# Analytical and Numerical Reliability Analysis of Certain Pratt Steel Truss
## 4 Stochastic Response of the Decisive Structural Connection and Its Discussion  
Additionally, the load capacity of structural connection nodes, designed according to the statements included in the designing code [30], was studied in the context of stochastic structural response and reliability. This part of the numerical study was based on the reduced von Mises stresses in their welds and some exemplary connections have been schematically shown in Figure 11, whereas the nodes with extreme stresses accounted for in further SFEM analysis have been marked in Figure 12 with the dashed line circles. Quite similarly to the SLS analysis in the previous section, one first needs to determine the response function of the reduced stresses with respect to the truss span via the LSM fitting on the basis of the results contained in Table 7. This is obtained in the form of the third-order polynomial having the following analytical form:  
\[\sigma_{rad}=0.007761l^{3}-0.421976l^{2}+23.5512l-45.2738 \tag{17}\]  
Furthermore, probabilistic characteristics of the structural response are determined with the Monte Carlo simulation and the stochastic perturbation method; these are: expectations (Figure 13), coefficients of variance (Figure 14), skewness (Figure 15) and kurtosis (Figure 16). This is conducted for three various probability distributions, but unfortunately, any analytical method is available here. Figure 13 documents that the expected values of the reduced stresses remarkably and nonlinearly decrease, contrary to the extreme deformations, while increasing input uncertainty. Furthermore, some small differences in between the Monte Carlo simulation and the stochastic perturbation method can be noticed for all PDFs, which additionally increase together with an increase in the input CoV. They are quite remarkable in Figure 13, but their realistic range is smaller than one thousandth, so they can be postponed in engineering practice. Variances of the reduced stresses quite \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline Length \(l\) [m] & 15.00 & 15.60 & 16.20 & 16.80 & 17.40 & 18.00 & 18.60 & 19.20 & 19.80 & 20.40 & 21.00 \\ \hline Stress \(\sigma\) [MPa] & 239.22 & 248.81 & 258.79 & 267.98 & 277.56 & 287.15 & 296.74 & 306.31 & 315.89 & 325.47 & 335.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Extreme reduced stresses in the external weld in the truss column. Figure 11: Axonometric view of the analyzed truss joints from the Autodesk ROBOT 2021.  
expectedly increase in Figure 13 while approaching the extreme values of the input \(\alpha(l)\) and now the differences between the statistical and perturbation approaches are invisible. The largest resulting statistical scattering is obtained for the uniformly distributed truss span, next, for the triangular PDF; and the smallest for the Gaussian distribution. This regularity partially validates the entire methodology and, specifically, the efficiency of the iterative generalized stochastic perturbation method for symmetric non-Gaussian distributions also. Figure 14: Variance _Var_ in the ULS. Figure 13: Expected Value \(E\) in the ULS. Skewness and kurtosis attached here for the completeness of probabilistic modeling (Figures 15 and 16) are all very close to 0, so these reduced stresses could be used in reliability analysis of Gaussian with relatively small error, so that the first two moments could be sufficient. Of course, this approximation is the most efficient when input uncertainty has a Gaussian character. Now, analogously to Figure 14, two stochastic approaches return the same values of both coefficients, which also confirms the applicability of the stochastic perturbation method for higher statistics. Finally, the reliability indices are contained in Table 8 and also compared in Figure 17 with the corresponding normative values [16]. Generally, one can observe that their values are remarkably smaller than for the structural elements analyzed before, see cf. Table 6. This  
Figure 16: Kurtosis _Kurt_ in the ULS.