability of the whole structure. The mechanisms connected with frame structures have parallel character, which means that all the elements in one set have to become unreliable to result in the failure of the whole structure. Therefore, the reliability for a single mechanism (failure path) is computed according to the following formula [1]:  
\[R_{p,n}=1-\sum_{i=1}^{n}(1-R_{i}) \tag{11}\]  
where \(R_{p,n}\)--reliability of the structure for the \(n\)-th failure path (parallel system); \(R_{i}\)--reliability of single node according to Equations (4)-(10); \(i\)--number of nodes that create the analyzed failure path. All analyzed mechanisms are connected with each other in a series way; therefore, final reliability for the structure with analyzed mechanism is computed as follows:  
\[R=\prod_{n=1}^{k}R_{p,n} \tag{12}\]  
where \(k\)--the total number of analyzed mechanisms.