# Reliability-Based Design of an Aircraft Wing Using a Fuzzy-Based Metaheuristic
## 5 Design Results and Discussions
For upper and lower skins using a laminated carbon fiber with three layers, from the calculated results, a symmetric and equivalent laminate for upper and lower skins in each PSI \(\pi_{\beta}{}^{max}\) = 0.1, 0.01 and 0.001 with the orientations of [30\({}^{\circ}\)/120\({}^{\circ}\)/30\({}^{\circ}\)], [105\({}^{\circ}\)/0\({}^{\circ}\)/120\({}^{\circ}\)] and [150\({}^{\circ}\)/0\({}^{\circ}\)/120\({}^{\circ}\)], respectively as displayed in Figure 5. The optimum ply orientation can eliminate shearing and bending twist coupling behaviors. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & **Mass (kg)** & **Flutter Speed (m/s)** & **Lift Effectiveness (\(\eta\))** \\ \hline \(\pi_{\beta}{}^{max}\) = 0.001 & 103.0109 & 302.5471 & 1.0272 \\ \(\pi_{\beta}{}^{max}\) = 0.01 & 98.1614 & 278.3650 & 1.0578 \\ \(\pi_{\beta}{}^{max}\) = 0.1 & 83.1111 & 251.2743 & 1.1149 \\ Deterministic & 72.4086 & 190.189 & 1.151 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Optimum solutions for the aircraft wing design. Figure 4: The search history of the composite aircraft wing design. The first 10 lowest natural frequency modes used in flutter analysis at each PSI value are presented in the Table 4. From the results shown, the natural frequencies are very high for a wing--even for wind tunnel model testing. This is caused by the strict deformation constraint of 0.1 m, which is less than 2% of half the wings pan. The plots of damping ratio versus velocity with various PSI \(\pi_{\beta}{}^{max}\) values are shown in Figure 6. The results show flutter speeds estimated by quasi-steady aerodynamics, which are affected by PSI \(\pi_{\beta}{}^{max}\) as the lower PSI \(\pi_{\beta}{}^{max}\) results in the lower flutter speeds. Furthermore, from comparing the design variables and the objective functions, the result from PSIBDO is larger than the deterministic solution. The results may indicate that skin thicknesses in Layers 1, 3, 4, and 6, and spar thickness to lower stress when the aircraft wing is subject to the aerodynamic loads. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Design Variables and** & \(\pi_{\beta}{}^{max}\) = 0.1 & \(\pi_{\beta}{}^{max}\) = 0.01 & \(\pi_{\beta}{}^{max}\) = 0.001 & **Deterministic** \\ \hline Upper skin Layer 1 (mm.) & 0.0005 & 0.0005 & 0.0005 & 0.0005 \\ Upper skin Layer 2 (mm.) & 0.0005 & 0.0005 & 0.001 & 0.0005 \\ Upper skin Layer 3 (mm.) & 0.001 & 0.001 & 0.001 & 0.0005 \\ Lower skin Layer 1 (mm.) & 0.0005 & 0.0005 & 0.0005 & 0.0005 \\ Lower skin Layer 2 (mm.) & 0.0005 & 0.001 & 0.001 & 0.0005 \\ Lower skin Layer 3 (mm.) & 0.001 & 0.001 & 0.001 & 0.0005 \\ Rib thickness (mm.) & 0.0015 & 0.0015 & 0.0015 & 0.001 \\ Spar thickness (mm.) & 0.001 & 0.001 & 0.001 & 0.001 \\ Lower skin Layer 1 (deg.) & 30 & 105 & 150 & 135 \\ Lower skin Layer 2 (deg.) & 120 & 0 & 0 & 60 \\ Lower skin Layer 3 (deg.) & 30 & 120 & 120 & 135 \\ Upper skin Layer 1 (deg.) & 30 & 105 & 150 & 135 \\ Upper skin Layer 2 (deg.) & 120 & 0 & 0 & 60 \\ Upper skin Layer 3 (deg.) & 30 & 120 & 120 & 135 \\ Mass (kg) & 83.1111 & 98.1614 & 103.0109 & 72.4086 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Design variables of optimum solutions for the aircraft wing design by PSIBDO. Figure 5: Layout of ply orientation angles on upper and lower wing skin in the composite laminate. Figure 6: The variations of the PSI at (a) \(\pi_{\beta}^{max}=0.1\), (b) \(\pi_{\beta}^{max}=0.01\) and (c) \(\pi_{\beta}^{max}=0.001\) due to the variation damping ratio vs. velocity.