# Reliability-Based Design of an Aircraft Wing Using a Fuzzy-Based Metaheuristic
## 4 Numerical Experiment  
Aeroelastic design of an aircraft wing structure by fuzzy uncertainties in material property and transverse displacement on the wing is used for a design demonstration. The PSIBDO with fuzzy uncertainty is used to quantify epistemic uncertainty due to lack of knowledge of material properties and allowable transverse displacement. For value of material consider only the Young's modulus in aluminum only since it has a greater impact on the mass of aircraft wing than laminated carbon fiber materials. In this research, the Goland wing is used as a model for design demonstration in Figure 2[21]. The general geometry of the Goland wing composed of chord length and semi-span wing as 1.216 m and 6.096 m, respectively and wing thickness of \(\pm\)0.0508 m. The aircraft wing structure is made of aluminum ribs and spars, while the skins are made from three layers of laminated carbon fiber. The properties of both materials are presented in Table 1.  
The optimization problem for minimize mass of an aircraft wing subject to aeroelastic and structural constraints is expressed as:  
\[Minf(\mathbf{x})=\text{\it mass} \tag{11}\]  
Subject to \(-u_{max}+u_{al}\geq\ 0\)  
\[ \begin{array}{l}-V_{\mathit{fall}}+V_{f}\ \geq\ 0\\ -\eta_{\mathit{Lall}}+\eta_{L}\ \geq\ 0\\ \mathbf{x}^{L}\ \leq\ \mathbf{x}\ \leq\ \mathbf{x}^{L}\end{array} \]  
where \(\mathbf{x}\) is \(\mathbf{a}\) design variable vectors, \(x^{L}\) and \(x^{L}\) is lower and upper bounds, \(u_{max}\) is the maximum transverse displacement on the aircraft wing structure, while \(u_{al}\) is allowable transverse displacement. \(\eta_{L}\ \ \text{and}\ \eta_{\mathit{Lall}}\) are wing lift effectiveness and allowable lift effectiveness, respectively. \(V_{f}\) is a flutter speed and \(V_{f,al}\) is an allowable flutter speed. Aerodynamic properties, aeroelastic and structural constraints are set as \(\rho_{\text{air}}=1.2\) kg/m\({}^{3}\), free stream velocity = 40 m/s, \(u_{al}=0.1\) m, \(\eta_{\mathit{Lall}}\ \geq 0.9\) and \(V_{\mathit{fall}}\ \geq 180\) m/s. The Goland wing opens angle of attach for \(3^{\circ}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Aluminum** & \\ \hline
**Properties** & **Value** & **Unit** \\ \hline Young’s modulus (\(E\)) & \(70\times 10^{9}\) & Pa \\ Poisson’s ratio (\(\
u\)) & 0.3 & - \\ Density (\(\rho\)) & 2700 & kg/m\({}^{3}\) \\ \hline
**Carbon fiber** & \\ \hline
**Properties** & **Value** & **Unit** \\ \hline Young’s modulus (\(E_{11}\)) & \(1.49\times 10^{11}\) & Pa \\ Young’s modulus (\(E_{22}\)) & \(8.83\times 10^{9}\) & Pa \\ Shear modulus (\(G_{12}\)) & \(5.38\times 10^{9}\) & Pa \\ Shear modulus (\(G_{13}\)) & \(5.38\times 10^{9}\) & Pa \\ Shear modulus (\(G_{23}\)) & \(2.98\times 10^{9}\) & Pa \\ Longitudinal tensile strength (\(Sty_{11}\)) & \(500\times 10^{6}\) & Pa \\ Transverse tensile strength (\(Sty_{22}\)) & \(5\times 10^{6}\) & Pa \\ In-plane tensile strength (\(Sty_{12}\)) & \(35\times 10^{6}\) & Pa \\ Longitudinal compressive strength (\(Scy_{11}\)) & \(350\times 10^{6}\) & Pa \\ Transverse compressive strength (\(Scy_{22}\)) & \(75\times 10^{6}\) & Pa \\ In-plane compressive strength (\(Scy_{12}\)) & \(35\times 10^{6}\) & Pa \\ [PERSON]’s ratio (\(v_{12}\)) & 0.342 & - \\ Density (\(\rho\)) & 1800 & kg/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of aluminum and carbon fiber. Figure 2: Geometry model of a Goland wing and aerodynamic grid. Aeroelastic phenomena in this design analysis of the composite aircraft wing can be categorized in two groups, which are static and dynamic aeroelastics. The most significant static aeroelastic parameters are lift effectiveness and divergence speed, while dynamic aeroelastic phenomenon is flutter speed. The divergence is less important when compared with the flutter speed, so it not included in the optimization design problem. Lift effectiveness is the ratio of total wing lift when considering flexibility to the rigid wing counterpart.