Wings and tails

Together with the normal-plane coordinate and chord relations, the shear and bending moment are related to the corresponding airplane-axes quantities and to the sweep angle $\Lambda$ as follows.

\[\begin{aligned} {\cal S}_{\scriptscriptstyle \perp}& = & {\cal S} %\label{Sperp} \\ {\cal M}_{\scriptscriptstyle \perp}& = & {\cal M}/ \cos \Lambda %\label{Mperp} \end{aligned}\]

The box is assumed to be the only structurally-significant element, with the slats, flaps, and spoilers (if any), represented only by added weight. It is convenient to define all dimensions as ratios with the local normal-plane chord $c_{\scriptscriptstyle \perp}$.

\[\begin{aligned} \bar{h} &\!\equiv\!& \frac{h_{\rm wbox}}{c_{\scriptscriptstyle \perp}} \\ \bar{w} &\!\equiv\!& \frac{w_{\rm wbox}}{c_{\scriptscriptstyle \perp}} \\ \bar{t}_{\rm cap}&\!\equiv\!& \frac{t_{\rm cap}}{c_{\scriptscriptstyle \perp}} \\ \bar{t}_{\rm web}&\!\equiv\!& \frac{t_{\rm web}}{c_{\scriptscriptstyle \perp}} \end{aligned}\]

The maximum height $h_{\rm wbox}$ at the box center corresponds to the airfoil thickness, so that $\bar{h}$ is the usual "$t/c$" airfoil thickness ratio. The height is assumed to taper off quadratically to a fraction $r_h$ at the webs, so that the local height $h {\scriptstyle (\xi)}$ is $\begin{aligned} h {\scriptstyle (\xi)}& = & h_{\rm wbox}\left[ \: 1 - (1\!-\!r_h) \xi^2 \: \right] \end{aligned}$

where $\xi = -1 \ldots 1$ runs chordwise over the sparbox extent. Typical metal wings and airfoils have $\bar{w} \simeq 0.5$, $r_h \simeq 0.75$, although these are left as input parameters. For evaluating areas and approximating the bending inertia, it's useful to define the simple average and r.m.s. average normalized box heights.

\[\begin{aligned} \bar{h}_{\rm avg}& = & \frac{1}{c_{\scriptscriptstyle \perp}} \int_0^1 h {\scriptstyle (\xi)}\; {\rm d}\xi \;=\; \bar{h} \left[ \: 1 - \frac{1}{3}(1\!-\!r_h) \, \right] \\ \bar{h}_{\rm rms}^2 & = & \frac{1}{c_{\scriptscriptstyle \perp}^2} \int_0^1 h^2 {\scriptstyle (\xi)}\; {\rm d}\xi \;=\; \bar{h}^2 \left[ \: 1 - \frac{2}{3}(1\!-\!r_h) + \frac{1}{5} (1\!-\!r_h)^2 \, \right] \end{aligned}\]

The areas and the bending and torsion inertias, all normalized by the normal chord, can now be determined.

\[\begin{aligned} \bar{A}_{\rm fuel}&\!\equiv\!& \frac{A_{\rm fuel}}{c_{\scriptscriptstyle \perp}^2} \;=\; (\bar{w} - 2 \bar{t}_{\rm web})(\bar{h}_{\rm avg}- 2 \bar{t}_{\rm cap}) \\ \bar{A}_{\rm cap}&\!\equiv\!& \frac{A_{\rm cap}}{c_{\scriptscriptstyle \perp}^2} \;=\; 2 \, \bar{t}_{\rm cap}\bar{w} \\ \bar{A}_{\rm web}&\!\equiv\!& \frac{A_{\rm web}}{c_{\scriptscriptstyle \perp}^2} \;=\; 2 \, \bar{t}_{\rm web}\, r_h \, \bar{h} \\ \bar{I}_{\rm cap}& \simeq & \frac{I_{\rm cap}}{c_{\scriptscriptstyle \perp}^4} \;=\; \frac{\bar{w}}{12} \left[ \bar{h}_{\rm rms}^3 - (\bar{h}_{\rm rms}\!\!-\!2\bar{t}_{\rm cap})^3 \right] \\ \bar{I}_{\rm web}&\!\equiv\!& \frac{I_{\rm web}}{c_{\scriptscriptstyle \perp}^4} \;=\; \frac{\bar{t}_{\rm web}\, r_h^3 \, \bar{h}^3}{6} \; \ll \; \bar{I}_{\rm cap} \hspace{2em} \mathrm{(typically)} \\ G\bar{J} &\!\equiv\!& \frac{4 (\bar{w} - \bar{t}_{\rm web})^2 (\bar{h}_{\rm avg}- \bar{t}_{\rm cap})^2} { \displaystyle 2 \frac{ r_h \bar{h} \!-\! \bar{t}_{\rm cap}}{G_{\rm web}\bar{t}_{\rm web}} \:+\: 2 \frac{ \bar{w} \!-\! \bar{t}_{\rm web}}{G_{\rm cap}\bar{t}_{\rm cap}} } \end{aligned}\]