Geometry

The wing or tail surface is assumed to have a two-piece linear planform with constant sweep $\Lambda$, shown in the figure below. The inner and outer surface planforms are defined in terms of the center chord $c_o$ and the inner and outer taper ratios.

\[\begin{aligned} \lambda_s & = & c_s/c_o \\ \lambda_t & = & c_t/c_o \end{aligned}\]

Similarly, the spanwise dimensions are defined in terms of the span $b$ and the normalized spanwise coordinate $\eta$.

\[\begin{aligned} \eta & = & 2y / b \\ \eta_o & = & b_o / b \\ \eta_s & = & b_s/b \end{aligned}\]

For generality, the wing center box width $b_o$ is assumed to be different from the fuselage width to allow possibly strongly non-circular fuselage cross-sections. It will also be different for the tail surfaces. A planform break inner span $b_s$ is defined, where possibly also a strut or engine is attached. Setting $b_s \!=\! b_o$ and $c_s \!=\! c_o$ will recover a single-taper surface.

It's convenient to define the piecewise-linear normalized chord function

\[\begin{aligned} \frac{c {\scriptstyle (\eta)}}{c_o} \; \equiv \; C {\scriptstyle (\eta \, ; \, \eta_o,\eta_s, \lambda_s,\lambda_t)} & = & \left\{ \begin{array}{lcl} \; 1 & , & 0 < \eta < \eta_o \\[0.5em] \displaystyle \; 1 \, + (\lambda_s\!-\!1 \,) \frac{\eta \!-\! \eta_o}{\eta_s\!-\!\eta_o} & , & \eta_o < \eta < \eta_s \\[0.25em] \displaystyle \lambda_s + (\lambda_t\!-\!\lambda_s) \frac{\eta \!-\! \eta_s}{1\!-\!\eta_s} & , & \eta_s < \eta < 1 \end{array} \right. %\label{ceta} \end{aligned}\]

The following integrals will be useful for area, volume, shear, and moment calculations.

\[\begin{aligned} \int_0^{\eta_o} C \:\: {\rm d}\eta & = \eta_o \\ \int_{\eta_o}^{\eta_s} C \:\: {\rm d}\eta & = \frac{1}{2} ( 1\!+\!\lambda_s)(\eta_s\!-\!\eta_o) \\ \int_{\eta_s}^1 C \:\: {\rm d}\eta & = \frac{1}{2} (\lambda_s\!+\!\lambda_t)(1\!-\!\eta_s) \\[0.25em] \int_0^{\eta_o} C^2\:\: {\rm d}\eta & = \eta_o \\ \int_{\eta_o}^{\eta_s} C^2\:\: {\rm d}\eta & = \frac{1}{3} ( 1\!+\!\lambda_s\!+\!\lambda_s^2)(\eta_s\!-\!\eta_o) \\ \int_{\eta_s}^1 C^2\:\: {\rm d}\eta & = \frac{1}{3} (\lambda_s^2\!+\!\lambda_s \lambda_t\!+\!\lambda_t^2)(1\!-\!\eta_s) \\ % \int_{\eta_o}^{\eta_s} C \: (\eta\!-\!\eta_o) \:\: {\rm d}\eta & = \frac{1}{6} ( 1\!+\!2\lambda_s)(\eta_s\!-\!\eta_o)^2 \\ \int_{\eta_s}^1 C \: (\eta\!-\!\eta_s) \:\: {\rm d}\eta & = \frac{1}{6} (\lambda_s\!+\!2\lambda_t)(1\!-\!\eta_s)^2 \\ \int_{\eta_o}^{\eta_s} C^2 \: (\eta\!-\!\eta_o) \:\: {\rm d}\eta & = \frac{1}{12} ( 1\!+\!2\lambda_s\!+\!3\lambda_s^2)(\eta_s\!-\!\eta_o)^2 \\ \int_{\eta_s}^1 C^2 \: (\eta\!-\!\eta_s) \:\: {\rm d}\eta & = \frac{1}{12} (\lambda_s^2\!+\!2\lambda_s\lambda_t\!+\!3\lambda_t^2)(1\!-\!\eta_s)^2 %\label{Cint2} \end{aligned}\]

The surface area $S$ is defined as the exposed surface area plus the fuselage carryover area.

\[\begin{aligned} S & = 2 \int_0^{b/2} \! c \:\: {\rm d}y \;=\; c_o \, b \, K_c %\label{Sdef} \\ \mathrm{where} \hspace{3em} K_c & = \int_0^1 C \:\: {\rm d}\eta \;=\; \eta_o + {\textstyle \frac{1}{2}}( 1 \!+\!\lambda_s)(\eta_s\!-\!\eta_o) + {\textstyle \frac{1}{2}}(\lambda_s\!+\!\lambda_t )(1 \!-\!\eta_s) \end{aligned}\]

The aspect ratio is then defined in the usual way. This will also allow relating the root chord to the span and the taper ratios.

\[\begin{aligned} {A\hspace{-0.5ex}R}& = & \frac{b^2}{S} %\label{ARdef} \end{aligned}\]

It is also useful to define the wing's mean aerodynamic chord $c_{\rm ma}$ and area-centroid offset ${\scriptstyle \Delta}x_{\rm wing}$ from the center axis.

\[\begin{aligned} \frac{c_{\rm ma}}{c_o} &\!= \frac{1}{c_o} \frac{2}{S} \int_0^{b/2} \! c^2 \; {\rm d}y \;=\; \frac{K_{cc}}{K_c} \\ {\scriptstyle \Delta}x_{\rm wing} &\!= \frac{2}{S} \int_{b_o/2}^{b/2} \! c \, (y \!-\! y_o) \tan\Lambda \; {\rm d}y \;=\; \frac{K_{cx}}{K_c} \, b \: \tan\Lambda \\ x_{\rm wing}& = x_{\rm wbox}\,+\, {\scriptstyle \Delta}x_{\rm wing} \\[0.25em] \mathrm{where} \hspace{1em} K_{cc} &\!= \int_0^1 C^2 \:\: {\rm d}\eta \nonumber \\ &\!= \eta_o + \frac{1}{3} ( 1 \!+\!\lambda_s\!+\!\lambda_s^2)(\eta_s\!-\!\eta_o) + \frac{1}{3} (\lambda_s^2\!+\!\lambda_s \lambda_t\!+\!\lambda_t^2)(1\!-\!\eta_s) \hspace{3em} \\[0.25em] K_{cx} &\!= \int_{\eta_o}^1 C \: (\eta\!-\!\eta_o) \:\: {\rm d}\eta \nonumber \\ &\!= \frac{1}{12} ( 1 \!+\!2\lambda_s)(\eta_s\!-\!\eta_o)^2 + \frac{1}{12} (\lambda_s\!+\!2\lambda_t)(1\!-\!\eta_s)^2 + \frac{1}{4} (\lambda_s\!+\!\lambda_t)(1\!-\!\eta_s)(\eta_s\!-\!\eta_o) \hspace{3em} \end{aligned}\]

The wing area centroid is used in the fuselage bending load calculations as described earlier.