# Analytical and Numerical Reliability Analysis of Certain Pratt Steel Truss
## 3 Numerical Results for Structural Elements and Their Discussion
The abbreviations in this table heading are consistent with the notation of the previous figure, i.e., MCS-NC  
Figure 9: Kurtosis _Kurt(u)_ in the SLS. denotes the results corresponding to the Monte Carlo simulation method while using the finite element method analysis. The data collected in this table confirm the well-known trend of the reliability index, which usually exponentially decreases together with an additional increase in the cumulated input uncertainty level. Monte Carlo simulation and the iterative generalized stochastic perturbation technique both return practically the same results, which, taking into account decisively smaller time and computer power effort of this second technique, means that the perturbation-based SFEM is preferred in this case. As is expected, the lowest values of this index were returned for the uniform distribution of the truss span, slightly larger values were obtained for the triangular PDF of this span and the extremely high reliability index was obtained in the case of the Gaussian distribution. It is observable that an arbitrary assumption of the Gaussian character of most structural parameters without a precise verification may lead to some inaccuracies or even to unsafe designing of the engineering structures. A very interesting observation can be made while comparing numerical and analytical analyses--these second series return the results each time with no more than a few percent larger than the FEM-based analysis. Nevertheless, this difference is almost negligible when the uncertainty level belongs to the interval \(\alpha\in[0.05,0.10]\), which is highly expected in most of the statistical parameters. Moreover, it is widely known that standard EN 1990 [16] defines the three reliability classes RC1, RC2 and RC3, which are associated with the consequences classes CC1, CC2 and CC3, describing the possible consequences of a failure of the entire structure or just its member. The table below (Table 4) contains the minimum values of the reliability index \(\beta\) recommended by EN 1990 (Table B2) [16], which were recalled to determine the admissibility level of input randomness for the given parameter of the Pratt truss span. The results of this study are presented by inserting numerical values from Tables 3 and 4 into a single graph--see Figure 10 below. \begin{table}
\begin{tabular}{c c c c c} \hline \(\alpha\) [-] & **MCS-NC** & **MCS-AC** & **PM-NC** & **PM-AC** \\ \hline  & 37.93 & 38.28 & 37.93 & 38.28 \\
0.025 & 37.40 & 37.83 & 37.40 & 37.83 \\  & 38.11 & 38.43 & 38.11 & 38.43 \\  & 18.95 & 19.12 & 18.95 & 19. Truss girders are structures that are expected to last at least 50 years and the described construction has been assigned to the third reliability class that allows resisting even if the coefficient of variance of the truss span slightly exceeds the value of \(\alpha>0.150\). An intersection of the reliability curves delivered by numerical and analytical approaches with the lines representing target values of the reliability index shows that this admissible value equals \(\alpha=0.175\). This value is enormously large for any measuring method for the span, so one may conclude that the presented method generally has no limitations coming from the engineering practice while investigating statistics in the truss spans. Taking into account the well-known limitations of the FORM approach, the new reliability index has been proposed following the relative entropy (probabilistic divergence) model proposed by [PERSON] [27]. Its values computed using the first two probabilistic moments have been collected above in Table 5, in the same manner as the results contained in Table 4. Despite the different range of variability caused by the fact that Equation (9) contains a linear form of the expectations of the functions R and E, one may see the same properties and interrelations in between different methods and input distributions.