Wing weight estimation in conceptual design: a method for strut-braced wings considering static aeroelastic effects

Abstract

This paper presents a method for the wing weight estimation of strut-braced wing aircraft in conceptual design. The method is simple to implement while still capturing important effects for early design estimates. Static aeroelastic loads, aeroelastic divergence and aileron reversal criteria are calculated directly with small matrices suitable for implementation in spreadsheet software. Maneuver, gust and ground cases are considered. A direct non-iterative method is used for the strut and wing internal loads calculation. The wing and strut load-carrying structures are sized with analytical box-beam equations for strength, buckling and fatigue criteria. Aluminum or composite laminates can be considered. Semi-empirical methods are presented for non-optimal mass components and the secondary structure. The aeroelastic effects and strut reaction estimations are compared for a wide range of design parameters with Nastran validating the proposed method. The weight estimations are verified with conventional aircraft data and strut-braced wing studies available in the literature, showing good accuracy. Design trade studies are presented illustrating typical applications of the method. A potential to reduce the wing mass in about 18 % or to increase the aspect ratio from 10 to 16 compared to a cantilever wing is identified.

Appendix: Inclusion of strut effect in the static aeroelastic equations

The static aeroelastic equations for the cantilever wing are extended to include the strut effect. The objective is to determine the correction matrices $[K_E]$ and $[K_F]$ in Eq. (3) accounting for the strut torsion and vertical reaction.

The torsion at each section due to the distributed lift and pitching moment loading plus the torsion due to the strut streamwise torsion moment $T_{st}$ and vertical force $S_{st}$ (see Fig. 2) is:

$$\begin{aligned}&q[E]\{cc_l\} + q[F]\{c^2c_m\} \nonumber \\&\quad +\, [C^{\theta \theta }]\{1_{st}\}(T_{st}+e_{st}S_{st})+[C^{\theta z}]\{1_{st}\}S_{st} = \{\theta \} \end{aligned}$$

(30)

where $\{1_{st}\}$ is a column vector with 1.0 at the wing section connected to the strut and zeros at the other elements. Note that the equation above is defined with aerodynamic loadings $\{cc_l\}$, $\{c^2c_m\}$ and displacements $\{\theta \}$ at the wing sections and not at the bays.

To keep the equations similar to the cantilever wing case, it is of interest to express $T_{st}$ and $S_{st}$ as functions of the aerodynamic loadings $\{cc_l\}$ and $\{c^2c_m\}$. This is achieved by writing the boundary conditions of equal torsion angle and vertical displacement at the LRA in the wing section connected to the strut:

$$\begin{aligned} (T_{st}+e_{st}S_{st})C_{ss}^{\theta \theta } + S_{st}C_{ss}^{\theta z} + A = c_TT_{st} \end{aligned}$$

(31)

$$\begin{aligned} S_{st}C_{ss}^{zz} + (T_{st}+e_{st}S_{st})C_{ss}^{\theta z} + A' = c_Z S_{st} - e_{st}c_TT_{st} \end{aligned}$$

(32)

where $C_{ss}^{\theta \theta }$, $C_{ss}^{\theta z}$ and $C_{ss}^{zz}$ are the wing structural influence coefficients at the wing section connected to the strut due to loadings at the same section. They are extracted directly from the matrices $[C^{\theta \theta }]$, $[C^{\theta z}]$ and $[C^{zz}]$ at the row and column of the section attached to the strut. $c_T$ and $c_Z$ are the torsional and vertical flexibility coefficients of the complete strut, see Eqs. (39) and (40). A and $A'$ are, respectively, the twist and vertical displacement at the wing LRA in the section attached to the strut due to aerodynamic loading for a cantilever wing:

$$\begin{aligned} A&= q\{1_{st}\}^T([E]\{cc_l\} + [F]\{c^2c_m\}) \nonumber \\ A'&= q\{1_{st}\}^T([E']\{cc_l\} + [F']\{c^2c_m\}) \end{aligned}$$

(33)

where $[E']$ and $[F']$ are similar to [E] and [F] but represent the vertical displacement z due to loading:

$$\begin{aligned} [E'] &= ([C^{zz}] + [C^{\theta z}][e])[\Delta y] \\ [F'] &= [C^{\theta z}][\Delta y] \end{aligned}$$

(34)

Equations (31) and (32) are fundamental in this development. The terms in the left-hand side provide the twist, Eq. (31), and displacement, Eq. (32), at the LRA in the section attached to the strut of a cantilever wing subjected to the strut reactions $T_{st}$ and $S_{st}$ and aerodynamic loadings $\{cc_l\}$ and $\{c^2c_m\}$. The right-hand side terms provide the streamwise twist, Eq. (31), and vertical displacement, Eq. (32), at the strut alone due to the loads $T_{st}$ and $S_{st}$ at one strut end. We have, therefore, a system with two equations and two unknowns for the strut reactions. Solving this linear system of equations for $T_{st}$ and $S_{st}$ we get:

$$\begin{aligned} \begin{array}{ll} &T_{st} = (-A\cdot B_{22} + A'\cdot B_{12})/B \\ &S_{st} = (-A'\cdot B_{11} + A\cdot B_{21})/B \\ \end{array} \end{aligned}$$

(35)

where,

• $B_{11} = C_{ss}^{\theta \theta } - c_{T}$

• $B_{12} = e_{st}C_{ss}^{\theta \theta } + C_{ss}^{\theta z}$

• $B_{21} = C_{ss}^{\theta z} + c_T e_{st}$

• $B_{22} = C_{ss}^{zz} + e_{st}C_{ss}^{\theta z} - c_{Z}$

• $B = B_{11}B_{22} - B_{12}B_{21}$

In Eq. (35) we have the strut reactions explicitly as functions of the aerodynamic loadings $\{cc_l\}$ and $\{c^2c_m\}$ (in the terms A and $A'$).

Inserting Eq. (33) in Eq. (35) and substituting for $T_{st}$ and $S_{st}$ in Eq. (30) we get finally:

$$\begin{aligned} q([E]+[K_E])\{cc_l\} + q([F]+[K_F])\{c^2c_m\} = \{\theta \} \end{aligned}$$

(36)

where,

$$\begin{aligned}[K_E]& = [C^{\theta \theta }][1_{st}]\left( [E]\tfrac{(-B_{22}+B_{21}e_{st})}{B} +[E']\tfrac{(B_{12}-B_{11}e_{st})}{B}\right) \nonumber \\&\quad + [C^{\theta z}][1_{st}]\left( [E]\tfrac{B_{21}}{B} +[E']\tfrac{-B_{11}}{B}\right) \end{aligned}$$

(37)

$$\begin{aligned}[K_F]&=\, [C^{\theta \theta }][1_{st}]\left( [F]\tfrac{(-B_{22}+B_{21}e_{st})}{B} +[F']\tfrac{(B_{12}-B_{11}e_{st})}{B}\right) \nonumber \\&\quad +\,[C^{\theta z}][1_{st}]\left( [F]\tfrac{B_{21}}{B} +[F']\tfrac{-B_{11}}{B}\right) \end{aligned}$$

(38)

where $[1_{st}] = \{1_{st}\}\{1_{st}\}^T$.

These are the final correction matrices accounting for the strut effect used in the method implementation (Sect. 2.2). Note that in the equations above [E] can be exchanged by $[E_I]$ from Eq. (4) to account for inertia relief effects.

The only remaining terms are the strut flexibility coefficients $c_T$ and $c_Z$. They are obtained after coordinate transformations of the strut flexibility from the strut coordinate system to the global coordinate system and are given by:

$$\begin{aligned} c_T&= -L_{st}/\left( EI_{yy,st}\cos ^2\Lambda _{st}\sin ^2\phi _{st}+EI_{xx ,st}\sin ^2\Lambda _{st}\right. \nonumber \\&\quad +\left. GJ_{st}\cos ^2\Lambda _{st}\cos ^2\phi _{st}\right) \end{aligned}$$

(39)

$$\begin{aligned} c_Z = -L_{st}/A_{st}E_{st} \end{aligned}$$

(40)

where $EI_{yy,st}$ is the strut in-plane bending stiffness, $EI_{xx,st}$ is the strut bending stiffness and $E_{st}$ is the strut material longitudinal modulus.