## 2 Materials and Methods  
There are many methods to assess the reliability of a structure. In the paper, the system approach is used for reliability analysis. In this case, structural characteristics of the truss structure are taken into account, i.e., interaction of the structural elements in the load-carrying system [19; 20; 21; 22; 23; 24]. In the reliability theory, there are three models: serial, parallel, and mixed (parallel-serial and serial-parallel) systems. To identify the reliability model, the kinematically admissible failure mechanisms (KAFMs) should be specified. Each KAFM contains the minimal critical sets of elements (MCSE), which have a particular number of tshe causative elements (\(l\)). The serial system consists of \(k\) minimal critical sets of elements (\(k\) MCSE), each of which has one causative element (\(l=1\)). The serial system (Figure 1a) is operable if all elements are operable. Reliability \(R\) of the serial system is calculated from the formula  
\[R=\prod_{i=1}^{k}R_{i} \tag{1}\]  
where \(R_{i}\)--reliability of a single element and \(k\)--number of the system elements. In the case of a parallel system, there is one minimal critical set of elements (1 MCSE) and it consists of \(k\) causative elements (\(l=k\)). The parallel system (Figure 1b) is operable if at least one element is operable. Reliability \(R\) of the parallel system is calculated from the formula  
\[R=1-\prod_{i=1}^{k}(1-R_{i}) \tag{2}\]  
For more complex truss structures forming mixed systems, the reliability is evaluated by assuming actual connections and interactions between the elements. To identify the reliability model, it is necessary to analyze how the elements are coupled, i.e., what relations hold between them. Mixed systems consist of parallel-serial and serial-parallel arrangements. The parallel-serial system (Figure 1c) consists of \(m\) MCSEs, each of which includes \(k\) causative elements (\(l=k\)). Reliability \(R\) of the parallel-serial system is calculated from the formula  
\[R=1-\prod_{j=1}^{m}\left[1-\prod_{i=1}^{k}\left(1-R_{ji}\right)\right] \tag{3}\]  
For the last serial-parallel system (Figure 1d), the number of MCSEs is \(m\cdot k\) and each MCSE consists of \(k\) causative elements (\(l=k\)). Reliability \(R\) of the serial-parallel system is calculated from the formula  
\[R=1-\prod_{j=1}^{m}\left[1-\prod_{i=1}^{k}\left(1-R_{ji}\right)\right] \tag{4}\]  
To carry out the spectral analysis of the linear stiffness matrix (identify the reliability model) and to calculate the reliability index, the program developed by the authors in the Mathematica environment was applied. To determine the reliability of the system, it is necessary to calculate the reliability of a single element. The procedure that leads to the determination of the reliability of a single element \(R_{i}\) was presented, for example, in [21; 22]. The first step is to determine the effect of action \(E_{i}\), and capacity \(N_{i}\). The next step is to determine the safety margin \(Z_{i}\). It is a difference between the effect of action \(E_{i}\) and capacity \(N_{i}\). To compute reliability of the single element, it is necessary to determine expected value \(\mu_{Zi}=\mu_{Ni}-\mu_{Ei}\) and standard deviation \(\sigma_{Zi}=\sqrt{\sigma_{Ei}^{2}+\sigma_{Ni}^{2}}\) of the safety margin. A subsequent step is computing the reliability index for a single element \(\beta_{i}=\mu_{Zi}/\sigma_{Zi}\) and probability of the element failure \(P_{fi}=\Phi(-\beta_{i})\), where \(\Phi\) is the Laplace function. Finally, the reliability of a single element is estimated from the formula \(R_{i}=1-P_{fi}\). To compute the reliability of the system \(R\), it is necessary to identify the reliability model. After that, the probability of failure of whole structure \(P_{f}=1-R\) and reliability index of whole structure \(\beta=-\Phi^{-1}\left(P_{f}\right)\) are computed. In this paper, the algorithm described above was used to assess the reliability of a steel truss tower.