# Analytical and Numerical Reliability Analysis of Certain Pratt Steel Truss
## 3 Numerical Results for Structural Elements and Their Discussion
This new approach is more convenient in the engineering sense, as it measures a distance in between two random distributions and no specific form of the limit function is required. Its value has been scaled to compare with the existing designing algorithm and this has been performed using the following semi-empirical formula:  
\[\beta^{\prime\prime}=\frac{\sqrt{\beta^{\prime}}}{2}. \tag{16}\]  
Such a formula has been established by a comparison of the main components in Equations (10) and (14) containing mean values of the structural response and its resis \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Reliability Class**} & \multicolumn{3}{c}{**Minimum Values for \(\beta\)**} \\ \cline{2-3}  & **1 Year Reference Period** & **50 Years Reference Period** \\ \hline RC3 & 5.2 & 4.3 \\ RC2 & 4.7 & 3.8 \\ RC1 & 4.2 & 3.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Recommended minimum values for reliability index \(\beta\). Figure 10: The reliability index \(\beta\) in the SLS. tance. It is important to mention that this proposed scaling enables keeping the same practical demands and values and to have a better mathematical and engineering justification of the entire procedure. It is evident after scaling that the new reliability index (Table 6) is more demanding and does not allow for such a large input uncertainty interval; nevertheless, the importance of the input PDF choice remains the same. Analytical and numerical approaches here return almost the same result, so one could recommend inserting them directly into the engineering designing codes to give practicing engineers a fast and widely available alternative to the variety of rather complex stochastic finite element method implementations. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\boldsymbol{\alpha}\) [-] & **MCS-NC** & **MCS-AC** & **PM-NC** & **PM-AC** \\ \hline \multirow{4}{*}{0.025} & 53.45 & 53.80 & 53.45 & 53.80 \\  & 53.44 & 53.79 & 53.44 & 53.79 \\  & 53.45 & 53.81 & 53.45 & 53.81 \\  & 13.38 & 13.47 & 13.38 & 13.47 \\
0.050 & 13.37 & 13.46 & 13.37 & 13.46 \\  & 13.38 & 13.47 & 13.38 & 13.47 \\  & 5.99 & 6.03 & 5.99 & 6.03 \\
0.075 & 5.98 & 6.02 & 5.98 & 6.02 \\  & 5.99 & 6.04 & 5.99 & 6.04 \\  & 3.44 & 3.46 & 3.44 & 3.46 \\
0.100 & 3.42 & 3.45 & 3.42 & 3.45 \\  & 3.44 & 3.47 & 3.44 & 3.47 \\  & 2.29 & 2.31 & 2.29 & 2.31 \\
0.125 & 2.27 & 2.29 & 2.27 & 2.29 \\  & 2.29 & 2.32 & 2.29 & 2.32 \\  & 1.69 & 1.71 & 1.69 & 1.71 \\
0.150 & 1.67 & 1.69 & 1.67 & 1.69 \\  & 1.70 & 1.72 & 1.70 & 1.72 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: A contrast of the estimated reliability indices \(\beta^{\prime\prime}\) in the SLS by numerical and analytical methods based upon the Monte Carlo simulation method and the stochastic perturbation technique for the triangular, uniform and Gaussian probability distributions. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\boldsymbol{\alpha}\) [-] & **MCS-NC** & **MCS-AC** & **PM-NC** & **PM-AC** \\ \hline \multirow{4}{*}{0.025} & 11,426.46 & 11,579.79 & 11,426.45 & 11,579.77 \\  & 11,422.07 & 11,575.61 & 11,421.98 & 11,575.53 \\  & 11,428.21 & 11,581.44 & 11,427.93 & 11,581.19 \\  &