# Aeroelastic Optimization Design of the Global Stiffness for a Joined Wing Aircraft
## 2 Materials and Methods  
### Beam Cross-Section Simplification  
As shown in Figure 1, the out-of-plane components of a joined wing aircraft tend to bend the wings about a tilted bending axis. To resist this, the wing's structural material must form a deep spar about this axis. This implies that the material must be concentrated near the upper leading edge and the lower trailing edge. As shown in Figure 2, to simulate the real material distribution of the beam section of a joined wing, three section shapes are simplified to facilitate calculation without losing rationality. Through a series of theoretical derivations, the best scheme for a joined wing is selected from the three section shapes. As shown in Figure 2a, section parameters \(X_{1}\)-\(X_{4}\) are used for the design parameters in each section of the beams. The parameter \(X_{1}\) represents the main diagonal flange size; \(X_{2}\) represents the secondary diagonal flange size; \(X_{3}\) represents the web thickness; \(X_{4}\) represents the skin thickness; and \(X_{5}\) and \(X_{6}\) represent the width and height of the wing box, respectively, and they are fixed in each beam section. The symbol \(\alpha\) represents the tilted angle, as shown in Figures 1 and 2. Figure 1: The structural layout, forces, and tilted bending axis of a three-dimensional joined-wing model. As shown in Figure 2a, the equations used to evaluate the principal inertia moments \(I_{z1}\) and \(I_{y1}\), the torsional stiffness coefficient \(I_{t1}\), and the inertial product \(I_{yz1}\) of the cross section are as follows:  
\[I_{z1}=\ \frac{1}{2}A_{t}X_{6}^{2}+\frac{X_{3}(-\overline{X}^{3}-2X_{6}^{3}+3X_{6 }\overline{X}^{2})}{6} \tag{1}\]  
\[I_{y1}=\frac{1}{2}A_{t}X_{5}^{2}+\frac{X_{4}(-\overline{X}^{3}-2X_{5}^{3}+3X_{5 }\overline{X}^{2})}{6} \tag{2}\]  
\[I_{t1}=\frac{4\Omega^{2}}{\oint\frac{\mathrm{d}\Omega}{\delta}}\approx\frac{2 X_{3}X_{4}(X_{5}X_{6})^{2}}{X_{4}X_{6}+X_{3}X_{5}} \tag{3}\]  
\[I_{yz1}=\frac{X_{5}X_{6}}{2}(A(X_{2})-A(X_{1})) \tag{4}\]  
where:  
\[A(X_{1})=X_{1}^{2},\quad A(X_{2})=X_{2}^{2},\quad A(X_{3})=X_{3}L_{w},\quad A (X_{4})=X_{4}L_{s}\]  
\[A_{t}=\sum\limits_{i=1}^{4}A(X_{i}),\quad L_{s}=X_{5}-\overline{X},\quad L_{w} =X_{6}-\overline{X},\quad\overline{X}=\frac{X_{1}+X_{2}}{2} \tag{5}\]  
In general, the flange size is larger than the size of the web thickness and skin thickness, and as mentioned earlier, to simulate the material distribution, the main diagonal flange size is larger than the secondary diagonal flange size shown in Figure 2a, so the relations of the cross section parameters are as in Equation (6):  
\[\max(X_{3},X_{4})<X_{2}<X_{1} \tag{6}\]  
The section area can be calculated as:  
\[S=2A_{t} \tag{7}\]  
According to the rotation axis formula of the moment of inertia in Equation (8), the moment of inertia of a beam section relative to the bending axis u\({}_{1}\) in Figure 2a can be derived as Equation (9):  
\[I_{u}=\frac{I_{z}+I_{y}}{2}+\frac{I_{z}-I_{y}}{2}\cos\ 2\alpha-I_{yz}\sin\ 2\alpha \tag{8}\]  
\[I_{z1}=\ \frac{1}{2}A_{t}X_{6}^{2}+\frac{X_{3}(-\overline{X}^{3}-2X_{6}^{3}+3X_{6 }\overline{X}^{2})}{6} \tag{1}\]0  
For the case of Figure 2b:  
\[I_{z1}=\ \frac{1}{2}A_{t}X_{6}^{2}+\frac{X_{3}(-\overline{X}^{3}-2X_{6}^{3}+3X_{6 }\overline{X}^{2})}{6} \tag{1}\]1  
\[I_{z1}=\ \frac{1}{2}A_{t}X_{6}^{2}+\frac{X_{3}(-\overline{X}^{3}-2X_{6}^{3}+3X_{6 }\overline{X}^{2})}{6} \tag{1}\]2 Simplified beam sections with a tilted bending axis. (**a**) Centrosymmetry beam section; (**b**) axial symmetry beam section; (**c**) same diagonal beam section. then  
\[I_{z1}=\ \frac{1}{2}A_{t}X_{6}^{2}+\frac{X_{3}(-\overline{X}^{3}-2X_{6}^{3}+3X_{6 }\overline{X}^{2})}{6} \tag{1}\]3  
For the case of Figure 2c, the four flange areas are equal.