# The Impact of Support Type on the Reliability of Steel Trusses Subjected to the Action of a Fire
## 3 Results
### Reliability Analysis  
As mentioned at the beginning of the article, the performance of a system analysis requires the identification of all kinematically admissible failure mechanisms for the given structure. This task was accomplished using a computing module prepared in the Mathematica environment [27; 28]. In the case of the analyzed subject, after each change in the static scheme it becomes necessary to determine new KAFMs, because they are related to the geometry of the truss that undergoes the change. For the analyzed trusses, there is a considerable number of KAFMs, which complicates the task and prevents the performance of calculations without the use of computer methods. The identified KAFMs were divided into groups based on the number of causative elements. Tables 1-3 list the quantity of KAFMs for individual static schemes. Table 1 presents results related to truss \"0\". Due to the capabilities of the computer and the time-consuming KAFM search algorithm, for truss \"A\" with an initial static scheme (Figure 4a), it was not possible to obtain an eight-element KAFM. The authors' experience indicates that mechanisms deciding about reliability include those with very small numbers of causative elements. Figure 7: The change of axial forces and resistance with fire duration (truss “B”): (**a**) elements 20, 31 and (**b**) elements 1, 6. Therefore, this mechanism was excluded from the calculations. The reliability of such a complex mechanism is very close to one, which does not affect the final result. It is not possible to present the whole reliability calculation procedure in the article due to the complexity of calculations. Nonetheless, a fragment of them will be presented in order to illustrate the method. It involves truss \"B\" after the first change of the static scheme (Figure 5a). Geometry of the truss along with the current numbering of rods are presented in Figure 8. Figure 9 presents the schemes of reliability system characteristics for the analyzed truss. Due to the fact that the fourth, fifth, and sixth KAFMs are characterized by a large number of causative elements (Table 3), the figure presents only the first and the last frame of the system. \begin{table} \begin{tabular}{c c c c c} \hline \hline
**Number of Causative Elements** & **Static Scheme I** & **Static Scheme II** & **Static Scheme III** & **Static Scheme IV** \\
**Elements** & **(Figure 2a)** & **(Figure 3a)** & **(Figure 3b)** & **(Figure 3c)** \\ \hline \(1\) & - & 8 & 16 & 25 \\ \(2\) & 44 & 32 & 20 & – \\ \(3\) & 88 & 56 & 25 & – \\ \(4\) & 72 & 40 & – & – \\ \(5\) & 56 & 25 & – & – \\ \(6\) & 40 & – & – & – \\ \(7\) & 25 & – & – & – \\ \(\Sigma\) & 325 & 161 & 61 & 25 \\ \hline \hline \end{tabular}  \end{_Appl. Sci._**2020**, _10_, 7916  
**I Kinematically Admissible Failure Mechanism (I KAFM)**  
**Figure 9. Schemes of the reliability system of truss \"B\" after the first reduction. **The first KAFM corresponds to a serial system; therefore, based on the Formula (9), its reliability is calculated as a product of the reliabilities of individual elements:  
\[R_{I}=R_{7}\cdot R_{12}\cdot R_{13}\cdot R_{19}\cdot R_{24}\cdot R_{25} \tag{12}\]  
A mixed system appears in the case of KAFM II. Therefore, at first it is necessary to calculate the reliability of individual frames. If elements connected in series appear in a given frame, it is necessary to calculate the product of their reliability, followed by the reliability of elements connected in parallel according to Formula (10). With the assumption about the separation of mechanisms with common elements, it can be presumed that individual frames are connected to each other in series [32]. Therefore, the reliability of KAFM II will be calculated in the following manner:  
\[