## 1 Introduction  
The reliability of structures is crucially important in the context of human health and safety. On the other hand, the building structures should be designed economically. The question is how to find the best method, which enables one to reconcile these two apparently inconsistent aims. Therefore, reliability analysis is now one of the most developed fields of science, not only in the context of civil engineering. There are numerous scientific papers and books according to the basis of the reliability method; some of them are now some kind of \"classic\" [1, 2, 3].  
Generally, for bar structures, a lot of reliability methods are useful, including approximation methods (FORM, SORM) [4, 5, 6, 7, 8], simulation methods (Monte Carlo, Importance Sampling) [9, 10, 11, 12, 13, 14] or even neural networks [15, 16, 17, 18]. However, it seems that for truss structures, the system reliability analysis could be the most appropriate [19, 20, 21], as the alternative to the stochastic finite element method SFEM [22, 23, 24]. Unfortunately, using this method, one needs to force the problem connected with searching the so-called _cut-sets_. These are such sets of elements that decide about transforming the structure into the mechanism if all of these set elements become unreliable (in all of them, the effect of action exceeds their load bearing capacity). The importance is the fact that each of such a set includes the minimum number of elements leading to making the structure geometrically variable (_minimal critical set_). Elements which create the single cut-set are called _causative elements_.  
To identify all the cut-sets, few methods could be used. In the following paper, the stiffness matrix spectral analysis was applied. The detailed description of the method can be found in [25].  
Generally, in this method, all combinations of \"removing\" elements from the stiffness matrix are checked. If some combination results in the fact that the determinant of the stiffness matrix is equal to zero, it means that the structure becomes the mechanism and the cut-set is identified. The difficulty is connected with that, then, in complex structures with many elements, and a huge number of cut-sets can be generated. To force this problem in the last years, various selective searching techniques have been proposed in scientific publications [26; 27; 28].  
Fortunately, in the case of the truss, it is known that taking into account elements in tension can be neglected. Of course, elements in compression are likely to become unreliable first. Furthermore, some groups of elements are made from the same profiles, which results in that part of them having a very low ratio effect of action to load-bearing capacity. These facts enable us to reduce the potential number of causative elements: what makes the whole process of searching cut-sets much more effective and faster.  
The proposed system reliability method was used to compute the reliability of structures in a persistent design situation and accidental-under-fire load. The previous experience of the author indicates that system analysis is very useful for analysis of statically indeterminate steel trusses. In structures like this, the temperature of fire gases heats the members of the structure and causes the redistribution of internal forces in the following minutes of fire duration. It was also observed that statically indeterminate structures are likely to create the new static scheme while exceeding bearing capacity in some elements, and it seems that the system reliability analysis is the only way to identify them.