## 7 Results and Discussion  
Figure 8a-e depict the probability density function (PDF) histogram and cumulative density function (CDF) of the output parameters for each design constraint from DesignXplorer(c). For the stress constraint case (ULS), the probability that maximum equivalent stress remains less than \(\sigma_{\mathit{VM,allow}}=323\) MPa, \(P_{f}(\sigma_{\mathit{VM}}\leq 323\mathit{MPa})\), is about \(0.99999867\) which means that there is \(1-0.99999867=1.327655\times 10^{-6}\) probability that the von Mises stress exceed the limit value of \(323\) MPa, i.e., \(P_{f}(\sigma_{\mathit{VM}}>323\mathit{MPa})=1.327655\times 10^{-6}\). Therefore, the probability of failure of the critical component of the structure is \(1.327655\times 10^{-6}\) considering the ULS (stress constraint) design criteria. The corresponding reliability index value is \(\beta=4.7\). For the fatigue constraint case (FLS), the probability that the minimum safety factor is less than \(f_{\mathit{sr,allow}}=1.15\), \(P_{f}(f_{\mathit{sr}}\leq 1.15)\), is about \(5.6461\times 10^{-7}\). Therefore, the probability of failure of the critical component of the structure is \(5.6461\times 10^{-7}\) considering FLS design criteria (using time-invariant analysis). The corresponding reliability index value is \(\beta=4.87\). For the deformation constraint (SLS), the DesignXplorer(c) simulation could not produce results within the region of \(P_{f}(d\leq 0.54)\), i.e., \(d_{\mathit{allow}}=0.54\). The value \(P_{f}(d>0.54)\), which is the value of the probability of failure, is expected to be very small, and this is identified as a limitation of the direct MCS requiring a prohibitively large number of simulations to estimate results of this magnitude. Similarly, for the buckling constraint case (ULS), the simulation could not produce the results within the region of \(P_{f}(L_{m}\leq 1.4)\), as this value is expected also to be very small. For the vibration constraint (SLS), the probability of failure of the critical component of the structure \(P_{f}(f_{1 st}\leq 0.133)\) is about \(0.1276\times 10^{-4}\), and the corresponding \(\beta=4.21\). Figure 8: _Cont._ \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Description**} & \multicolumn{2}{c}{**Ultimate Load Case**} & \multicolumn{2}{c}{**Fatigue Load Case**} \\ \cline{2-5}  & **Mean** & **Standard** & **Mean** & **Standard** \\  & **Mean** & **Deviation** & **Mean** & **Deviation** \\ \hline Wind thrust (kN) & 1700 & 170 & 203 & 20.3 \\ Tilting moment (kN-m) & 38,567 & 3856.7 & 3687 & 368.7 \\ Torsional moment (kN-m) & 7876 & 787.6 & 3483 & 348.3 \\ RNA mass (kg) & 1,072,000 & 107,200 & 1,072,000 & 107,200 \\ Young’s modulus of soil stratum 1 (MPa) & 30 & 3 & 30 & 3 \\ Young’s modulus of soil stratum 2 (MPa) & 50 & 5 & 50 & 5 \\ Young’s modulus of soil stratum 3 (MPa) & 80 & 8 & 80 & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Characteristics of design variables. The results obtained from the non-intrusive formulation for each design constraint are depicted in Figure 9 and Table 10. The results obtained from a time-invariant fatigue reliability assessment employing the FORM are the reliability index, \(\beta=5.077\), and probability of failure, \(P_{f}=0.192\times 10^{-6}\). Figure 8: Cumulative density function (CDF) and histogram for the output parameter: (**a**) maximum von Mises stress (stress constraint case); (**b**) fatigue safety factor minimum (fatigue constraint case); (**c**) maximum deformation; (**d**) total deformation load multiplier (buckling constraint case);, (**e**) first eigen-frequency (vibration constraint case)-DesignXplorer(). As evident from the results above, the reliability assessment performed on the structure shows that the present model, as designed and for the modeling of the stochastic variables considered, satisfies the recommended reliability assessment criteria, as the reliability indices for all the design constraints considered are within the design thresholds.