# Analytical and Numerical Reliability Analysis of Certain Pratt Steel Truss
## 3 Numerical Results for Structural Elements and Their Discussion
Three different symmetric probability distributions were compared and these are triangular (TD), uniform (UD) and Gaussian (normal) distribution (ND). These distributions  
Figure 4: Structural effort variations of the Pratt truss structural components w.r.t. Pratt truss span. A general conclusion which can be drawn from these data is that all probabilistic characteristics monotonously increase together with an additional increase in the input uncertainty level. This is quite an expected result, as is the fact that the first two moments reach the largest values when the input distribution has a uniform PDF. The second general conclusion is that the results of both probabilistic methods coincide almost perfectly for all of the first four probabilistic characteristics, which is a very promising result in the context of the relatively large input uncertainty level and the fact that a nonlinear BFGS solver was used. It should be noticed at this point that a situation may be more complex for large-scale engineering structures and the input CoV interval permitting such a perfect agreement may become somewhat smaller, i.e., \(\alpha(I)\in[0.00,0.10]\); this should be verified through more complex optimization problems. Figure 8: Skewness \(\textit{Skew}(u)\) in the SLS. Figure 7: Variance \(\textit{Var}(u)\) in the SLS. detailed FEM experiments. The next important information, which can be rather useful in civil engineering practice, is that expectations computed via analytical formulas (Figure 6) are remarkably smaller than those obtained from the FEM experiments. This may follow the fact that analytics is available in the linear elastic regime only, whereas the FEM analysis is conducted including geometrical nonlinearity; nevertheless, the first approach seems to be insufficient in all these cases, where nonlinear effects may play a remarkable role during exploitation time. Higher order statistics such as skewness (Figure 8) and kurtosis (Figure 9) differ from 0, so that the resulting distributions of extreme displacements are slightly different from the Gaussian PDF, and this divergence increases while increasing the input uncertainty level. It is noticeable that the largest values of both coefficients are obtained for the Gaussian PDF of the truss span, slightly smaller values are obtained for the triangularly distributed span and the smallest are obtained in the case of the uniform distribution. Skewness keeps positive values for the entire variability interval of truss span uncertainty for all chosen probability distributions, while kurtosis values are more complex. They are all positive for the normal PDF, start from negative and tend towards positive while having a triangularly distributed parameter \(l\) and are all negative when the uniform distribution is considered. These higher-order statistics are usually postponed during the final reliability assessment; however, these results demonstrate that the initial choice of PDF type remarkably affects the resulting stochastic structural response. The first two probabilistic moments are contained in Figures 6 and 7 and resulting from the Gaussian distribution of the truss span were used to calculate final reliability values in the SLS state. Having computed the probabilistic response of the Pratt truss, its reliability indices were computed in the computer algebra software MAPLE 2019 according to Equations (9) and (14), sequentially. This was also carried out for three different probability distributions: triangular, uniform and Gaussian, to see an influence of the PDF choice on their final values. Numerical values relevant to Equation (9) have been collected in Table 3, where an influence of the input uncertainty of the truss span in the form of the coefficient of variance \(\alpha\) has been also included. Each cell of this table includes in turn the reliability index for these three PDFs--the first one corresponds to the triangular probability function, the middle to the uniform one and the lowest to the Gaussian distribution.