2.2.1. Aircraft geometry

Many aircraft layouts are possible, but due to the mission requirements of ACC aircraft, the most common is the conventional layout with an unswept or low-sweep high-aspect-ratio wing and an aft tail at the end of a slender tail boom (Fig. 1). The main parameters that define the aircraft are the wingspan, b, wing chord, c, tail arm, l _ht, tail boom length, l _tb, wing planform area, S _wing, horizontal tail area, S _ht, and vertical tail area, S _vt. An orthogonal xyz coordinate system is selected with its origin placed at the leading edge of the wing root chord. The x-axis points towards the tail, the y-axis points to the right wing and the z-axis points upwards.

Figure 1. Typical ACC aircraft geometry.

Since different tail arms may be used during the design process, it is convenient to define some characteristic non-dimensional parameters based on a reference wing with given chord length c _wing and span b _wing. The ratio b _wing/c _wing is the aspect ratio A _wing. It is then possible to define two coefficients that are directly proportional to the reference wing aspect ratio, one for the tail arm and another for the tail boom length, in the form

(5) \begin{align}K_{ht}^{} = \left( {\frac{{{l_{ht}}}}{{{c_{wing}}}}} \right){A_{wing}}\end{align}

(6) \begin{align}{K_{tb}} = \left( {\frac{{{l_{tb}}}}{{{c_{wing}}}}} \right){A_{wing}}\end{align}

From Eq. (5) and Eq. (6), it is possible to obtain the tail arm and the tail boom length, respectively, for any sized wing from

(7) \begin{align}{l_{ht}} = K_{ht}^{}c\end{align}

(8) \begin{align}{l_{tb}} = K_{tb}^{}c\end{align}

where c is the actual wing mean chord. In general terms, the wing chord may be defined by its mean chord. If the chord of the wing is constant along the span and equal to the mean chord, it may be assumed that only one wing panel exists as in Fig. 2(a). In this case, the characteristics of the wing cross-section may be defined at the root of the wing and are constant spanwise.

Figure 2. Wing planform shapes: (a) constant-chord (single panel) wing example and (b) tapered (multiple panels) wing example.

However, in most ACC aircraft, the wings have taper. This implies a spanwise chord change, which might have a constant taper or a varying taper. In both cases, the wing is composed of panels (Fig. 2(b)). In this case, the sizing may be done for the first panel, with the characteristics of the remaining sections being determined based on the first panel size and local loads. In general, the sizing is done at the panel root, so the characteristics are constant throughout the panel.

2.2.2. Wing structural concepts and cross-section geometry

Based on the observation of the different wing structures used in ACC aircraft, three different wing structural concepts are considered in this study, namely load-bearing skin concept, D-box wing concept and circular tube spar wing concept. The concepts of these wing structures and the corresponding geometric and sizing parameters are shown in Fig. 3.

Figure 3. Structural wing concepts representation: (a) load-bearing skin wing, (b) D-box wing and (c) circular tube spar wing.

The load-bearing skin wing concept (Fig. 3(a)) represents a monocoque skin wing with a single spar. The complete skin is made of a sandwich with two faces of constant thickness t _{skin_face} each and a core of constant thickness t _{skin_core}. The two spar caps are a unidirectional laminate with cross-section area A _spar, while the spar web is also a composite sandwich with two faces of thickness t _{spar_face} each and a core of thickness t _{spar_core}. This configuration forms a two-cell closed-section beam.

The D-box wing concept (Fig. 3(b)) is divided into two distinct parts. The forward part, from the leading edge to the single spar, is similar to the previous concept and forms a closed section, the so-called D-box. The aft part is made of balsa wood ribs with thickness t _rib and lateral area A _rib spaced at constant intervals b _rib and a balsa wood trailing-edge stringer of cross-section area A _te. The complete wing is covered by a thin film of thickness t _film that closes the aft part and gives the wing its aft aerodynamic shape. This configuration forms a single-cell closed-section beam.

In the circular tube spar wing concept (Fig. 3(c)), the composite circular tube spar resists all wing loads. Its cross-section dimensions are wall thickness of t _tube, radius of r _tube and section area of A _tube = Πr _tube². The aerodynamic shape of the wing is given by balsa wood ribs and leading-edge and trailing-edge stringers with cross-section area of A _le and A _te, respectively. In this concept, the complete wing is also covered by a thin film of thickness t _film to close it and form the aerodynamic shape.

There are further cross-section parameters that are required for the full definition of the cross-section, namely the aerofoil’s chord, c, thickness-to-chord ratio, t/c, cross-section area, A _aerofoil, and perimeter P _aerofoil. In the load-bearing skin wing concept, the forward cell has perimeter p _I and cross-section area A _I while the aft cell has perimeter P _II and cross-section area A _II (Fig. 4(a)). Finally, the perimeter and cross-section area of the D-box are P _box and A _box, respectively (Fig. 4(b)).

Figure 4. Further geometric parameters for the composite sandwich skin concepts: (a) load-bearing skin wing and (b) D-box wing.

The aerofoil selected for the wing provides the geometry of the wing cross-section. Since different aerofoil sizes may be used in the wing during the design process, it is convenient to define some characteristic non-dimensional coefficients for the aerofoils based on a reference aerofoil with given chord length c _aerofoil, thickness-to-chord ratio (t/c)_aerofoil, cross-section area A _aerofoil and perimeter P _aerofoil. Based on these reference parameters, it is possible to define two coefficients. The first coefficient is associated with the cross-section area and is inversely proportional to the chord squared and to the thickness-to-chord ratio. It is defined by

(9) \begin{align}{K_A} = {{{A_{airfoil}}} \over {{{(t/c)}_{airfoil}}c_{airfoil}^2}}\end{align}

Assuming that the thickness-to-chord ratio varies only slightly with the aerofoil perimeter, the perimeter-associated coefficient is only a function of the chord, thus

(10) \begin{align}K_P^{} = \frac{{{P_{airfoil}}}}{{{c_{airfoil}}}}\end{align}

Based on the coefficients K _A and K _P, it is possible to obtain coefficients for any cross-section area or perimeter ratio between 0 and 1 based on the total cross-section area and perimeter, respectively, of any geometrically identical aerofoil. Thus, general area and perimeter coefficients may be used in the form

(11) \begin{align}K_{A,i}^{} = \left( {\frac{{{A_i}}}{{{A_{airfoil}}}}} \right){K_A}\end{align}

(12) \begin{align}K_{P,i}^{} = \left( {\frac{{{P_i}}}{{{P_{airfoil}}}}} \right){K_P}\end{align}

where i represents any element of the wing cross-section, such as a trailing-edge stringer or a D-box. If i is omitted, then the complete aerofoil is considered.

For the load-bearing skin wing concept, i in Eq. (11) and Eq. (12) represents either cell I or II. The ratios A _i/A _aerofoil depend on the x-position of the wing spar and A _I/A _aerofoil + A _II/A _aerofoil = 1. If A _I/A _aerofoil is specified, then

(13) \begin{align}\left( {\frac{{{A_{II}}}}{{{A_{airfoil}}}}} \right) = 1 - \left( {\frac{{{A_I}}}{{{A_{airfoil}}}}} \right)\end{align}

In the case of the D-box wing concept, there is only one cell, the D-box itself. Coefficients in Eq. (11) and Eq. (12) are used for the D-box by substituting i with box, as shown in Fig. 4(b), giving K _A,box and K _P,box. Also, the rib and the trailing-edge stringer require a geometric non-dimensional coefficient associated with the cross-section area. In these components, Eq. (7) is used by substituting i with le or te for the leading- and trailing-edge stringer, respectively, giving K _A,rib and K _A,te. Note that A _box/A _aerofoil + A _rib/A _aerofoil + A _te/A _aerofoil = 1. Given A _box/A _aerofoil and A _te/A _aerofoil, then

(14) \begin{align}\left( {\frac{{{A_{rib}}}}{{{A_{airfoil}}}}} \right) = 1 - \left( {\frac{{{A_{box}}}}{{{A_{airfoil}}}}} \right) - \left( {\frac{{{A_{te}}}}{{{A_{airfoil}}}}} \right)\end{align}

In the circular tube spar wing concept, it is necessary to use four geometric non-dimensional coefficients to account for different aerofoil sizes. The first coefficient is associated with the spar radius and is given by

(15) \begin{align}K_{tube}^{} = \frac{{{r_{tube}}}}{{{c_{airfoil}}}}\end{align}

The other three coefficients are associated with the rib and the leading-edge and trailing-edge stringer cross-section areas. Therefore, Eq. (11) can be used in this case by substituting i with rib, le or te for the rib or the leading- and trailing-edge stringers, respectively, giving K _A,rib, K _A,le and K _A,te. Note that A _tube/A _aerofoil + A _rib/A _aerofoil + A _rib/A _aerofoil + A _te/A _aerofoil = 1. Given A _le/A _aerofoil and A _te/A _aerofoil, then

(16) \begin{align}\left( {\frac{{{A_{rib}}}}{{{A_{airfoil}}}}} \right) = 1 - \left( {\frac{{{A_{le}}}}{{{A_{airfoil}}}}} \right) - \left( {\frac{{{A_{te}}}}{{{A_{airfoil}}}}} \right) - \frac{{\pi r_{tube}^2}}{A}\end{align}

where A is the cross-section area of the aerofoil under consideration.

From Eqs. (9) and (11) and Eqs. (10) and (12), it is possible to obtain the cross-section area and the perimeter, respectively, for any element of the cross-section in the form

(17) \begin{align}{A_i} = {K_{A,i}}(t/c){c^2}\end{align}

(18) \begin{align}{P_i} = K_{P,i}^{}c\end{align}

Another parameter that can be derived from the cross-section geometry is the position of the elastic centre. For simplicity, the following is assumed: the elastic centre coincides with the shear centre; the skin and spar web have the same thickness, Young’s modulus, and shear modulus; the cross-section does not deform under load; and the cross-section can be idealised to a combination of direct load-carrying booms at the spar caps and shear load-carrying skins. The derivation is based on the structural idealisation theory in Ref. (21).

For the load-bearing skin wing concept, the non-dimensional elastic centre position is

(19) \begin{align}{\bar{x}_e} = {\bar x_{spar}} - \frac{{2\left[ {\left( {\frac{{{K_{P,II}}}}{{t/c}} - 1} \right){K_{A,I}} - \left( {\frac{{{K_{P,I}}}}{{t/c}} - 1} \right){K_{A,iI}}} \right]}}{{\frac{{{K_{P,I}}{K_{P,II}}}}{{{{\left( {t/c} \right)}^2}}} - 1}}\end{align}

where ${\bar{x}_{sp}} = {x_{spar}}/c$ is the non-dimensional position of the spar, measured from the leading edge of the wing to the vertical centreline of the spar.

In the D-box wing concept, the non-dimensional elastic centre position is

(20) \begin{align}{\bar x}_{e} = {\bar {x}}_{spar} - {{2{K_{A,box}}\left( {t/c} \right)} \over {{K_{P,box}}}}\end{align}

and in the circular tube spar wing concept, the elastic centre position is just the position of the centre of the circle. Thus, ${\bar{x}_e} = {\bar{x}_{spar}}$.

In the following mass models, the parameter c refers to the wing mean geometric chord.

2.2.3. Load-bearing skin wing concept mass

The mass of each component of the wing is obtained by multiplying its volume by the corresponding material density, ${\rho}$. In a simple way, the individual volumes may be calculated from the cross-section area of each component multiplied by the span of the wing. Correction factors are added to take into account extra mass due to bonding of components, wing panel connections and supporting elements.

For the wing skin, the mass is given by

(21) \begin{align}m_{skin}^{} = {f_{sandwich}}\left( {2{t_{skin\_face}}{\rho _{skin\_face}} + {t_{skin\_core}}{\rho _{skin\_core}}} \right){K_P}cb\end{align}

where f _sandwich is the correction factor due to the resin absorption by the core material when it is bonded to the skin faces during the curing process. Depending on the core material, f _sandwich may take a value between 1.2 and 1.5. Highly porous core materials, such as EPS or PVC foams, should have larger values, whilst less porous materials, such as balsa wood, should have smaller values. This factor may also be increased slightly to account for wing surface painting. The skin core thickness is selected by the designer based on stiffness requirements.

The spar web mass may be obtained in a similar manner, thus

(22) \begin{align}m_{spar\_web} = {f_{sandwich}}\left( {2{t_{spar\_face}}{\rho _{spar\_face}} + {t_{spar\_core}}{\rho _{spar\_core}}} \right)\left(t/c\right)\!cb\end{align}

where the thickness of the spar core is defined by the designer considering stiffness requirements and (t/c)c is the spar height, which is equivalent to the aerofoil thickness.

The spar cap mass is given by

(23) \begin{align}m_{spar\_cap}^{} = 2{A_{spar}}b{\rho _{spar\_cap}}\end{align}

Finally, the total mass of the wing is

(24) \begin{align}{m_{wing}} = {f_{wing}}^{\left( {{n_{panels}} - 1} \right)}\left( {{m_{skin}} + {m_{spar\_web}} + {m_{spar\_cap}}} \right)\left( {1 + \frac{{{S_{winglet}}}}{{{S_{wing}}}}} \right)\end{align}

where f _wing is a correction factor to account for the connecting elements between the various wing panels and for other supporting structure (for holding servos, wires, control surfaces, flaps or access doors), and ${S_{winglet}}/{S_{wing}}$ is the winglet area ratio (winglet area divided by wing area). In the ACC, there is a requirement that the full aircraft must be disassembled and put in a transportation box of limited sized. To fulfil this requirement, the wing must be split into a number of panels, n _panels, which, depending on the design, may be as high as 6. Naturally, the value of f _wing is affected by the number of wing panel interface connections n _panels – 1 and structural arrangement, but average values of 1–1.2 are typical.

2.2.4. D-box wing concept mass

The wing mass model of the D-box wing concept is partly similar to the skin load-bearing wing as far as the spar is concerned. The differences lie in the skin, trailing-edge ribs and stringer and covering film.

The mass of the skin may be obtained from Eq. (21) by substituting the perimeter of the aerofoil by the perimeter of the D-box subtracted by the height of the spar, thus

(25) \begin{align}{m_{skin}} = {f_{sandwich}}(2{t_{skin\_face}}{\rho _{skin\_face}} + {t_{skin\_core}}{\rho _{skin\_core}})[{K_{p,box}} - (t/c)]cb\end{align}

Equation (22) and Eq. (23) may be used for the spar web and spar cap masses, respectively.

The ribs mass can be estimated by

(26) \begin{align}{m_{rib}} = {n_{rib}}{t_{rib}}{K_{A,rib}}(t/c){c^2}{\rho _{rib}}\end{align}

where n _rib is the number of ribs used in the wing design and can be estimated from b/b _rib.

The mass of the trailing-edge stringer is obtained from

(27) \begin{align}{m_{te}} = {K_{A,te}}(t/c){c^2}b{\rho _{te}}\end{align}

The final element in the wing is the covering film. Its mass may be estimated by

(28) \begin{align}m_{film}^{} = {t_{film}}{K_P}cb{\rho _{film}}\end{align}

Finally, the total mass of the wing is

(29) \begin{align}{m_{wing}} = {f_{wing}}^{\left( {{n_{panels}} - 1} \right)}\left( {{m_{skin}} + {m_{spar\_web}} + {m_{spar\_cap}} + {m_{rib}} + {m_{te}} + {m_{film}}} \right)\left( {1 + \frac{{{S_{winglet}}}}{{{S_{wing}}}}} \right)\end{align}

2.2.5. Circular tube wing concept mass

This wing concept has five components to account for in the mass model. The most important structural component is the spar tube. Its mass is calculated from

(30) \begin{align}m_{tube}^{} = \pi \left[ {r_{tube}^2 - {{\left( {{r_{tube}} - {t_{tube}}} \right)}^2}} \right]b{\rho _{tube}}\end{align}

where t _tube is the wall thickness of the tube and ${\rho}$_tube is the density of the material used in the tube.

The contribution of the ribs, trailing edge and covering film to the mass can be obtained from Eq. (23), Eq. (27) and Eq. (28), respectively.

The final component is the leading-edge stringer. Its mass representation is similar to the trailing-edge equation and is given by

(31) \begin{align}{m_{le}} = {K_{A,le}}(t/c){c^2}b{\rho _{le}}\end{align}

The total mass of the wing then becomes

(32) \begin{align}{m_{wing}} = {f_{wing}}^{\left( {{n_{panels}} - 1} \right)}\left( {{m_{tube}} + {m_{rib}} + {m_{te}} + {m_{le}} + {m_{film}}} \right)\left( {1 + \frac{{{S_{winglet}}}}{{{S_{wing}}}}} \right)\end{align}

2.2.6 Wing mass of tapered wings

In Sections 2.2.3 through 2.2.5, a rectangular wing has been considered. In this case, the wing chord and thickness-to-chord ratio are constant. However, some wings may have different chord lengths or thickness-to-chord ratios at different spanwise positions. The wing mass models previously presented may be extended to incorporate such aerofoil spanwise changes and produce a more accurate estimate. If the wing is divided into a given number of panels, then the mass of the wing may be computed from

(33) \begin{align}m_{wing}^{} = \sum\limits_{j = 1}^{{n_{panel}}} {{m_{panel,j}}} \end{align}

where n _panel is the number of panels and m _panel,j is the mass of the j ^th panel, which may be calculated with Eq. (24), Eq. (29) or Eq. (32), depending on the structural concept. In computing each panel mass, the parameters b, c and (t/c) must be substituted with b _j, c _j and (c/t)_j corresponding to span, representative chord length and representative thickness-to-chord ratio of the j ^th panel.

2.2.7 Tail mass

The tail is composed of the horizontal tail and the vertical tail. Usually, these components are small when compared with the wing, and the loads applied on them are an order of magnitude smaller than those applied on the wing. Therefore, a convenient way to estimate their mass is to scale the wing mass by the area ratio of each aerodynamic surface and apply a correction factor to account for the tail layout and structural concept. The mass of the horizontal and vertical tails can be estimated from

(34) \begin{align}m_{ht}^{} = {f_{ht}}\frac{{{S_{ht}}}}{{{S_{wing}}}}{m_{wing}}\end{align}

and

(35) \begin{align}m_{vt}^{} = {f_{vt}}\frac{{{S_{vt}}}}{{{S_{wing}}}}{m_{wing}}\end{align}

respectively, where f _ht and f _vt are the correction factors for the horizontal and vertical tail, respectively, which may be selected based on existing data for similar aircraft, S _ht and S _vt are the planform areas of the horizontal and vertical tails, respectively, and S _wing is the wing planform reference area. Typical values of ${f_{ht}}$ are in the range 0.2–0.8, and those of ${f_{vt}}$ are in the range 0.2–1.2.

2.2.8 Fuselage mass

The fuselage may be divided into two main components: the tail boom and the cargo bay. The determination of the tail boom mass is independent of the wing configuration. The tail boom mass is estimated from

(36) \begin{align}{m_{tb}} = {f_{tb}}\pi \left[ {r_{tb}^2 - {{\left( {{r_{tb}} - {t_{tb}}} \right)}^2}} \right]{l_{tb}}{\rho _{tb}}\end{align}

where r _tb is the outer radius of the tail boom, t _tb is its wall thickness, l _tb is its length and ${\rho}$_tb the material density used in the tail boom. The correction factor f _tb serves to correct the mass of the tail boom to provide the mass for the complete fuselage length from nose to tail. Typical values for this correction factor range from 10 to 30.

The cargo bay mass is determined from its surface area, S _cb, the wall thickness, t _cb, and the material density, $\rho_{cb}$, defined for this component in the form

(37) \begin{align}{m_{cb}} = {f_{cb}}{S_{cb}}\end{align}

where f _cb is a mass per unit area factor that takes into account the cargo bay skin and the internal support structure necessary to transmit all the loads from the payload to the fuselage and wing. This factor may have a value ranging from 0.4 to 1.5kg/m².

The fuselage mass is computed as the sum of the tail boom and cargo bay masses obtained earlier as

(38) \begin{align}{m_{fus}} = {m_{tb}} + {m_{cb}}\end{align}

2.2.9 Landing gear mass

The landing gear structure depends on the aircraft layout and landing gear type. Therefore, its mass is considered to be a function of the total mass in the form

(39) \begin{align}{m_{gear}} = {f_{gear}}m\end{align}

The correction factor f _gear is selected based on known data for similar aircraft. Typical values are 0.003–0.015.

2.2.10 Aircraft mass

To determine the total mass of the aircraft, it is necessary to know the mass of all components as described in the previous sections. The aircraft is divided into wing, horizontal and vertical tails, fuselage, landing gear, systems and payload. The systems and payload masses are defined directly by the designer, based on experience and/or the competition requirements. Usually, the systems comprise a motor and a propeller, an electronic speed controller and servo actuators, a battery for the motor and another for the servos and a radio receiver.

The empty mass is given by

(40) \begin{align}{m_{empty}} = {m_{wing}} + {m_{ht}} + {m_{vt}} + {m_{fus}} + {m_{gear}} + {m_{sys}}\end{align}

where m _sys is the mass of the systems.

Finally, the take-off mass of the aircraft is

(41) \begin{align}m = {m_{empty}} + {m_{pay}}\end{align}

where m _pay is the payload mass. It is possible to verify that the components’ mass depend on the take-off mass via the size of the structural components since the loads are functions of the aircraft mass, making the determination of the empty and total aircraft masses an iterative process.

In most situations, the maximum take-off mass is limited by the propulsion and aerodynamic characteristics of the aircraft. In that case, Eq. (41) is solved for the payload mass, given m and m _empty obtained from Eq. (40).

3.2.1 Loads

A representation of the flight wing loading and ground test wing loading is shown in Fig. 5. In the flight load condition, only wing lift is considered since the drag is usually small. The spanwise lift distribution is simplified into a uniformly distributed load of magnitude

(43) \begin{align}w_z^{} = \frac{{nW}}{b}\end{align}

where $n$ is the limit load factor and $W$ is the aircraft weight. Both bending moment and shear force have their maximum values at the wing root and their minimum value of zero at the tip. The flight shear force at a given position y along the semi-span is the negative of the integral of Eq. (43) from y to the tip (y = b/2) and is given by

(44) \begin{align}Q_z^{fl} = \frac{{nW}}{b}\left( {\frac{b}{2} - y} \right)\end{align}

Figure 5. Wing loading conditions: (a) spanwise lift distribution and (b) ground test loads.

The flight bending moment at a given position y along the semi-span is the integral of the shear force, thus

(45) \begin{align}M_x^{fl} = \frac{{nW}}{{2b}}{\left( {\frac{b}{2} - y} \right)^2}\end{align}

The flight torsion moment around the quarter chord line for an unswept wing is a function of the dynamic pressure, q, aerofoil pitching moment coefficient, C _m, and chord length. Assuming that the spanwise torsion moment distribution is uniform, then the torsion moment at a position y along the semi-span is

(46) \begin{align}M_y^{fl} = q\left| {{C_m}} \right|{c^2}\left( {\frac{b}{2} - y} \right)\end{align}

The dynamic pressure can be computed for the maximum level design flight speed.

Aeroelastic phenomena, such as divergence or flutter, may occur at higher speeds if the wing is not stiff enough in torsion. Flutter is a complex coupling between bending and torsion modes and is, therefore, not considered explicitly here. The torsion moment due to wing twist, $\theta$, is important to estimate the divergence speed. This moment may be approximated, for a single wing, by

(47) \begin{align}M_y^{div} = q\frac{{b{c^2}}}{2}\left( {{{\mathop x\limits^ - }_e} - \frac{1}{4}} \right)\frac{{d{C_L}}}{{d\alpha }}\theta \end{align}

where ${\bar{x}_e}$ is the non-dimensional position of the elastic centre relative to the wing leading edge, $d{C_L}/d\alpha $ is the lift curve slope of the wing and the aerodynamic centre is assumed to be at the quarter chord position. The elastic centre position needs to be estimated from the torsion-resisting cross-section, or Eq. (21) and Eq. (22) may be used. The dynamic pressure in this case may be computed for the maximum level design flight speed multiplied by a safety factor typically equal to 1.2.

The ground structural validation test consists in holding the aircraft by its wing tips and loading it with the flight payload. In this case, the bending moment is also maximum at the wing root and zero at the wing tip. However, the shear force is constant along the wingspan. The ground shear force and bending moment at a given position y along the semi-span, for a tip load of W/2, are respectively

(48) \begin{align}Q_z^{gd} = \frac{W}{2}\end{align}

(49) \begin{align}M_x^{gd} = \frac{W}{2}\left( {\frac{b}{2} - y} \right)\end{align}

The ground torsion moment is a function of the distance from the quarter chord line, x _c/4, to the point where the aircraft is held during the ground test, x _gd. Thus, the torsion moment is

(50) \begin{align}M_y^{gd} = \frac{W}{2}\left( {{x_{c/4}} - {x_{gd}}} \right)\end{align}

The critical sizing loads for strength requirements are obtained by selecting the maximum value from the flight and ground load conditions. Therefore, the shear force, bending moment and torsion moment are given by

(51) \begin{align}Q_z^{} = \max \left( {Q_z^{fl},Q_z^{gd}} \right)\end{align}

(52) \begin{align}M_x^{} = \max \left( {M_x^{fl},M_x^{gd}} \right)\end{align}

(53) \begin{align}M_y^{} = \max \left( {M_y^{fl},M_y^{gd}} \right)\end{align}

For stiffness sizing, only the flight loads are considered because they affect the aerodynamic shape, hence flight performance, and may give rise to undesired aerodynamic phenomena such as flutter or divergence. Flutter can be avoided by indirectly sizing for bending and torsional stiffness, while divergence can be accounted for by using $M_y^{div}$.

There are also flight loads on the tail boom, resulting mainly from the lift on the horizontal tail. The horizontal tail lift is obtained from the longitudinal moment balance and the need to manoeuvre the aircraft in pitch. Thus, the wing pitching moment must be balanced by the horizontal tail moment. Using Eq. (46) with y = 0 and knowing that the tail moment is given by the tail lift multiplied by the tail arm, this relationship leads to the value of the tail lift or the tail boom shear force in the form

(54) \begin{align}Q_z^{tb} = \frac{{q{C_m}b{c^2}}}{{{l_{ht}}}}\end{align}

Using Eq. (7) for l _ht, the above equation becomes

(55) \begin{align}Q_z^{tb} = \frac{{q{C_m}bc}}{{{K_{ht}}}}\end{align}

Since the above force is constant along the length of the tail boom, the bending moment at the root of the tail boom is obtained by multiplying the shear force with l _tb. Using Eq. (8) for l _tb, the bending moment becomes

(56) \begin{align}M_y^{tb} = \frac{{q{C_m}{K_{tb}}b{c^2}}}{{{K_{ht}}}}\end{align}

3.2.2 Deflections

Large deflections are only critical during flight because drastic geometric changes may alter the flight behaviour of the aircraft and compromise safety. Limiting wing tip deflection and twist during flight is important to guarantee both static stiffness and to prevent dynamic aeroelastic effects at higher air speeds. Also, limiting tail boom rotation at the horizontal tail position is important to avoid pitch trim problems and control problems at higher speeds.

Figure 6 illustrates wing tip deflection due to bending, wing tip twist due to torsion and tail boom rotation due to bending.

Figure 6. Wing and tail boom deflections: (a) wing tip deflection due to bending and (b) wing tip twist due to torsion and tail boom rotation due to bending.

3.3.1 Load-bearing skin wing concept

For this first case, the structure can be considered as being predominantly manufactured in carbon fibre composite with a light core, typically made of a foam or balsa wood, resulting in a sandwich configuration as exemplified in Fig. 3(a).

Some simplifications are assumed in the calculation of the required material to withstand the applied stresses. It is considered that only the spar caps resist bending loads (for both flight and ground test conditions). The parameter A _spar is sized taking into account as limiting factors the allowable direct stress and tip deflection. The load-bearing thickness of the spar web, 2t _{spar_face}, results from the sum of the amount of material required to withstand the shear stresses produced by shear forces and torsion moments. Finally, the load-bearing thickness of the skin, 2t _{skin_face}, is sized for torsion loads only, considering the allowable shear stress and tip twist angle. For the sake of simplicity, the thickness of the skin is assumed constant in both cells; this results in oversizing of the cell where the shear flow is smaller. Both skin and spar web core thicknesses are provided by the designer to ensure adequate stiffness to prevent structural instability or damage under aircraft ground handling. The thickness considered for mass calculations is selected from the highest value obtained for the different loading cases, so that resistance to the critical condition is ensured.

The following equations are presented as the basis for the calculation of the parameters A _spar, t _{skin_face} and t _{spar_face}. The sizing procedure follows classical thin-walled beam theory^{(Reference Megson21)}.

Spar cap cross-section area

Regarding the bending strength, the section properties are first derived. The second moment of area, ${I_{xx}}$, according to the axes system represented in Fig. 1, is given by

(57) \begin{align}I_{xx}^{} = \int\limits_A {{z^2}dA} \end{align}

where dA is an infinitesimal area of the arbitrary shape at a vertical distance z from the section centroid. Neglecting the geometry of the section and assuming that the entire bending-resisting area is concentrated at the two spar caps, the second moment of area can be simplified to

(58) \begin{align}{I_{xx}} = 2{A_{spar}}{(t/2)^2}\end{align}

where t is the aerofoil maximum thickness. In this case, it is assumed that the spar is located at the maximum thickness position. The aerofoil maximum thickness is given by

(59) \begin{align}t = (t/c)c\end{align}

Substituting Eq. (59) into Eq. (58), the expression for the second moment of area becomes

(60) \begin{align}I_{xx}^{} = \frac{{{A_{spar}}}}{2}{\left( {\frac{t}{c}} \right)^2}{c^2}\end{align}

For the calculation of the required spar cap area, the limiting criteria are the bending moment and the tip deflection.

The direct stress at the spar caps when the wing is subject to a bending moment M _x is given by

(61) \begin{align}{\sigma _y} = {{{M_x}(t/2)} \over {{I_{xx}}}}\end{align}

Substituting Eq. (59) and Eq. (60) into Eq. (61) and solving for A _spar gives

(62) \begin{align}A_{spar}^\sigma = {{{M_x}} \over {{\sigma _{y,\max }}(t/c)c}}\end{align}

where $\sigma_{y,max}$ is the maximum allowable direct stress in the spar caps and M _x is given by Eq. (52). At the root of the wing, M _x in Eq. (62) is calculated with y = 0, giving the minimum required spar cap area, which is the highest spanwise value.

Regarding the bending stiffness, it is necessary to determine the spar cap area which limits the maximum tip deflection during flight conditions. To simplify this analysis, considering that the cross-section of the spar caps may exhibit spanwise variation, it is assumed that the bending curvature occurs mainly close to the wing root. Figure 6 represents the bending deflection at position y ₁ and at the wing tip considering two different wing panels. The wing tip deflection can, thus, be approximated by

(63) \begin{align}{\delta _{tip}} = {\delta _1} + (b/2 - {y_1}){\beta _1} + {\delta _2}\end{align}

where $\delta_{1}$ is the deflection at the first panel tip, $\beta_{1}$ is the slope at the first panel tip and $\delta_{2}$ is the deflection for the remaining wing. The slope and the deflection of a cantilevered beam of length y are given by

(64) \begin{align}\beta = \frac{{{w_z}{y^3}}}{{6{E_y}{I_{xx}}}} + \frac{{{M_x}y}}{{{E_y}{I_{xx}}}}\end{align}

(65) \begin{align}\delta = \frac{{{w_z}{y^4}}}{{8{E_y}{I_{xx}}}} + \frac{{{M_x}{y^2}}}{{2{E_y}{I_{xx}}}}\end{align}

where E _y is the Young’s modulus and M _x is the bending moment at the tip of the beam.

Assuming that the spar cap cross-section area is proportional to the span squared, the second moment of area for the outboard part of the wing is assumed to be

(66) \begin{align}{I_{xx,2}} = {(1 - 2{y_1}/b)^2}{I_{xx,1}}\end{align}

Using Eq. (45) to determine the bending moment at position y = y ₁ and at y = b/2 and substituting into Eq. (64) and Eq. (65), noting that from Eq. (43) w _z = nW/b and using Eq. (66), the values for $\delta_{1},\beta_{1}$ and $\delta_{2}$ are computed as

(67) \begin{align}{\delta _1} = \frac{{nW}}{{4b{E_y}{I_{xx,1}}}}\left[ {{{\left( {\frac{b}{2}} \right)}^2}y_1^2 - 2\left( {\frac{b}{2}} \right)y_1^3 + \frac{3}{2}y_1^4} \right]\end{align}

(68) \begin{align}{\beta _1} = \frac{{nW}}{{2b{E_y}{I_{xx,1}}}}\left[ {{{\left( {\frac{b}{2}} \right)}^2}y_1^{} - 2\left( {\frac{b}{2}} \right)y_1^2 + \frac{4}{3}y_1^3} \right]\end{align}

(69) \begin{align}{\delta _2} = \frac{{nW}}{{8b{E_y}{I_{xx,2}}}}\left[ {{{\left( {\frac{b}{2}} \right)}^4} - 2{{\left( {\frac{b}{2}} \right)}^3}y_1^{} + {{\left( {\frac{b}{2}} \right)}^2}y_1^2} \right]\end{align}

Now, using Eq. (63) and substituting for I _xx,1 using Eq. (60), the wing tip deflection is obtained from

(70) \begin{align}{\delta _{tip}} = {{nW} \over {192b{E_y}{A_{spar}}{{(t/c)}^2}{c^2}}}\left( {3{b^4} + 24{b^3}{y_1} - 120{b^2}y_1^2 + 224by_1^3 - 112y_1^4} \right)\end{align}

Then, the cross-section area of the spar cap necessary to sustain a given maximum wing tip deflection $\delta_{tip,max}$ is given by

(71) \begin{align}A_{spar}^\delta = {{nW} \over {192b{E_y}{\delta _{tip,\max }}{{(t/c)}^2}{c^2}}}\left( {3{b^4} + 24{b^3}{y_1} - 120{b^2}y_1^2 + 224by_1^3 - 112y_1^4} \right)\end{align}

The value of $\delta$_{tip_max} is specified by the designer to provide adequate bending stiffness to avoid large wing shape changes in dihedral, which would affect lift distribution and latero-directional stability and may result in the onset of the flutter vibration mode. Typical upper values for $\delta$_{tip_max}/b are 0.1, though slower aircraft may have higher ratios provided the vertical tail area is compatible with the increased wing dihedral.

The final spar cap cross-section area is the maximum value among those calculated for the bending moment, Eq. (62), and for the maximum allowable wing tip deflection, Eq. (71), and is given by the expression

(72) \begin{align}A_{spar}^{} = \max \left( {A_{spar}^\sigma ,A_{spar}^\delta } \right)\end{align}

Skin thickness

The aerofoil cross-section is divided into two cells, as shown in Fig. 4. It is assumed that the skin and the spar web together support the applied torsion moment while only the spar web supports the shear loads. Therefore, assuming for the time being that the spar web supports only torsion, the following expressions are used to represent the torsion moment M _y and twist rate $d\theta/dy$, respectively:

(73) \begin{align}M_y^{} = \sum\limits_{R = 1}^2 {2{A_R}{q_R}} \end{align}

(74) \begin{align}{\left( {\frac{{d\theta }}{{dy}}} \right)_R} = \frac{1}{{2{A_R}G}}\oint {{q_R}\frac{{ds}}{t}} \quad ,\quad R = 1,2\end{align}

where A _R and q _R are the area and shear flow of the R ^th cell, respectively, G is the shear modulus, ds is the infinitesimal distance around the cell wall and t is the cell wall thickness. Based on Eq. (73) and Eq. (74) and knowing that the shear flow is defined as 2t _{skin_face}$\tau$, since initially the spar web thickness is assumed to be equal to the skin thickness, the three-equation system is defined as

(75) \begin{align}\left.\begin{array}{c} {{d\theta } \over {dy}} = {1 \over {2{A_I}G}}{\rm{[}}{p_I}{\tau _I} - (t/c)c{\tau _{II}}] \\[9pt] {{d\theta } \over {dy}} = {1 \over {2{A_{II}}G}}{\rm{[}} - (t/c)c{\tau _I} + {p_{II}}{\tau _{II}}] \\[9pt] {{M_y} = 4{t_{skin\_face}}({A_1}{\tau _1} + {A_{II}}{\tau _{II}})} \end{array}\right\}\end{align}

where $\tau$_I and $\tau$_II are the shear stresses in cell I and II, respectively. The unknowns in Eq. (75) are the two shear stresses in the cells and the skin face thickness required to ensure that a maximum specified twist angle is achieved at the wing tip.

To solve the system of equations in Eq. (75), the following geometric parameters are defined, using Eq. (17) and Eq. (18):

(76) \begin{align}{C_1} = \frac{{{K_{P,II}} + \frac{{{K_{A,II}}}}{{{K_{A,I}}}}\left( {\frac{t}{c}} \right)}}{{\frac{{{K_{A,II}}}}{{{K_{A,I}}}}{K_{P,I}} + \left( {\frac{t}{c}} \right)}}\end{align}

(77) \begin{align}{C_2} = \frac{1}{{4\left( {{K_{A,I}}{C_1} + {K_{A,II}}} \right)\left( {t/c} \right){c^2}}}\end{align}

Assuming that the rate of twist is the same for cell I and cell II, the solution of the system of Eqs. (75) is

(78) \begin{align}\left.\begin{array}{c} {\tau _I} = {{{C_1}{C_2}{M_y}} \over {{t_{skin\_face}}}} \\[10pt] {\tau _{II}} = {{{C_2}{M_y}} \over {{t_{skin\_face}}}} \\[10pt] {{d\theta } \over {dy}} = {{[{K_{P,I}}{C_1} - (t/c)]{C_2}M_y^{fl}} \over {2{K_{A,I}}(t/c)cG{t_{skin\_face}}}} \end{array} \right\}\end{align}

Solving each of the above equations for the skin face thickness, using Eq. (46) for the torsion moment in the third equation and integrating it from root, y = 0, to tip, y = b/2, gives the following three values:

(79) \begin{align}t_{skin\_face}^I = \frac{{{C_1}{C_2}{M_y}}}{{{\tau _{\max }}}}\end{align}

(80) \begin{align}t_{skin\_face}^{II} = \frac{{{C_2}{M_y}}}{{{\tau _{\max }}}}\end{align}

(81) \begin{align}t_{skin\_face}^\theta = \frac{{\left[ {{K_{P,I}}{C_1} - \left( {t/c} \right)} \right]{C_2}bM_y^{fl}}}{{8{K_{A,I}}\left( {t/c} \right)cG{\theta _{max}}}}\end{align}

where $\tau_{max}$ is the maximum allowable shear stress in the sandwich faces and $\theta_{max}$ is the maximum allowable wing tip twist angle. The torsion moment M _y is given by Eq. (53), while the torsion moment in the twist case is given by Eq. (46) with y = 0. At the root of the wing, M _y in Eqs (79) through (81) are calculated with y = 0. The value of $\theta$_max is specified by the designer to provide adequate torsion stiffness to avoid wing shape variations, which would degrade the aerodynamic performance or allow the onset of flutter oscillations. Typical values for $\theta$_max are around 2° at the level flight maximum speed. Slower aircraft may have slightly higher values.

At the divergence condition, the aerodynamic torsion moment given by Eq. (47) must be equal to the structural torsion moment due to twist, for any arbitrary twist angle, $\theta$. Using the last of Eqs. (78), substituting $M_y^{fl}$ with $M_y^{div}$ and integrating over the semi-span from root to tip gives the condition

(82) \begin{align}\frac{{2GJ}}{b}\theta - M_y^{div} = 0\end{align}

since $d\theta /dy = M_y^{div}/\left( {GJ} \right),$ and where $M_y^{div}$ is a linear function of $\theta$. Using the last of Eqs. (78) to obtain GJ, and solving Eq. (82) for t _{skin_face} gives

(83) \begin{align}t_{skin\_face}^{div} = {{[{K_{p,1}}{C_1} - (t/c)]{c_2}b} \over {4{K_{A,I}}(t/c)cG}}q{{b{c^2}} \over 2}({\bar x_e} - {1 \over 4}){{d{c_L}} \over {d\alpha }}\end{align}

where ${\bar x_e}$ may be obtained from Eq. (20).

The final skin face thickness is selected from the maximum value among those determined to withstand the torsion moment in cell I (Eq. (79)) and cell II (Eq. (80)), to guarantee the allowable wing tip twist, Eq. (81), and to prevent divergence, Eq. (83), and is obtained from

(84) \begin{align}{t_{skin\_face}} = \max \left( {t_{skin\_face}^I,t_{skin\_face}^{II},t_{skin\_face}^\theta ,t_{skin\_face}^{div}} \right)\end{align}

Spar web thickness

The maximum spar web shear stress due to a shear force Q _z must also be considered. Its maximum value occurs at the neutral axis and is calculated using the expression

(85) \begin{align}\tau = \frac{{3{Q_z}}}{{2{A_{spar\_face}}}}\end{align}

where A _{spar_face} is the spar web resisting cross-section area, given by 2t _{spar_face}(t/c)c. Substituting this into Eq. (85) and solving for the web face thickness gives

(86) \begin{align}t_{spar\_face}^\tau = {{3{Q_Z}} \over {4{\tau _{\max }}(t/c)c}}\end{align}

where $\tau_{max}$ is the maximum allowable shear stress in the spar web. The spar web thickness due to the shear force is obtained by substituting Eq. (51) into Eq. (86). At the root of the wing, Q _z in Eq. (86) is calculated with y = 0, giving the minimum required spar web thickness, which is the highest spanwise value.

The final web face thickness is the sum of the value required to withstand the shear stress given by Eq. (86) and the thickness required to withstand the torsion, obtained in Eq. (84). The web face thickness is, thus, given by

(87) \begin{align}t_{spar\_face}^{} = t_{spar\_face}^\tau + t_{skin\_face}^{}\end{align}

The designer needs to select the spar cap areas and sandwich face thicknesses from commercially available materials provided they are greater than or equal to the values obtained in the above equations, taking into consideration also handling loads during ground and transportation operations.

3.3.2 D-box wing concept

This second configuration consists of a D-box structure from the leading edge until the spar. As before, a sandwich structure made of carbon fibre composite sandwich is used, as shown in Fig. 3(b). The rear section of the aerofoil has ribs to transmit loads from the thin covering film to the spar and to guarantee that the aerofoil shape is maintained. A reinforcement at the trailing edge made of balsa wood maintains the shape of the trailing edge and provides stiffness and tension to the covering film.

The following equations are presented as the basis for the calculation of parameters A _spar, t _{skin_face} and t _{spar_face}. As before, the sizing procedure follows classical thin-walled beam theory^{(Reference Megson21)}.

Skin thickness

Again, there is a need for the structure to withstand the torsion moment and a limiting wing tip twist angle. Since only one cell exists in this configuration, Eq. (73) reduces to

(88) \begin{align}M_y^{} = 2{A_{box}}{q_{box}}\end{align}

where q _box is the shear flow around the D-box. Using the definition of cross-section area from Eq. (17), and the definition of shear flow 2t _{skin_face}$\tau$, Eq. (88) can be solved for the sandwich skin face thickness as

(89) \begin{align}t_{skin\_face}^T = {{{M_y}} \over {4{K_{A,box}}(t/c){c^2}{\tau _{\max }}}}\end{align}

where $\tau_{max}$ is the maximum allowable shear stress in the sandwich faces and the torsion moment M _y is given by Eq. (53). At the root of the wing, M _y is calculated with y = 0.

For the twist case, according to the axes represented in Fig. 1, the twist rate for a single cell due to the aerodynamic torsion moment, which here is assumed constant along the semi-span, is given by

(90) \begin{align}\frac{{d\theta }}{{dy}} = \frac{{M_y^{fl}}}{{GJ}}\end{align}

where GJ is the torsion rigidity given by

(91) \begin{align}GJ = {{4A_{box}^2} \over {\oint {ds/(2G{t_{skin\_face}})} }}\end{align}

where, for a constant thickness and constant shear modulus skin, the integral of ds is the D-box perimeter, P _box, obtained from Eq. (18). Substituting Eq. (91) into Eq. (90) and solving the above equation for the skin face thickness and integrating it from root to tip gives

(92) \begin{align}t_{skin\_face}^\theta = \frac{{{K_{P,box}}bM_y^{fl}}}{{32K_{A,box}^2{{\left( {t/c} \right)}^2}{c^3}G{\theta _{max}}}}\end{align}

where the torsion moment, $M_y^{fl}$, is given by Eq. (46) with y = 0.

At the divergence condition, the aerodynamic torsion moment given by Eq. (47) must be equal to the structural torsion moment due to twist, for any arbitrary twist angle, $\theta$. Using the same approach as in the load-bearing skin wing configuration case, where, in this case, GJ is given by Eq. (91) with the integral of ds being the perimeter of the D-box, the t _{skin_face} value is obtained from

(93) \begin{align}t_{skin\_face}^{div} = {{{K_{P,box}}b} \over {16K_{A,box}^2{{(t/c)}^2}{c^2}G}}q{{b{c^2}} \over 2}({\bar x_e} - {1 \over 4}){{d{C_L}} \over {d\alpha }}\end{align}

where $\bar{x}_{e}$ may be obtained from Eq. (21).

The final skin face thickness is the maximum value among those determined to resist the torsion moment, Eq. (89), that to guarantee the maximum wing tip twist, Eq. (91), and that for the divergence condition, Eq. (93), and is given by

(94) \begin{align}{t_{skin\_face}} = \max \left( {t_{skin\_face}^T,t_{skin\_face}^\theta ,t_{skin\_face}^{div}} \right)\end{align}

Wing rib thickness

Wing ribs in the aft portion of the wing need to transmit the aerodynamic loads from the thin film skin to the spar. The ribs work essentially as small beams that need to withstand shear and bending loads.

The simplified chordwise lift distribution^{(Reference Raymer1)} presented in Fig. 7 is considered to derive the loads on the wing ribs. This approximation assumes a uniform chordwise load distribution from the leading edge to 15% of the chord with value $\omega$ and a linear distribution from $\omega$ at this position to zero at the trailing edge.

Figure 7. Real and approximate aerofoil lift distribution^{(Reference Raymer1)}.

The chordwise load distribution, p(x), is then given by

(95) \begin{align}p(x) = \left\{ \begin{array}{cc} \omega & \quad {0 \le x \le 0.15c} \\[5pt] {\omega (1 - x/c)/0.85} & \quad {0.15c \lt x \le c} \end{array}\right.\end{align}

Assuming for simplicity that the spanwise load distribution is uniform, the total lift of the wing can be obtained by integrating the chordwise load distribution such that

(96) \begin{align}nW = b\int_0^c {p\left( x \right)} dx\end{align}

Substituting for p(x) from Eq. (95) in Eq. (96) and solving for $\omega$ gives

(97) \begin{align}\omega = \frac{{40}}{{23}}\frac{{nW}}{{bc}}\end{align}

The maximum shear force, Q _z, at the rib/spar interface is determined by integrating the distributed load given by Eq. (95) from the trailing edge to the spar and multiplying by the span, resulting in the following equation:

(98) \begin{align}{Q_z} = \frac{{400}}{{391}}nW{\left( {\frac{{{c_{rib}}}}{c}} \right)^2}\end{align}

where c _rib is the wing rib chord. Using Eq. (85) with the wing rib cross-section area t _rib(t/c)c, the shear stress at the rib/spar interface is

(99) \begin{align}\tau = {3 \over 2}{{{Q_Z}} \over {{n_{rib}}{t_{rib}}(t/c)c}}\end{align}

where n _rib is the number of ribs. Substituting the shear force from Eq. (98) into Eq. (99) and solving for the total wing rib thickness, n _ribt _rib, given a maximum allowable shear stress, $\tau_{max}$, in the rib gives

(100) \begin{align}{n_{rib}}t_{rib}^\tau = {{600} \over {391}}{{nW{{({c_{rib}}/c)}^2}} \over {{\tau _{\max }}(t/c)c}}\end{align}

The maximum bending moment, M _y, at the rib$/$spar interfaces is determined by integrating twice the distributed load given by Eq. (95) from the trailing edge to the spar and multiplying by the span, resulting in the following equation:

(101) \begin{align}{M_y} = \frac{{400}}{{1173}}nWc{\left( {\frac{{{c_{rib}}}}{c}} \right)^3}\end{align}

Using Eq. (61) with the wing rib cross-section second moment of area t _rib(t/c)³c ³/12, the maximum direct stress at the rib/spar interface is

(102) \begin{align}{\sigma _x} = {{{M_y}{{(t/c)c} \over 2}} \over {{n_{tib}}{t_{rib}}{{{{(t/c)}^3}{c^3}} \over {12}}}}\end{align}

Substituting the bending moment from Eq. (101) into Eq. (102) and solving for the total wing rib thickness, given a maximum allowable direct stress, $\sigma_{max}$, in the rib gives

(103) \begin{align}{n_{tib}}t_{rib}^\sigma = {{2400} \over {1173}}{{nWc{{({c_{rib}}/c)}^3}} \over {{\sigma _{\max }}{{(t/c)}^2}{c^2}}}\end{align}

The total rib thickness is obtained from the maximum of Eq. (100) and Eq. (103), thus

(104) \begin{align}{n_{rib}}t_{rib}^{} = \max \left( {{n_{rib}}t_{rib}^\tau ,{n_{rib}}t_{rib}^\sigma } \right)\end{align}

The designer needs to select the spar cap areas, sandwich face thicknesses and rib thickness from commercially available materials provided they are greater than or equal to the values obtained in the above equations, taking into consideration also handling loads during ground and transportation operations.

3.3.3 Tube spar wing concept

In this final configuration, illustrated in Fig. 3(c), the wing is composed by a circular thin-walled section spar, with ribs to transmit the stresses and support the thin covering film, which provides the desired aerofoil shape, with reinforcements in the leading and trailing edges made of balsa wood.

Tube spar thickness

Contrary to the presentation above, where the loads were divided by the different wing parts, in this situation, the required tube spar thickness is calculated to resist bending, shear and torsion. The final thickness is the sum of all individual thicknesses obtained for the above-mentioned loads.

The second moment of area is given by the following simplified expression for thin-walled tubes:

(105) \begin{align}{I_{xx}} = \pi r_{tube}^3{t_{tube}}\end{align}

where t _tube is the tube thickness.

As before, the spanwise lift distribution is assumed uniform for the calculation of the bending moment. The required thickness is determined by the critical case when comparing the results for the bending moment (in flight and ground test) and the tip deflection.

The direct stress at the farthest point from the tube neutral axis is

(106) \begin{align}\sigma _y^{} = \frac{{{M_x}{r_{tube}}}}{{{I_{xx}}}}\end{align}

Substituting Eq. (105) and Eq. (15) into Eq. (106) and solving for the circular tube spar thickness, t _tube, gives

(107) \begin{align}t_{tube}^\sigma = \frac{{{M_x}}}{{{\sigma _{y,\max }}\pi K_{tube}^2{c^2}}}\end{align}

where $\sigma_{y,max}$ is the maximum allowable direct stress in the spar and M _x is given by Eq. (52). At the root of the wing, M _x in Eq. (59) is calculated with y = 0, giving the minimum required tube spar thickness, which is the highest spanwise value.

Regarding the bending stiffness, it is necessary to determine the tube spar thickness that limits the maximum tip deflection during flight conditions. As above, it is assumed that the bending curvature occurs mainly close to the wing root. Assuming also that the tube spar thickness is proportional to the span squared, the thickness for the outboard part of the wing is assumed to be

(108) \begin{align}{t_{tube,2}} = {(1 - 2{y_1}/b)^2}{t_{tube,1}}\end{align}

Using the approach described from Eqs (63) through Eq. (69) and taking into account the thickness variation from Eq. (108), the wing tip deflection is approximated as

(109) \begin{align}{\delta _{tip}} = \frac{{nW}}{{384b{E_y}\pi K_{tube}^3{c^3}{t_{tube}}}}\left( {3{b^4} + 24{b^3}y_1^{} - 120{b^2}y_1^2 + 224by_1^3 - 112y_1^4} \right)\end{align}

Then, the thickness of the tube spar necessary to sustain a given maximum wing tip deflection $\delta_{tip,max}$ is given by

(110) \begin{align}t_{tube}^\delta = \frac{{nW}}{{384b{E_y}{\delta _{tip,\max }}\pi K_{tube}^3{c^3}}}\left( {3{b^4} + 24{b^3}y_1^{} - 120{b^2}y_1^2 + 224by_1^3 - 112y_1^4} \right)\end{align}

The expression for the shear stress is given by Eq. (82), where the area is replaced by the tube cross-section solid area:

(111) \begin{align}{A_{tube}} = 2\pi r_{tube}^{}{t_{tube}}\end{align}

Substituting the expression in Eq. (85) and solving for the tube spar thickness gives

(112) \begin{align}t_{tube}^\tau = \frac{{3{Q_z}}}{{4{\tau _{\max }}\pi K_{tube}^{}c}}\end{align}

where $\tau_{max}$ is the maximum allowable shear stress in the spar. The spar web thickness due to the shear force is obtained by substituting Eq. (51) into Eq. (112). At the root of the wing, Q _z in Eq. (112) is calculated with y = 0, giving the minimum required spar web thickness, which is the highest spanwise value.

The expressions used for the torsion load are identical to those for the D-box wing configuration. The circle area of the tube is given by

(113) \begin{align}A = \pi r_{tube}^2\end{align}

Using Eq. (88) for the torsion moment, substituting the area with Eq. (113) and defining the shear flow as t _tube$\tau$, the required thickness to resist the torsion moment is obtained as

(114) \begin{align}t_{tube}^T = \frac{{M_y^{}}}{{2\pi K_{tube}^2{c^2}{\tau _{\max }}}}\end{align}

where $\tau_{max}$ is the maximum allowable shear stress in the tube spar and the torsion moment M _y is given by Eq. (53). At the root of the wing, M _y is calculated with y = 0.

For the twist case, Eq. (90) and Eq. (91) can be used to obtain the twist angle. Substituting Eq. (113) into Eq. (91) and using Eq. (15) gives the torsion rigidity as

(115) \begin{align}GJ = 4\pi K_{tube}^3{c^3}G{t_{tube}}\end{align}

Now, substituting Eq. (115) into Eq. (90), integrating from root to tip and solving for the tube spar thickness gives

(116) \begin{align}t_{tube}^\theta = \frac{{bM_y^{fl}}}{{16\pi K_{tube}^3{c^3}G{\theta _{max}}}}\end{align}

where the torsion moment, $M_y^{fl}$, is given by Eq. (46) with y = 0.

For the divergence case, considering the derivation presented for the previous structural concepts, Eq. (116) can be adapted to give

(117) \begin{align} t_{tube}^{div} = \frac{b}{{8\pi K_{tube}^3{c^3}G}}q\frac{{b{c^2}}}{2}\left( {{{\mathop x\limits^ - }_{spar}} - \frac{1}{4}} \right)\frac{{d{C_L}}}{{d\alpha }}\end{align}

Often the centre of the tube spar is located at the quarter chord position. In that case, the aerodynamic and elastic centres are coincident, and Eq. (117) becomes unnecessary.

The final tube spar thickness is selected from the sum of all individual thicknesses required to withstand the bending moment, Eq. (107) and Eq. (110), the shear force, Eq. (112), and the torsion moment, Eq. (114), Eq. (116) and Eq. (117). For each stress, the maximum value, among those calculated for each limiting criterion, is considered. The expression for the tube spar thickness is thus

(118) \begin{align}{t_{tube}} = \max \left( {t_{tube}^\sigma ,t_{tube}^\delta } \right) + t_{tube}^\tau + \max \left( {t_{tube}^T,t_{tube}^\theta ,t_{tube}^{div}} \right)\end{align}