Biplanes, Triplanes & Flying Wings

J. Gordon Leishman

62 Biplanes, Triplanes & Flying Wings

Introduction

The history of aircraft design includes many configurations that differ substantially from the familiar single-wing airplane with a fuselage and tail. Among the most important are biplanes, triplanes, and flying wings. Each represents a different answer to the same basic design problem of how to produce sufficient lift, provide adequate stability and control, keep the structure light and strong, and achieve acceptable aerodynamic efficiency. Their differences show that aircraft are shaped by aerodynamics, structures, materials, control requirements, mission needs, and the state of technology at the time.

Biplanes and triplanes were dominant during the early decades of aviation. Their arrangement of two or three vertically separated wings allowed designers to obtain a large wing area without requiring a long, heavy, highly stressed span. This compact arrangement was especially valuable when aircraft structures were made from wood, fabric, wires, and struts, and when fully cantilevered monoplane wings were difficult to build with sufficient strength and stiffness. The Royal Aircraft Factory SE5a, shown in Figure 1, is representative of the successful WWI biplane layout. The penalty was aerodynamic interference between the wings, together with the drag of struts, bracing wires, and structural junctions. Nevertheless, biplanes and triplanes were practical and effective solutions for their time, and biplanes continue to appear in specialized roles such as aerobatic and recreational airplanes.

Early biplanes, such as the SE5a, used vertically separated wings to obtain large wing area and structural stiffness within a compact span.

Flying wings represent a very different design philosophy. Instead of dividing the airplane into separate lifting, fuselage, and tail components, a flying wing integrates most of these functions into the wing itself. The B-2 Spirit, shown in Figure 2, is a modern example of this tailless arrangement. This configuration can reduce parasitic drag and improve structural and payload integration, but it also creates difficult design problems. Without a conventional horizontal or vertical tail, the wing must provide lift, trim, longitudinal stability, directional stability, roll control, and yaw control. These requirements strongly influence the choice of planform, sweep, twist, airfoil section, control-surface arrangement, and spanwise loading.

The B-2 Spirit integrates the lifting surface, structure, payload volume, and control surfaces into a single tailless configuration.

The finite-wing theory developed in Chapter 38 provides the foundation for analyzing these configurations. Concepts such as circulation, downwash, induced angle of attack, and induced drag remain essential, but they must be applied with additional care. For multiple lifting surfaces, such as biplanes and triplanes, each wing can be modeled as a lifting line that interacts aerodynamically with the others. The mutual downwash changes the local effective angle of attack, alters the spanwise distribution of circulation, and modifies the induced drag relative to isolated wings. These interference effects depend on gap, stagger, decalage, chord ratio, relative loading, and wing spacing. Therefore, comparisons with monoplanes must explicitly state the reference span, reference area, and total lift. In a flying wing, the spanwise loading must satisfy induced-drag, trim, stability, control, bending-moment, stall-behavior, and aeroelastic constraints.

These configurations provide useful examples of the conditional nature of aerodynamic optima. An elliptical spanwise loading minimizes induced drag for a wing of fixed lift and span, but that result alone does not determine the best aircraft. A biplane may accept interference drag to gain structural compactness. A triplane may accept even more interference to obtain a large wing area and maneuverability within a short span. A flying wing may depart from elliptical loading to obtain favorable yawing characteristics, reduced bending moment, or acceptable trim and control. The best configuration is the one that satisfies the full set of aerodynamic, structural, stability, control, and mission requirements.

Learning Objectives

Be able to identify the primary geometric parameters of biplanes and triplanes, including gap, stagger, and decalage.
Understand how to extend classical finite-wing theory to biplanes, triplanes, and flying wings.
Explain how aerodynamic interference between multiple wings affects lift, downwash, lift-curve slope, and induced drag.
Interpret Munk’s stagger theorem and its implications for biplane and multiplane induced drag.
Compare elliptical and bell-shaped spanwise loadings in terms of induced drag, root bending moment, and flying-wing design tradeoffs.
Describe the aerodynamic design challenges of flying wings, including trim, stability, control, twist, and airfoil selection.

Biplanes

Biplanes use two main wings arranged one above the other. This arrangement was especially important during the early development of aviation because it allowed designers to obtain a large lifting area without requiring a long, heavy, flexible, and highly stressed wing. The configuration also provided convenient structural depth for struts, wires, and bracing, which was important when aircraft were built mainly from wood, fabric, and other relatively flexible structural members.

The aerodynamics of a biplane are notable because the two wings do not behave as independent lifting surfaces. Each wing produces its own bound circulation and trailing vortex system, and each modifies the flow field seen by the other. The result is mutual aerodynamic interference, which affects the lift curve slope, spanwise loading, downwash, induced drag, and stall behavior of the complete wing system.

A simple biplane consists of two wings spaced apart by a vertical distance called the gap, ${h}$ , as shown in Figure 3. Each wing can be assumed to have a span ${b}$ and a chord ${c}$ , although the upper and lower wings may have different spans and chords. The wings may also be horizontally offset by a geometric parameter called the stagger, denoted by $St$ . Another geometric parameter, the decalage angle, is the angular offset between the incidences (pitch angles) of the upper and lower wings, which affects the lift balance between them. If the upper wing has a pitch angle ${\theta_u}$ and the lower wing has a pitch angle ${\theta_l}$ , then the decalage angle, ${\Delta \theta_d}$ , is defined as

(1) $\begin{equation*} \Delta \theta_d = \theta_u - \theta_l \end{equation*}$

Besides the usual wing parameters, the geometry of a biplane is described by three additional geometric parameters, namely, gap, stagger, and decalage.

History

Biplanes were used for many different purposes during the early development of aviation, from trainers and reconnaissance aircraft to fighters and aerobatic airplanes. The Curtiss JN-4 “Jenny” was used primarily as a trainer during WWI. Its relatively small wing gap and limited stagger produced more aerodynamic interference than in some higher-performance biplanes, but its docile handling qualities made it well-suited for training. Its modest performance reflected the broader compromise in biplane design between structural compactness, stability, ease of handling, and aerodynamic efficiency.

The Fokker D.VII was a notably successful WWI fighter and was unusual in having relatively little stagger but a comparatively large wing gap. This arrangement helped reduce aerodynamic interference between the wings and contributed to its strong performance. The D.VII was among the more aerodynamically refined biplanes of its era, illustrating how careful choices of gap, stagger, airfoil section, and structural arrangement could significantly improve a biplane’s efficiency.

A modern example of a biplane is the Pitts Special, a compact, high-performance aerobatic aircraft. Unlike early military and training biplanes, the Pitts Special prioritizes extreme maneuverability and rapid control response. It uses low-aspect-ratio wings, positive stagger, and symmetric airfoils, which help produce rapid roll rates and good performance in both upright and inverted flight. In such designs, minimum induced drag is not the primary objective; instead, the aerodynamic layout is optimized for agility, maneuverability, and precise control during high-load maneuvers. The Pitts Special shown in Figure 4 demonstrates why the biplane configuration remains useful in specialized roles where compactness and maneuvering capability are more important than cruise efficiency.

Single-seat Pitts S-1S (left) and a modified two-seat S-2AE (right).

Other biplanes, such as the Christen (Aviat) Eagle and the Waco YMF-5, further demonstrate the continued relevance of the biplane layout in specialized applications. The Christen Eagle, like the Pitts Special, emphasizes aerobatic performance and structural compactness. The Waco YMF-5, in contrast, is a modern reproduction of a 1930s-era sport biplane, emphasizing nostalgic appeal and stable touring qualities rather than maximum aerodynamic efficiency. These examples show that although monoplanes dominate mainstream civil and military aviation, biplanes remain attractive where compactness, handling qualities, aerobatic capability, or historical character are important design goals.

Aerodynamics of Biplanes

In a biplane configuration, each wing generates a trailing-vortex system that induces downwash at the other wing. This mutual interference changes the effective angle of attack, the spanwise distribution of circulation, the total lift produced at a given geometric angle of attack, and the induced drag. The effect becomes stronger as the vertical gap between the wings decreases. In a real biplane, the aerodynamic behavior is also influenced by stagger, decalage, pitch angles, planform, taper, twist, chord distribution, airfoil section, and the relative loading of the two wings.

The overall aerodynamic efficiency of a practical biplane is usually lower than that of an efficient monoplane because interference is accompanied by low aspect ratio, struts, bracing wires, and additional wetted area. The inviscid induced-drag comparison alone, however, must be made carefully because the result depends on the reference monoplane, the chosen span and area definitions, and the assumed lift sharing between the wings. In the classical biplane theory presented by von Mises,^[1] as discussed next, the induced drag is written as the sum of two self-induced drag terms and a mutual-interference term. The mutual term is represented by an interference coefficient that depends on the gap and span ratio of the two lifting systems. Stagger is a separate ideal-induced-drag issue, addressed by Munk’s stagger theorem.

A complete biplane theory is a coupled aerodynamic problem. The upper and lower wings usually carry different lift and spanwise loading. The circulation on each wing depends on its own geometry, pitch angles, and twist, as well as on the induced velocity field produced by the other wing. This coupling is central to understanding historical biplane performance and the trade-off between structural compactness and aerodynamic efficiency.

Classic Theory

The classical theory of biplane induced drag is usually associated with Ludwig Prandtl and the subsequent presentation by von Mises. In this theory, the biplane is represented by two lifting lines with spanwise-distributed circulation, as shown in Figure 5. The two wings carry total lift values $L_1$ and $L_2$ and have spans $b_1$ and $b_2$ . The vertical gap between the lifting lines is $h$ . Each wing leaves behind it a trailing-vortex sheet, and each sheet induces downwash on the other wing. These mutual downwash effects give rise to the induced-drag terms $D_{12}$ and $D_{21}$ . In the basic von Mises biplane problem treated here, the lifting lines lie in a plane normal to the freestream. Stagger is a separate, ideal-induced-drag issue and is addressed later under the section on Munk’s stagger theorem.

Biplane represented by two lifting systems with spanwise circulation distributions $\Gamma_1(y_1)$ and $\Gamma_2(y_2)$ .

For a single finite wing with elliptical spanwise loading, the induced angle of attack is uniform across the span and is

(2) $\begin{equation*} \alpha_i = \frac{C_L}{\pi \, A\!R} \end{equation*}$

Using $L=q_\infty S C_L$ and $A\!R=b^2/S$ , this result may also be written as

(3) $\begin{equation*} \alpha_i = \frac{L}{\pi \, q_\infty \, b^2} \end{equation*}$

Therefore, the induced drag of the single wing is

(4) $\begin{equation*} D_i = L\alpha_i = \frac{L^2}{\pi \, q_\infty \, b^2} \end{equation*}$

The biplane problem differs from the monoplane because each wing is affected by the trailing-vortex system of the other wing. Let $\Gamma_1(y_1)$ and $\Gamma_2(y_2)$ be the circulation distributions on wings 1 and 2. The trailing-vortex sheet shed by wing 2 has strength proportional to $-d\Gamma_2/dy_2$ . Using the usual lifting-line sign convention, the downwash induced at a point $y_1$ on wing 1 by the trailing-vortex sheet of wing 2 may be written as

(5) $\begin{equation*} w_{12}(y_1) = -\frac{1}{4\pi} \int_{-b_2/2}^{b_2/2} \frac{y_2-y_1} {h^2+(y_2-y_1)^2} \left( \frac{d\Gamma_2}{dy_2} \right) dy_2 \end{equation*}$

This equation relates the spanwise variation of circulation on one wing to the induced velocity at the other wing. Integrating by parts and noting that the circulation vanishes at the tips for an elliptic loading gives the equivalent form

(6) $\begin{equation*} w_{12}(y_1) = \frac{1}{4\pi} \int_{-b_2/2}^{b_2/2} \Gamma_2(y_2) \frac{ h^2-(y_2-y_1)^2 } { \bigg( h^2+(y_2-y_1)^2 \bigg)^2 } \, dy_2 \end{equation*}$

The mutual induced-drag contribution is obtained by multiplying this induced angle by the lift distribution on wing 1. Because

(7) $\begin{equation*} dL_1 = \varrho_\infty V_\infty \Gamma_1(y_1) \, dy_1 \end{equation*}$

and

(8) $\begin{equation*} \alpha_{i,12} = \frac{w_{12}}{V_\infty} \end{equation*}$

the mutual drag term is

(9) $\begin{equation*} D_{12} = \varrho_\infty \int_{-b_1/2}^{b_1/2} \Gamma_1(y_1) w_{12}(y_1) \, dy_1 \end{equation*}$

Substituting the expression for $w_{12}$ gives

(10) $\begin{equation*} D_{12} = \frac{\varrho_\infty}{4\pi} \int_{-b_1/2}^{b_1/2} \int_{-b_2/2}^{b_2/2} \Gamma_1(y_1)\Gamma_2(y_2) \left( \frac{ h^2-(y_2-y_1)^2 } { \bigg( h^2+(y_2-y_1)^2 \bigg)^2 } \right) dy_2 \, dy_1 \end{equation*}$

This is the mutual induced-drag term. It is the part of the drag caused by the downwash of wing 2 acting on the lift distribution of wing 1. By reciprocity, the corresponding term for wing 1 acting on wing 2 has the same value.

For elliptically loaded wings, the circulation distribution on a wing of span $b$ and lift $L$ is

(11) $\begin{equation*} \Gamma(y) = \frac{4L}{\pi \varrho_\infty V_\infty b} \bigg( 1-\left(\frac{2y}{b}\right)^2 \bigg)^{1/2} \end{equation*}$

Substitution of the two elliptic circulation distributions into the mutual-drag integral reduces the mutual term to the von Mises form

(12) $\begin{equation*} D_{12} = \frac{\sigma L_1L_2} {\pi \, q_\infty \, b_1 b_2} \end{equation*}$

where $\sigma$ is the biplane interference coefficient. Therefore, the coefficient $\sigma$ is the nondimensional interference coefficient obtained from the mutual downwash integral. With

(13) $\begin{equation*} \eta_1 = \frac{2y_1}{b_1} \qquad \text{and} \qquad \eta_2 = \frac{2y_2}{b_2} \end{equation*}$

one corresponding nondimensional form is

(14) $\begin{equation*} \sigma = \frac{b_1 b_2}{2\pi^2 h^2} \int_{-1}^{1} \int_{-1}^{1} \bigg( (1-\eta_1^2)(1-\eta_2^2) \bigg)^{1/2} \left( \dfrac{ 1-\left( \dfrac{b_2\eta_2-b_1\eta_1}{2h} \right)^2 } { \bigg( 1+\left( \dfrac{b_2\eta_2-b_1\eta_1}{2h} \right)^2 \bigg)^2 } \right) d\eta_2 \, d\eta_1 \end{equation*}$

This expression shows explicitly where the gap $h$ and the span ratio enter the biplane interference theory. In von Mises’ presentation, the resulting values of $\sigma$ are given as interference curves using the gap ratio

(15) $\begin{equation*} \frac{2h}{b_1+b_2} \end{equation*}$

and the span ratio

(16) $\begin{equation*} \mu = \frac{b_2}{b_1} \end{equation*}$

as the independent parameters.

The two self-induced drag terms have the ordinary monoplane form, i.e.,

(17) $\begin{equation*} D_{11} = \frac{L_1^2}{\pi \, q_\infty \, b_1^2} \qquad \text{and} \qquad D_{22} = \frac{L_2^2}{\pi \, q_\infty \, b_2^2} \end{equation*}$

The total biplane induced drag is the sum of the two self-induced terms and the two equal mutual terms,

(18) $\begin{equation*} D_{\text{bi}} = D_{11} + 2D_{12} + D_{22} \end{equation*}$

Therefore,

(19) $\begin{equation*} D_{\text{bi}} = \frac{1}{\pi \, q_\infty} \left( \frac{L_1^2}{b_1^2} + 2\sigma \frac{L_1L_2}{b_1 b_2} + \frac{L_2^2}{b_2^2} \right) \end{equation*}$

This is the von Mises biplane induced-drag formula. The first and third terms are the induced-drag contributions of the two wings acting on themselves. The middle term is the mutual induced-drag contribution arising from aerodynamic interference between the two lifting systems.

For the common special case of equal spans, $b_1=b_2=b$ , and equal lift sharing, $L_1=L_2=L/2$ , the induced drag becomes

(20) $\begin{equation*} D_{\text{bi}} = \frac{1}{\pi \, q_\infty} \left( \frac{L^2}{4b^2} + 2\sigma\frac{L^2}{4b^2} + \frac{L^2}{4b^2} \right) \end{equation*}$

or

(21) $\begin{equation*} D_{\text{bi}} = \frac{(1+\sigma)L^2} {2\pi \, q_\infty \, b^2} \end{equation*}$

For comparison, a monoplane of the same span carrying the same total lift has

(22) $\begin{equation*} D_{\text{mono}} = \frac{L^2}{\pi \, q_\infty \, b^2} \end{equation*}$

Therefore,

(23) $\begin{equation*} \frac{D_{\text{bi}}}{D_{\text{mono}}} = \frac{1+\sigma}{2} \end{equation*}$

If the two lifting systems are so close that their trailing-vortex systems nearly coincide, then $\sigma \rightarrow 1$ and

(24) $\begin{equation*} \frac{D_{\text{bi}}}{D_{\text{mono}}} \rightarrow 1 \end{equation*}$

so the biplane has the same induced drag as a monoplane of the same span and total lift. If the two lifting systems are widely separated, then $\sigma \rightarrow 0$ and

(25) $\begin{equation*} \frac{D_{\text{bi}}}{D_{\text{mono}}} \rightarrow \frac{1}{2} \end{equation*}$

which is the ideal result for two independent lifting systems, each carrying one-half of the total lift.

The same result can be expressed using Munk’s equivalent span factor. Letting $\mu = \dfrac{b_2}{b_1}$ and $\nu = \dfrac{L_2}{L_1}$ , then the equivalent span factor $M$ is

(26) $\begin{equation*} M = \frac{(1+\nu)\mu} {\sqrt{\mu^2+2\mu\nu\sigma+\nu^2}} \end{equation*}$

This factor gives the span of the equivalent monoplane relative to the span $b_1$ . It shows how the aerodynamic benefit of a biplane depends on the lift ratio, span ratio, and interference coefficient.

The result can also be written in coefficient form. Let the total biplane wing area be

(27) $\begin{equation*} S_{\text{tot}} = S_1 + S_2 \end{equation*}$

and define the total lift coefficient as

(28) $\begin{equation*} C_L = \frac{L}{q_\infty S_{\text{tot}}} \end{equation*}$

For two identical wings, each with area $S$ , span $b$ , and aspect ratio $A\!R=b^2/S$ , the total biplane area is $S_{\text{tot}}=2S$ . For equal lift sharing, the induced-drag coefficient becomes

(29) $\begin{equation*} C_{D_{i,\text{bi}}} = \frac{D_{\text{bi}}}{q_\infty S_{\text{tot}}} = \frac{(1+\sigma)L^2} {2\pi q_\infty^2 b^2 S_{\text{tot}}} \end{equation*}$

Substituting $L=q_\infty S_{\text{tot}} C_L$ gives

(30) $\begin{equation*} C_{D_{i,\text{bi}}} = \frac{(1+\sigma)C_L^2 S_{\text{tot}}} {2\pi b^2} \end{equation*}$

or, because $S_{\text{tot}}=2S$ , then

(31) $\begin{equation*} C_{D_{i,\text{bi}}} = \frac{(1+\sigma)C_L^2} {\pi \, A\!R} \end{equation*}$

where $A\!R=b^2/S$ is the aspect ratio of one wing alone. If the aspect ratio is instead based on the total biplane area, so that

(32) $\begin{equation*} A\!R_{\text{tot}} = \frac{b^2}{S_{\text{tot}}} = \frac{A\!R}{2} \end{equation*}$

then the same result becomes

(33) $\begin{equation*} C_{D_{i,\text{bi}}} = \frac{(1+\sigma)C_L^2} {2\pi \, A\!R_{\text{tot}}} \end{equation*}$

Therefore, the von Mises biplane theory is an induced-drag theory built from the circulation distributions on the two lifting lines. The circulation on one wing determines the trailing-vortex sheet, which induces downwash at the other wing, and the product of that downwash with the other wing’s lift distribution gives the mutual induced-drag term. For elliptically loaded lifting lines, the mutual term is summarized by the interference coefficient $\sigma$ . The theory applies to the inviscid induced-drag component. The total drag of a real biplane must still include profile drag, strut-and-wire drag, junction drag, viscous interference, and structural constraints.

Effect of Gap on Biplane Interference

The biplane interference coefficient $\sigma$ measures the strength of the mutual induced-drag term between the two lifting systems. In the von Mises formulation, $\sigma$ depends on the nondimensional gap, i.e.,

(34) $\begin{equation*} \frac{2h}{b_1+b_2} \end{equation*}$

and on the span ratio,

(35) $\begin{equation*} \mu = \frac{b_2}{b_1} \end{equation*}$

Thus,

(36) $\begin{equation*} \sigma = \sigma \left( \frac{2h}{b_1+b_2}, \mu \right) \end{equation*}$

where $\mu=1$ corresponds to equal spans. For equal spans, $b_1=b_2=b$ , the nondimensional gap reduces to

(37) $\begin{equation*} \frac{2h}{b_1+b_2} = \frac{h}{b} \end{equation*}$

Figure 6 shows the biplane interference coefficient as a function of the nondimensional gap for several span ratios. The interference is strongest when the gap is small because the trailing-vortex systems of the two wings strongly interact. As the gap increases, the mutual downwash decreases, and $\sigma$ decreases. For the cases shown, the equal-span case, $\mu=1$ , gives the largest interference coefficient.

Biplane interference coefficient $\sigma$ as a function of the nondimensional gap $2h/(b_1+b_2)$ for several span ratios $\mu=b_2/b_1$ .

For the equal-span, equal-lift case, the induced-drag ratio relative to a monoplane of the same span and total lift is

(38) $\begin{equation*} \frac{D_{\text{bi}}}{D_{\text{mono}}} = \frac{1+\sigma}{2} \end{equation*}$

This ratio is independent of aspect ratio because the comparison is made at the same total lift and the same span. The aspect ratio appears in the induced-drag coefficient, but it cancels from the ratio $D_{\text{bi}}/D_{\text{mono}}$ .

Figure 7 shows the corresponding induced-drag ratio for the equal-span, equal-lift case. At a small gap, $\sigma$ is large, and the biplane behaves almost like a single lifting system, so the induced drag approaches that of a monoplane of the same span and total lift. As the gap increases, the two wings interfere less strongly, and the induced drag approaches the value for two independent lifting systems, each carrying one-half of the total lift.

Induced-drag ratio $D_{\text{bi}}/D_{\text{mono}}$ for an equal-span, equal-lift biplane. The ratio is $(1+\sigma)/2$ , so it decreases from unity toward one half as the gap-to-span ratio $h/b$ is increased.

Induced-drag ratio $D_{\text{bi}}/D_{\text{mono}}$ for an equal-span, equal-lift biplane. The ratio is $(1+\sigma)/2$ , so it decreases from unity toward one half as the gap-to-span ratio $h/b$ is increased.

These results show only the ideal inviscid induced-drag effect. They do not imply that increasing the gap is always beneficial in a complete aircraft design. A larger gap can reduce lifting-line interference, but it also requires longer struts, longer bracing wires, greater structural weight, and additional parasite drag. Therefore, the gap of a practical biplane is an aerodynamic and structural compromise, not an induced-drag optimization alone.

General Theory

The von Mises biplane formula provides a compact expression for induced drag in two idealized interacting lifting systems. It is useful because it separates the two self-induced drag terms from the mutual-interference term. However, it does not by itself determine how the total lift is shared between the two wings, nor does it determine the actual spanwise circulation distributions. A more general biplane theory must solve for the circulation on the upper and lower wings simultaneously.

Let the spanwise circulation distributions on the upper and lower wings be $\Gamma_u(y)$ and $\Gamma_l(y)$ , respectively. These distributions depend on the local chord, twist, pitch angles, airfoil section, gap, stagger, decalage, and the induced velocity field produced by both wings. Therefore, two biplanes with the same gap can still exhibit different lift sharing, lift-curve slope, induced drag, and stall behavior if their planforms, twist distributions, pitches, or staggers differ.

For each wing, the local section lift relation may be written as

(39) $\begin{equation*} C_l(y) = C_{l_\alpha}(y) \bigg( \alpha_{\rm geom}(y) - \alpha_i(y) - \alpha_0(y) \bigg) \end{equation*}$

where $\alpha_{\rm geom}(y)$ is the local geometric angle of attack, $\alpha_i(y)$ is the induced angle of attack, and $\alpha_0(y)$ is the local zero-lift angle of attack. The section lift coefficient is also related to the bound circulation by

(40) $\begin{equation*} C_l(y) = \frac{2\Gamma(y)} {V_\infty c(y)} \end{equation*}$

where $c(y)$ is the local chord. Therefore, the circulation must satisfy

(41) $\begin{equation*} \frac{2\Gamma(y)} {V_\infty c(y)} = C_{l_\alpha}(y) \bigg( \alpha_{\rm geom}(y) - \alpha_i(y) - \alpha_0(y) \bigg) \end{equation*}$

For the upper and lower wings, the local geometric angles of attack may be written as

(42) $\begin{equation*} \alpha_{{\rm geom},u}(y) = \alpha + \theta_u + \theta_{{\rm tw},u}(y) \end{equation*}$

and

(43) $\begin{equation*} \alpha_{{\rm geom},l}(y) = \alpha + \theta_l + \theta_{{\rm tw},l}(y) \end{equation*}$

where $\alpha$ is the airplane angle of attack, $\theta_u$ and $\theta_l$ are the pitch angles of the upper and lower wings, and $\theta_{{\rm tw},u}(y)$ and $\theta_{{\rm tw},l}(y)$ are their twist distributions. The difference between $\theta_u$ and $\theta_l$ is the decalage.

In a biplane, the induced angle of attack on each wing contains a contribution from its own trailing-vortex system and a contribution from the other wing’s trailing-vortex system. Therefore,

(44) $\begin{equation*} \alpha_{i,u}(y) = \alpha_{i,uu}(y) + \alpha_{i,ul}(y) \end{equation*}$

and

(45) $\begin{equation*} \alpha_{i,l}(y) = \alpha_{i,ll}(y) + \alpha_{i,lu}(y) \end{equation*}$

where the first subscript identifies the wing on which the induced angle is evaluated and the second subscript identifies the wing producing the induced velocity. The self-induced terms $\alpha_{i,uu}(y)$ and $\alpha_{i,ll}(y)$ describe the usual finite-wing downwash of each wing. The mutual terms $\alpha_{i,ul}(y)$ and $\alpha_{i,lu}(y)$ describe the aerodynamic interference between the two wings.

Therefore, the coupled lifting-line equations for the upper and lower wings may be written as

(46) $\begin{equation*} \frac{2\Gamma_u(y)} {V_\infty c_u(y)} = C_{l_{\alpha},u}(y) \bigg( \alpha_{{\rm geom},u}(y) - \alpha_{i,uu}(y) - \alpha_{i,ul}(y) - \alpha_{0,u}(y) \bigg) \end{equation*}$

and

(47) $\begin{equation*} \frac{2\Gamma_l(y)} {V_\infty c_l(y)} = C_{l_{\alpha},l}(y) \bigg( \alpha_{{\rm geom},l}(y) - \alpha_{i,ll}(y) - \alpha_{i,lu}(y) - \alpha_{0,l}(y) \bigg) \end{equation*}$

These equations show why gap and stagger cannot be separated from the loading problem in a general biplane theory. Therefore, the gap $h$ changes the vertical separation of the lifting systems and so the mutual downwash in the Trefftz-plane sense. Stagger does not directly change the ideal induced drag if the same Trefftz-plane circulation distribution is preserved, as stated by Munk’s stagger theorem, but it can change the actual lift sharing, trim, and circulation distributions on a real biplane.

The total lift is obtained by integrating the circulation over both wings, i.e.,

(48) $\begin{equation*} L = \varrho_\infty V_\infty \bigg( \int_{-b_u/2}^{b_u/2} \Gamma_u(y) \, dy + \int_{-b_l/2}^{b_l/2} \Gamma_l(y) \, dy \bigg) \end{equation*}$

The individual wing lift values are

(49) $\begin{equation*} L_u = \varrho_\infty V_\infty \int_{-b_u/2}^{b_u/2} \Gamma_u(y) \, dy \end{equation*}$

and

(50) $\begin{equation*} L_l = \varrho_\infty V_\infty \int_{-b_l/2}^{b_l/2} \Gamma_l(y) \, dy \end{equation*}$

The induced drag is obtained from the product of lift and the induced angle of attack over both spans, i.e.,

(51) $\begin{equation*} D_i = \varrho_\infty V_\infty \bigg( \int_{-b_u/2}^{b_u/2} \Gamma_u(y) \, \alpha_{i,u}(y) \, dy + \int_{-b_l/2}^{b_l/2} \Gamma_l(y) \, \alpha_{i,l}(y) \, dy \bigg) \end{equation*}$

This expression is the general counterpart of the simplified von Mises biplane formula. In the simplified formula, the detailed dependence on $\Gamma_u(y)$ , $\Gamma_l(y)$ , $h$ , pitches, twist, planform, and loading is condensed into the biplane interference coefficient $\sigma$ and into the assumed lift sharing. In the complete theory, these quantities must be solved as part of the coupled lifting-line problem.

For a monoplane, departures from elliptical loading are often represented by a correction factor $\delta_{\text{mono}}$ , giving

(52) $\begin{equation*} C_{D_{i,\text{mono}}} = \frac{(1+\delta_{\text{mono}})C_L^2}{\pi \, A\!R} \end{equation*}$

For a biplane, the corresponding factor must account for both non-elliptic loading and the aerodynamic interference between the two lifting systems. For symmetric lift sharing, equal spans, and total biplane area $S_{\text{tot}}=2S$ , the result may be written as

(53) $\begin{equation*} C_{D_{i,\text{bi}}} = \frac{(1+\delta_{\text{bi}})C_L^2}{\pi \, A\!R} \end{equation*}$

where $A\!R=b^2/S$ is the aspect ratio of one wing. If the aspect ratio is based on the total biplane area, then

(54) $\begin{equation*} A\!R_{\text{tot}} = \frac{b^2}{S_{\text{tot}}} = \frac{A\!R}{2} \end{equation*}$

and the same result becomes

(55) $\begin{equation*} C_{D_{i,\text{bi}}} = \frac{(1+\delta_{\text{bi}})C_L^2}{2\pi \, A\!R_{\text{tot}}} \end{equation*}$

The definitions of reference area and aspect ratio must be stated explicitly when comparing biplanes and monoplanes. The effective value of $\delta_{\text{bi}}$ includes the effects of spanwise loading on each wing, lift sharing, gap, stagger-induced changes in loading, decalage, planform, twist, pitches, and airfoil selection.

Munk’s Stagger Theorem

Munk’s Stagger Theorem is a classical result in biplane and multiplane theory formulated by Max Munk in the 1920s. It states that the ideal induced drag of a system of lifting lines is unchanged if the lifting lines are displaced fore or aft relative to one another, provided the same spanwise circulation distribution is preserved in the Trefftz plane. Therefore, a pure streamwise displacement, or stagger, has no direct effect on the ideal induced drag if the far-wake vortex system is unchanged.

The reason is that ideal induced drag is determined by the trailing-vortex system far downstream, not by the streamwise location of the bound lifting lines. For a system of lifting lines, the induced drag may be written as

(56) $\begin{equation*} D_i = \varrho_\infty V_\infty \sum_n \int_{-b_n/2}^{b_n/2} \Gamma_n(y) \, \alpha_{i,n}(y) \, dy \end{equation*}$

or, using $\alpha_{i,n}(y)=w_n(y)/V_\infty$ ,

(57) $\begin{equation*} D_i = \varrho_\infty \sum_n \int_{-b_n/2}^{b_n/2} \Gamma_n(y) \, w_n(y) \, dy \end{equation*}$

where $\Gamma_n(y)$ is the circulation distribution on the $n$ th lifting line and $w_n(y)$ is the induced downwash at that lifting line. These expressions follow from $dL=\varrho_\infty V_\infty \Gamma(y)\,dy$ and $dD_i=dL\,\alpha_i$ .

For two elliptically loaded wings with prescribed lift distributions, the biplane induced drag may be written as

(58) $\begin{equation*} D_{\text{bi}} = \frac{1}{\pi q_\infty} \left( \frac{L_1^2}{b_1^2} + 2\sigma \frac{L_1L_2}{b_1b_2} + \frac{L_2^2}{b_2^2} \right) \end{equation*}$

where $\sigma$ is the biplane interference coefficient. In the basic von Mises formulation, $\sigma$ depends on the gap ratio $2h/(b_1+b_2)$ and on the span ratio $\mu=b_2/b_1$ , but not on stagger. The absence of a stagger term in this expression is the result described by Munk’s theorem. It is a fixed-loading result, not a complete statement about the induced drag of a real biplane after the lift distribution has changed.

The distinction is important. Munk’s theorem does not say that forward and rearward stagger have the same aerodynamic effect on a real biplane. It says that fore-and-aft displacement does not change the ideal induced drag if the spanwise circulation distributions in the Trefftz plane are held fixed. In the actual lifting-line problem, stagger changes the local induced velocities at the two wings. These induced velocities change the effective angles of attack, the lift sharing, and the circulation distributions. Once the circulation distributions change, the induced drag can also change.

The gap remains the primary geometric variable governing the classical biplane induced-drag interference. The gap changes the vertical separation of the trailing-vortex systems in the Trefftz plane, and so changes the interference coefficient $\sigma$ directly. Stagger is different. It does not enter the fixed-loading Trefftz-plane result directly, but it can alter the computed lift sharing by changing the local downwash experienced by each wing. Therefore, gap primarily sets the strength of the biplane interference, while stagger, decalage, pitch, twist, chord ratio, and airfoil section determine how the total lift is shared between the two wings.

Local Lifting-Line Solution

To see where stagger enters the local aerodynamic problem, consider the downwash induced at wing 1 by the trailing-vortex sheet of wing 2. In the unstaggered case, the two lifting lines have the same streamwise location and are separated only by the vertical gap $h$ . The induced downwash at a point $y_1$ on wing 1 may then be written as

(59) $\begin{equation*} w_{12}(y_1) = -\frac{1}{4\pi} \int_{-b_2/2}^{b_2/2} \frac{y_2-y_1} {h^2+(y_2-y_1)^2} \frac{d\Gamma_2}{dy_2} \, dy_2 \end{equation*}$

This is the expression used in the basic unstaggered biplane theory. Now let wing 2 be displaced streamwise relative to wing 1 by the stagger distance $St=x_2-x_1$ , where a positive $St$ means that wing 2 is behind wing 1. The Biot-Savart expression for the semi-infinite trailing-vortex sheet gives the staggered influence form

(60) $\begin{equation*} w_{12}(y_1;St) = -\frac{1}{4\pi} \int_{-b_2/2}^{b_2/2} \frac{y_2-y_1} {h^2+(y_2-y_1)^2} \bigg( 1 - \frac{St} {\sqrt{St^2+h^2+(y_2-y_1)^2}} \bigg) \frac{d\Gamma_2}{dy_2} \, dy_2 \end{equation*}$

When $St=0$ , the bracketed factor is unity and the unstaggered expression is recovered.

This result explicitly shows that the local downwash at one wing depends on the stagger. If $St>0$ , then wing 2 is behind wing 1 and

(61) $\begin{equation*} 1 - \frac{St} {\sqrt{St^2+h^2+(y_2-y_1)^2}} < 1 \end{equation*}$

so the rearward wing has a weaker downwash influence on the forward wing. If $St<0$ , then wing 2 is ahead of wing 1 and the bracketed factor is greater than unity, so the forward wing has a stronger downwash influence on the rearward wing. Therefore, forward and rearward stagger are not equivalent in the local lifting-line problem.

The corresponding influence of wing 1 on wing 2 is obtained by reversing the sign of the streamwise offset. In the same notation, this influence may be written as

(62) $\begin{equation*} w_{21}(y_2;-St) = -\frac{1}{4\pi} \int_{-b_1/2}^{b_1/2} \frac{y_1-y_2} {h^2+(y_1-y_2)^2} \bigg( 1 + \frac{St} {\sqrt{St^2+h^2+(y_1-y_2)^2}} \bigg) \frac{d\Gamma_1}{dy_1} \, dy_1 \end{equation*}$

These two equations show that stagger asymmetrically changes the mutual interference in that one wing may experience a reduced downwash contribution while the other experiences an increased contribution. Therefore, the two wings do not generally retain the same lift sharing when the stagger is changed.

The local effective angles of attack may be written as

(63) $\begin{equation*} \alpha_{{\rm eff},1}(y_1) = \alpha_{{\rm geom},1}(y_1) - \frac{ w_{11}(y_1)+w_{12}(y_1;St) } {V_\infty} - \alpha_{0,1}(y_1) \end{equation*}$

and

(64) $\begin{equation*} \alpha_{{\rm eff},2}(y_2) = \alpha_{{\rm geom},2}(y_2) - \frac{ w_{22}(y_2)+w_{21}(y_2;-St) } {V_\infty} - \alpha_{0,2}(y_2) \end{equation*}$

where $w_{11}$ and $w_{22}$ are the self-induced downwash terms of the two wings. The corresponding section lift relations may be written as

(65) $\begin{equation*} \Gamma_1(y_1) = \frac{1}{2} V_\infty c_1(y_1) C_{l_{\alpha},1}(y_1) \alpha_{{\rm eff},1}(y_1) \end{equation*}$

and

(66) $\begin{equation*} \Gamma_2(y_2) = \frac{1}{2} V_\infty c_2(y_2) C_{l_{\alpha},2}(y_2) \alpha_{{\rm eff},2}(y_2) \end{equation*}$

These equations show how stagger enters the coupled lifting-line biplane problem. The staggered downwash kernels change the effective angles of attack, which then change the circulation distributions $\Gamma_1(y_1)$ and $\Gamma_2(y_2)$ .

Lift Sharing

A reduced lift-sharing model can make this dependence explicit. Let $a_1$ and $a_2$ be the lift sensitivities of the two wings, including dynamic pressure, wing area, and lift-curve slope. Let $k_{11}$ and $k_{22}$ represent the self-induced angle effects, and let $k_{12}(St)$ and $k_{21}(St)$ represent the span-averaged mutual induced-angle effects. Then the lift on the two wings may be modeled as

(67) $\begin{equation*} L_1 = a_1 \bigg( \alpha_1 - k_{11}L_1 - k_{12}(St)L_2 \bigg) \end{equation*}$

and

(68) $\begin{equation*} L_2 = a_2 \bigg( \alpha_2 - k_{22}L_2 - k_{21}(St)L_1 \bigg) \end{equation*}$

where $\alpha_1$ and $\alpha_2$ are effective incidence terms that include pitch, representative twist, and zero-lift-angle effects. Their difference affects lift sharing. Decalage changes these incidence terms, whereas stagger changes the mutual downwash coefficients.

The staggered downwash expressions imply that, in general,

(69) $\begin{equation*} k_{12}(St) \ne k_{21}(St) \end{equation*}$

For positive $St$ , corresponding to wing 2 behind wing 1, the rearward wing has a reduced influence on the forward wing, while the forward wing has an increased influence on the rearward wing. Therefore, the mutual interference coefficients are not equal in the local lifting-line problem.

Solving the two lift equations gives

(70) $\begin{equation*} L_1(St) = \frac{ a_1\alpha_1(1+a_2k_{22}) - a_1a_2k_{12}(St)\alpha_2 } { (1+a_1k_{11})(1+a_2k_{22}) - a_1a_2k_{12}(St)k_{21}(St) } \end{equation*}$

and

(71) $\begin{equation*} L_2(St) = \frac{ a_2\alpha_2(1+a_1k_{11}) - a_1a_2k_{21}(St)\alpha_1 } { (1+a_1k_{11})(1+a_2k_{22}) - a_1a_2k_{12}(St)k_{21}(St) } \end{equation*}$

The lift fraction carried by wing 1 is then

(72) $\begin{equation*} f(St) = \frac{L_1(St)} {L_1(St)+L_2(St)} \end{equation*}$

This expression gives the design link between stagger and induced drag. Stagger changes the mutual downwash coefficients $k_{12}(St)$ and $k_{21}(St)$ ; these coefficients affect $L_1$ and $L_2$ , and the altered lift balance, which in turn affects the induced drag.

For equal spans, $b_1 = b_2 = b$ , the induced-drag ratio may be written as

(73) $\begin{equation*} \frac{D_{\text{bi}}}{D_{\text{mono}}} = f^2(St) + \big[1-f(St)\big]^2 + 2\sigma f(St)\big[1-f(St)\big] \end{equation*}$

where $D_{\text{mono}}$ is the induced drag of a monoplane of the same span carrying the same total lift. This expression uses the same biplane interference coefficient $\sigma$ as the classical von Mises gap-dependent theory, but it allows the lift fraction to change with stagger, decalage, pitch, twist, chord ratio, and airfoil choice.

The same expression can be written in a form that emphasizes the penalty for unequal lift sharing. Let

(74) $\begin{equation*} f(St) = \frac{1}{2} + \Delta f(St) \end{equation*}$

where $\Delta f(St)$ is the departure from equal lift sharing. Substitution gives

(75) $\begin{equation*} \frac{D_{\text{bi}}}{D_{\text{mono}}} = \frac{1+\sigma}{2} + 2(1-\sigma)\big(\Delta f(St)\big)^2 \end{equation*}$

This equation shows that, for a fixed value of $\sigma$ , the induced drag is minimized when $\Delta f = 0$ , i.e., when the two equal-span wings share the lift equally. If stagger, decalage, or another design variable moves the lift sharing toward equality, the induced drag decreases. If it shifts the lift-sharing away from equality, the induced drag increases.

Notice that for the usual biplane case, where $0 \leq \sigma < 1$ , the equal-sharing value is

(76) $\begin{equation*} \frac{D_{\text{bi}}}{D_{\text{mono}}} = \frac{1+\sigma}{2} \end{equation*}$

Therefore, the equal-lift case is the minimum induced-drag condition for equal spans and a fixed interference coefficient. This result does not imply that zero stagger is best, but that the best arrangement is the one that provides the desired lift-sharing. If zero stagger already gives $f = 1/2$ , then zero stagger gives the minimum induced drag. If decalage or local interference causes unequal lift sharing at zero stagger, then stagger may either improve or worsen the induced drag depending on whether it moves $\scriptstyle f$ toward or away from 0.5.

Illustrative Model

A simple illustrative model for the lift-sharing variation is

(77) $\begin{equation*} f(St) = f_0 - m\frac{St}{b} \end{equation*}$

where $f_0$ is the lift fraction at zero stagger and $m$ is a lift-sharing sensitivity. Substituting this model into the induced-drag expression gives a family of drag curves as a function of $St/b$ , shown in Figure 8. The curves show that the minimum induced drag occurs when the stagger brings the lift sharing to $f=1/2$ . Unequal lift sharing increases the induced drag above the equal-sharing value. Such a plot is not a universal biplane design chart, but it clearly shows the design mechanism that the induced drag reaches a minimum when the stagger produces equal lift sharing.

Representative induced-drag ratio $D_{\text{bi}}/D_{\text{mono}}$ as a function of stagger ratio $St/b$ for several zero-stagger lift fractions $f_0$ .

Figure 8 emphasizes that stagger itself is not an independent induced-drag-reduction parameter in the fixed-loading Munk theorem. Instead, stagger matters because it can change the lift balance. The lowest curve point occurs when the lift sharing is equal, not necessarily when the stagger is zero. Decalage can produce the same kind of effect by changing the geometric pitch of one wing relative to the other. Therefore, stagger and decalage should be understood as lift-sharing controls, whereas the gap is the primary geometric parameter governing the strength of biplane-induced drag interference.

The trim balance is also affected by stagger, as moving one wing forward or aft changes the moment arms of the wing’s lift forces. Therefore, stagger can change both the aerodynamic lift sharing and the pitching-moment balance of the airplane. These trim effects are important in design, but they are separate from the induced-drag mechanism shown above.

The practical conclusion is that Munk’s theorem is a fixed-loading theorem, not a complete biplane-design rule. The gap-to-span ratio enters directly into the classical biplane interference coefficient. Stagger enters the local lifting-line problem through the downwash kernels. Decalage enters through the geometric angles of attack. Both stagger and decalage can alter the lift balance, which in turn affects the induced drag. Therefore, the induced drag of a real biplane need not be independent of stagger, even though Munk’s theorem shows that stagger does not change the ideal induced drag when the Trefftz-plane circulation distributions are held fixed.

Design Tradeoffs for Biplanes

The biplane configuration was a rational engineering solution for early aircraft. It provided the required wing area without a large span, thereby reducing bending loads and keeping the structure compact. The vertical separation between the wings also gave the structure depth, allowing the wings, interplane struts, and bracing wires to act as a stiff truss-like cellule. This arrangement was well matched to wood-and-fabric construction, where fully cantilevered wings of comparable span and stiffness were difficult to build at acceptable weight.

The aerodynamic penalty was that the two wings did not act as independent lifting surfaces. Each wing operated partly in the induced-velocity field of the other, reducing the effective lift-curve slope and increasing the induced drag relative to two widely separated wings. The close spacing also increased wetted area, junction losses, and parasite drag from struts, bracing wires, fittings, and exposed structural members. Therefore, the biplane layout involved a direct tradeoff between structural efficiency and aerodynamic efficiency.

The principal geometric variables were gap, stagger, decalage, chord, span, and relative wing loading. Increasing the gap reduced the ideal lifting-line interference, but it also increased strut length, bracing loads, structural weight, and sometimes drag. Stagger affected the local flow field, lift sharing, pitching moments, cockpit visibility, structural arrangement, and stall progression. Decalage determines how the lift is shared between the two wings at a given airplane angle of attack. Twist, including washout, was used to control spanwise loading, reduce tip loading, preserve lateral control near stall, and influence which wing or wing panel stalled first.

The preceding theory shows that the induced-drag tradeoff can be separated into two parts. For equal-span wings, the induced-drag ratio may be written as

(78) $\begin{equation*} \frac{D_{\text{bi}}}{D_{\text{mono}}} = \frac{1+\sigma}{2} + 2(1-\sigma) \left( f-\frac{1}{2} \right)^2 \end{equation*}$

where $f=L_1/L$ is the fraction of the total lift carried by one wing. The first term contains the effect of the gap through the interference coefficient $\sigma$ , so the gap is the primary geometric variable controlling the ideal biplane interference. The second term is the penalty for unequal lift sharing. Stagger, decalage, twist, chord ratio, and airfoil selection are important because they can shift the lift-sharing toward or away from $f=1/2$ .

This result explains why there is no universal answer to whether forward or rearward stagger is best. Stagger is beneficial from an induced-drag standpoint only if it shifts the lift distribution toward equality. If it moves the lift-sharing closer to $f=1/2$ , it can reduce the induced drag relative to an unequally loaded biplane. If it moves the lift-sharing farther from equality, it can increase the induced drag. Decalage plays a similar design role by altering the relative geometric angles of attack of the two wings and, therefore, how the total lift is shared.

The eventual dominance of the monoplane came from changes in the structural and speed regimes. As airfoils, materials, metal construction, and stressed-skin design improved, cantilever monoplanes could provide the required strength and stiffness without external bracing. At higher speeds, the parasite drag of the biplane cellule became increasingly costly, while the structural advantage of the externally braced arrangement became less decisive. The biplane remained useful where compact span, maneuverability, low-speed performance, or historical configuration mattered, but the cantilever monoplane became the more efficient solution for most airplanes.

Check Your Understanding #1 – Biplane aerodynamic performance

A biplane has two identical wings, each with span $b$ = 6 m and chord $c$ = 1.2 m. The total mass of the airplane is $M$ = 1,000 kg. The aircraft flies at MSL ISA conditions at an airspeed of $V_\infty$ = 80 mph. Assume: 1. The two wings share the lift equally. 2. The von Mises biplane interference coefficient for the specified gap ratio is $\sigma = 0.30$ . 3. The wings are otherwise elliptically loaded. 4. Profile drag, strut drag, bracing-wire drag, and fuselage interference are neglected.

Determine the lift coefficient $C_L$ based on the total reference area $S_{\text{tot}} = 2S$ .
Determine the induced drag coefficient for two aerodynamically independent wings, using the same total reference area convention.
Determine the induced drag coefficient for the biplane using the specified interference coefficient.
Comment on the aerodynamic efficiency of the biplane compared with two aerodynamically independent wings of the same individual span and chord.

Show solution/hide solution.

The area of one wing is
$S = b \, c = 6~\text{m} \times 1.2~\text{m} = 7.2~\text{m}^2$

so the aspect ratio of each wing is

$A\!R = \frac{b^2}{S} = \frac{6^2}{7.2} = 5.0$

The total reference area for the biplane is

$S_{\text{tot}} = 2S = 2 \times 7.2 = 14.4~\text{m}^2$

The total lift required equals the airplane’s weight, i.e.,

${L} = M g = 1{,}000 \times 9.81 = 9{,}810~\text{N}$

The airspeed is ${V_\infty}$ = 80 mph = 35.76 m/s, so the dynamic pressure is

$q_\infty = \frac{1}{2} \varrho_\infty V_\infty^2 = \frac{1}{2} \times 1.225 \times (35.76)^2 = 783~\text{Pa}$

The lift coefficient based on the total biplane reference area is then

$C_L = \frac{L}{q_\infty S_{\text{tot}}} = \frac{9{,}810}{783 \times 14.4} = 0.87$
For two aerodynamically independent wings sharing the total lift equally, the induced drag coefficient based on the total biplane reference area is
$C_{D_{i,\text{ind}}} = \frac{C_L^2}{\pi \, A\!R}$

where $A\!R=b^2/S$ is the aspect ratio of one wing. Therefore,

$C_{D_{i,\text{ind}}} = \frac{(0.87)^2}{\pi \times 5.00} = 0.048$
For the equal-span, equal-lift biplane, the von Mises form gives
$C_{D_{i,\text{bi}}} = \frac{(1+\sigma)C_L^2}{\pi \, A\!R}$

where the reference area is $S_{\text{tot}}=2S$ and $A\!R=b^2/S$ is the aspect ratio of one wing. With $\sigma=0.30$ ,

$C_{D_{i,\text{bi}}} = \frac{(1+0.30)(0.87)^2}{\pi \times 5.00} = 0.063$
Under the assumptions used here, the biplane has a larger induced drag coefficient than two aerodynamically independent wings, i.e.,
$C_{D_{i,\text{bi}}}=0.063 \qquad \text{versus} \qquad C_{D_{i,\text{ind}}}=0.048$

The increase is caused by the mutual downwash interference between the upper and lower lifting systems. In this simplified example, the interference is represented by $\sigma=0.30$ , which means that the biplane induced drag is 30% higher than the value for two independent wings using the same total reference area convention. A real biplane would also include profile drag, strut drag, bracing-wire drag, junction drag, and other viscous interference effects.

Triplanes

A triplane has three main wings arranged vertically, usually as an extension of the biplane concept with a third lifting surface added above, between, or below the other two wings. The motivation for a triplane is to increase the total wing area without requiring a large span, thereby improving structural compactness, stiffness, and maneuverability. However, the added wing also increases aerodynamic interference, structural complexity, wetted area, and parasite drag.

History

Triplanes appeared during WWI as designers searched for ways to improve climb rate, maneuverability, and pilot visibility while keeping the wing span short. Aircraft engines of the period had limited power, and aircraft structures were still relatively light, flexible, and highly dependent on external bracing. Adding a third wing allowed designers to increase the total lifting area without increasing the span as much as would be required for a monoplane or even a biplane of the same area. The result was a compact airplane with relatively low wing loading, good climb capability, and strong maneuvering performance.

As shown in Figure 9, the Sopwith Triplane was a British single-seat fighter aircraft designed and manufactured by the Sopwith Aviation Company during WWI. It was among the first military triplanes to enter operational service and demonstrated the appeal of the triplane configuration for climb and maneuverability within a short span. Its success attracted considerable attention because it showed that a compact multi-wing arrangement could produce excellent agility, especially at the relatively low speeds of WWI combat.

A flying reproduction of the Sopwith Triplane, one of the first military triplanes to see operational service.

The Fokker Dr.I Triplane became the most famous example of the type, partly because of its association with Manfred von Richthofen, the “Red Baron.” It used short span, closely spaced wings to produce a compact and highly maneuverable fighter. Its climb capability and turning agility were impressive, but its closely spaced three-wing arrangement also produced high aerodynamic interference and parasite drag. These penalties limited its maximum speed, which became increasingly important as aircraft engines improved and air combat placed greater emphasis on speed and maneuverability.

The triplane represents a short-lived but important stage in aircraft design. It was a logical solution for designers who needed a large wing area, low wing loading, a compact span, and structural stiffness, given the materials and construction methods of the time. However, the aerodynamic penalties of the third wing were significant. As cantilever monoplane structures improved and more powerful engines became available, the triplane’s advantages diminished. The configuration largely disappeared after WWI because monoplanes and improved biplanes offered better overall performance with reduced interference drag, lower structural complexity, and fewer drag-producing struts and bracing wires.

Aerodynamics of Triplanes

The aerodynamic behavior of a triplane follows the same finite-wing principles as a biplane, but the mutual-interference problem is more complex. Each wing generates bound circulation and a trailing-vortex system. The flow at any one wing is affected by its own trailing vortex sheet and by the wakes from the other two wings. The middle wing is usually the most strongly affected because it lies between the upper and lower wings and is immersed in the induced-velocity field produced by both of them.

A complete triplane theory must also recognize that the lift carried by each wing is not determined by the number of wings alone. The planform, taper, chord distribution, twist, airfoil section, and decalage of each wing all influence the spanwise circulation distribution. Therefore, the upper, middle, and lower wings usually carry different lift and have different spanwise loadings. The interference problem is fundamentally a coupled lifting-line problem in which the circulation on each wing depends on the induced velocity field produced by all three wings.

For the upper, middle, and lower wings, the circulation distributions may be denoted by $\Gamma_u(y)$ , $\Gamma_m(y)$ , and $\Gamma_l(y)$ , respectively. For each wing, the local section lift relation may be written as

(79) $\begin{equation*} C_l(y) = C_{l_\alpha}(y) \bigg( \alpha_{\rm geom}(y) - \alpha_i(y) - \alpha_0(y) \bigg) \end{equation*}$

where $\alpha_{\rm geom}(y)$ includes the local pitch and twist, $\alpha_i(y)$ is the induced angle of attack, and $\alpha_0(y)$ is the local zero-lift angle of attack. In a triplane, the induced angle of attack on each wing includes contributions from all three lifting lines. Therefore,

(80) $\begin{eqnarray*} \alpha_{i,u}(y) & = & \alpha_{i,uu}(y) + \alpha_{i,um}(y) + \alpha_{i,ul}(y) \\[6pt] \alpha_{i,m}(y) & = & \alpha_{i,mu}(y) + \alpha_{i,mm}(y) + \alpha_{i,ml}(y) \\[6pt] \alpha_{i,l}(y) & = & \alpha_{i,lu}(y) + \alpha_{i,lm}(y) + \alpha_{i,ll}(y) \end{eqnarray*}$

where the first subscript identifies the wing on which the induced angle is evaluated and the second subscript identifies the wing producing the induced velocity. The self-induced terms describe the usual finite-wing downwash of each wing, while the cross terms describe the mutual aerodynamic interference between the wings.

This coupled formulation shows why planform and twist must be considered in a complete triplane theory. Taper, unequal spans, different chord distributions, washout, airfoil changes, and decalage all alter $\Gamma_u(y)$ , $\Gamma_m(y)$ , and $\Gamma_l(y)$ . These changes modify the trailing-vortex system and so the effective interference factors used in simplified formulas. Consequently, two triplanes with the same gap arrangement can exhibit different lift sharing, lift-curve slope, induced drag, and stall behavior if their planforms, twist distributions, or decalage differ. Stagger can also change the actual loading and trim of a real triplane, although Munk’s stagger theorem shows that fore-and-aft displacement alone does not change the ideal induced drag if the Trefftz-plane loading is preserved.

In an idealized analysis, a triplane can be represented by three lifting lines separated vertically by the gaps between adjacent wings. If the upper, middle, and lower wings carry lift values ${L_u}$ , $L_m$ , and $L_l$ , respectively, then the triplane induced drag may be written as

(81) $\begin{equation*} D_{i,\text{tri}} = \frac{1}{\pi q_\infty} \left( \frac{L_u^2}{b_u^2} + \frac{L_m^2}{b_m^2} + \frac{L_l^2}{b_l^2} + 2\sigma_{um}\frac{L_u L_m}{b_u b_m} + 2\sigma_{ml}\frac{L_m L_l}{b_m b_l} + 2\sigma_{ul}\frac{L_u L_l}{b_u b_l} \right) \end{equation*}$

where $b_u$ , $b_m$ , and $b_l$ are the spans of the upper, middle, and lower wings, and $\sigma_{um}$ , $\sigma_{ml}$ , and $\sigma_{ul}$ are pairwise interference coefficients between the corresponding lifting systems. This is the triplane analog of the von Mises biplane form. The adjacent-wing interference coefficients $\sigma_{um}$ and $\sigma_{ml}$ are usually larger than the upper-to-lower interference coefficient $\sigma_{ul}$ , because a greater vertical distance separates the upper and lower wings.

For the special case of three identical wings of span $b$ , with equal lift sharing, i.e., $L_u = L_m = L_l = \dfrac{L}{3}$ , and with symmetric spacing so that

(82) $\begin{equation*} \sigma_{um} = \sigma_{ml} = \sigma_1, \qquad \sigma_{ul} = \sigma_2 \end{equation*}$

the induced drag becomes

(83) $\begin{equation*} D_{i,\text{tri}} = \frac{L^2}{9\pi q_\infty b^2} \left( 3 + 4\sigma_1 + 2\sigma_2 \right) \end{equation*}$

If the three wings were very widely separated, then $\sigma_1$ and $\sigma_2$ would approach zero, and the induced drag would approach

(84) $\begin{equation*} D_{i,\text{tri}} \rightarrow \frac{L^2}{3\pi q_\infty b^2} \end{equation*}$

This is the ideal result for three independent lifting systems, each carrying one-third of the total lift. If the three wings were brought very close together so that their vortex systems nearly coincided, then $\sigma_1$ and $\sigma_2$ would approach unity, giving

(85) $\begin{equation*} D_{i,\text{tri}} \rightarrow \frac{L^2}{\pi q_\infty b^2} \end{equation*}$

which is the same induced drag as a single monoplane of the same span carrying the same total lift. Real triplanes lie between these limits, but practical triplanes are usually much closer to the strongly interfering case than to the widely separated ideal case.

The preceding result also explains why triplanes were not aerodynamically superior to monoplanes in the general sense. Adding a third wing increases the total lifting area and reduces the lift carried by each wing, but it also places the wings in each other’s downwash fields. The middle wing is especially compromised because the induced velocities from both the upper and lower wings reduce its effective angle of attack. Consequently, the lift carried by the three wings is not necessarily equal in practice, even when the wings have the same planform. Decalage, stagger, twist, and airfoil selection may be used to redistribute lift, but these adjustments usually entail a compromise among lift sharing, trim, stall progression, visibility, and structural arrangement.

In coefficient form, let the total wing reference area be

(86) $\begin{equation*} S_{\text{tot}} = S_u + S_m + S_l \end{equation*}$

For three identical wings, $S_{\text{tot}}=3S$ , where $S$ is the area of one wing and $A\!R=b^2/S$ is the aspect ratio of one wing. Using the symmetric equal-loading result gives

(87) $\begin{equation*} C_{D_{i,\text{tri}}} = \frac{D_{i,\text{tri}}}{q_\infty S_{\text{tot}}} = \frac{C_L^2}{3\pi A\!R} \left( 3 + 4\sigma_1 + 2\sigma_2 \right) \end{equation*}$

where $C_L=L/(q_\infty S_{\text{tot}})$ . This reference-area convention is important. When $\sigma_1=\sigma_2=0$ , the coefficient becomes

(88) $\begin{equation*} C_{D_{i,\text{tri}}} = \frac{C_L^2}{\pi A\!R} \end{equation*}$

which is the result for three independent wings using the total triplane area as the reference area. When $\sigma_1=\sigma_2=1$ , the coefficient becomes

(89) $\begin{equation*} C_{D_{i,\text{tri}}} = \frac{3C_L^2}{\pi A\!R} \end{equation*}$

which corresponds to a single lifting system of span $b$ carrying the same total lift, but with the coefficient still referenced to the total area $3S$ .

The result may also be written as

(90) $\begin{equation*} C_{D_{i,\text{tri}}} = \frac{(1+\delta_{\text{tri}})C_L^2}{\pi A\!R} \end{equation*}$

where, for the symmetric equal-loading idealization,

(91) $\begin{equation*} 1+\delta_{\text{tri}} = \frac{3+4\sigma_1+2\sigma_2}{3} \end{equation*}$

In more practical use, $\delta_{\text{tri}}$ is an effective factor that also absorbs departures from the ideal assumptions, including non-elliptic loading, unequal lift sharing, unequal spans, finite chord effects, twist, decalage, and detailed airfoil characteristics. Its value is specific to the assumed gap arrangement, span, chord, twist, airfoil section, loading distribution, and reference-area convention.

This formulation also clarifies the design tradeoff. A triplane can provide a large total wing area within a short span, resulting in low wing loading, a compact structure, and good maneuverability at low speeds. These were important advantages for WWI aircraft, whose structures were light, externally braced, and limited by the strength and stiffness of available materials. However, the close vertical spacing of the three wings increases interference, reduces the effective lift-curve slope of the complete lifting system, and adds profile, junction, strut, and bracing-wire drag. Therefore, the triplane was attractive when structural compactness and maneuverability were more important than maximum speed or cruise efficiency, but it became less attractive as cantilever monoplane structures and more powerful engines became practical.

Check Your Understanding #2 – Triplane induced-drag estimate

A triplane has three identical wings, each with span $b = 6$ m and chord $c = 1.0$ m. The airplane has a total mass of $M = 900$ kg and flies at MSL ISA conditions at $V_\infty$ = 40 m s $^{-1}$ . Assume the total lift is shared equally by the three wings and use an effective triplane induced-drag factor of $\delta_{\text{tri}} = 0.35$ , referenced to the total wing area.

Determine the total wing reference area, $S_{\text{tot}}$ .
Determine the lift coefficient $C_L$ based on $S_{\text{tot}}$ .
Determine the aspect ratio $A\!R$ of one wing.
Estimate the induced drag coefficient using
$C_{D_{i,\text{tri}}} = \frac{(1+\delta_{\text{tri}})C_L^2}{\pi A\!R}$

where $A\!R$ is based on one wing.

Show solution/hide solution.

The area of one wing is
$S = b\,c = 6 \times 1.0 = 6.0~\text{m}^2$

so the total wing reference area is

$S_{\text{tot}} = 3S = 18.0~\text{m}^2$
The lift required in steady level flight is
$L = Mg = 900 \times 9.81 = 8{,}829~\text{N}$

At MSL ISA conditions, take $\varrho_\infty = 1.225$ kg m $^{-3}$ . The dynamic pressure is

$q_\infty = \frac{1}{2}\varrho_\infty V_\infty^2 = \frac{1}{2}(1.225)(40)^2 = 980~\text{Pa}$

Therefore, the lift coefficient based on the total wing area is

$C_L = \frac{L}{q_\infty S_{\text{tot}}} = \frac{8{,}829}{980 \times 18.0} = 0.50$
The aspect ratio of one wing is
$A\!R = \frac{b^2}{S} = \frac{6^2}{6.0} = 6.0$
The induced drag coefficient is
$C_{D_{i,\text{tri}}} = \frac{(1+\delta_{\text{tri}})C_L^2}{\pi A\!R} = \frac{1.35(0.50)^2}{\pi(6.0)} = 0.0179$

Therefore, the estimated induced drag coefficient of the triplane, based on the total wing reference area, is approximately

$C_{D_{i,\text{tri}}} = 0.018$

The factor $\delta_{\text{tri}}$ represents an effective induced-drag penalty from mutual interference and departures from the ideal independent-wing loading assumed in this simplified estimate.

Canards, Tandem Wings, & Other Coupled Lifting Surfaces

The ideas developed for biplanes and triplanes also apply more broadly to aircraft with multiple lifting surfaces. A canard airplane, such as the Rutan Long-EZ, as shown in Figure 10, is not usually called a biplane because its lifting surfaces are separated primarily in the longitudinal direction rather than stacked vertically. However, from an aerodynamic perspective, it remains a coupled lifting-surface configuration. The forward canard and the aft main wing both produce lift and trailing vortex systems, contributing to the airplane’s total lift, induced drag, pitching moment, and stability.

The Rutan Long-EZ is not a biplane in the usual geometric sense because its lifting surfaces are separated longitudinally rather than stacked vertically.

This distinction is important because a canard is a forward lifting surface with its own trim, stability, control, and stall constraints. In many canard aircraft, the forward surface carries significant positive lift in trimmed flight. Therefore, the total lift of the airplane may be written as

(92) $\begin{equation*} L = L_c + L_w \end{equation*}$

where $L_c$ is the lift on the canard and $L_w$ is the lift on the main wing. The trim condition determines how this lift is shared between the two surfaces. With moments taken about the center of gravity, one possible sign convention gives

(93) $\begin{equation*} L_c x_c + L_w x_w + M_c + M_w = 0 \end{equation*}$

where $x_c$ and $x_w$ are the moment arms of the canard and main-wing lift forces relative to the center of gravity, and $M_c$ and $M_w$ are the aerodynamic pitching moments of the two surfaces. Therefore, the lift carried by the canard and the main wing cannot be selected independently; the design solution is constrained by the airplane’s center-of-gravity position, airfoil pitching moments, elevator deflection, and stability requirements.

The induced-flow problem is also coupled. The canard creates a trailing vortex system that alters the effective flow experienced by the main wing. Depending on the relative vertical position, longitudinal spacing, span, and loading of the two surfaces, the main wing may experience a nonuniform upwash or downwash field from the canard wake. Likewise, the main wing contributes to the airplane’s overall induced velocity field. The analysis is similar in spirit to biplane theory, but with an important distinction that the lifting surfaces are separated mainly by stagger (i.e., in the fore-and-aft direction) rather than primarily by gap.

Although the longitudinal separation of the canard and main wing resembles stagger in the geometric sense, the canard problem is not primarily a stagger problem. The forward and aft lifting surfaces are part of a trim-constrained aircraft system. Their circulation distributions are determined by the center-of-gravity location, pitching-moment balance, incidence angles, elevator deflection, vertical offset, airfoil selection, and stall constraints. Therefore, the induced drag of a canard or tandem-wing airplane cannot be inferred from longitudinal separation alone. It must be determined from the coupled loading of the complete lifting system.

A useful idealization is to treat the canard and main wing as two lifting lines with circulation distributions $\Gamma_c(y)$ and $\Gamma_w(y)$ . The local section lift relation on each surface may be written as

(94) $\begin{equation*} C_l(y) = C_{l_\alpha}(y) \bigg( \alpha_{\rm geom}(y) - \alpha_i(y) - \alpha_0(y) \bigg) \end{equation*}$

where $\alpha_{\rm geom}(y)$ includes the local twist and control deflection, $\alpha_i(y)$ is the induced angle of attack, and $\alpha_0(y)$ is the local zero-lift angle of attack. For the canard and main wing, the induced angles may be written as

(95) $\begin{equation*} \alpha_{i,c}(y) = \alpha_{i,cc}(y) + \alpha_{i,cw}(y) \qquad \text{and} \alpha_{i,w}(y) = \alpha_{i,ww}(y) + \alpha_{i,wc}(y) \end{equation*}$

where the first subscript identifies the surface on which the induced angle is evaluated and the second subscript identifies the surface producing the induced velocity. The self-induced terms describe the usual finite-wing downwash of each surface, while the cross terms describe the mutual aerodynamic interference between the canard and the main wing.

The total lift is obtained from the circulation carried by both lifting surfaces, i.e.,

(96) $\begin{equation*} L = \varrho_\infty V_\infty \bigg( \int_{-b_c/2}^{b_c/2} \Gamma_c(y) \, dy + \int_{-b_w/2}^{b_w/2} \Gamma_w(y) \, dy \bigg) \end{equation*}$

and the induced drag may be written as

(97) $\begin{equation*} D_i = \varrho_\infty V_\infty \bigg( \int_{-b_c/2}^{b_c/2} \Gamma_c(y) \, \alpha_{i,c}(y) \, dy + \int_{-b_w/2}^{b_w/2} \Gamma_w(y) \, \alpha_{i,w}(y) \, dy \bigg) \end{equation*}$

This form shows why canard and tandem-wing aircraft must be analyzed as complete lifting systems. The ideal induced drag depends on the combined trailing-vortex system far downstream, but the actual circulation distributions are set by trim, stability, geometry, and control requirements.

Canard airplanes require especially careful analysis because lift sharing, trim, stability, stall behavior, and induced drag are all linked. A conventional airplane can often use its horizontal tail to provide a trimming moment with relatively small lift. A canard airplane usually relies on the forward surface to carry significant positive lift and to provide pitch control. For safe stall behavior, the canard is often designed to reach its maximum lift before the main wing. When the canard stalls first, the nose drops, and the main wing is prevented from reaching a deeper stall. This requirement strongly influences the canard airfoil, area, aspect ratio, pitch angle, elevator authority, and allowable center-of-gravity range.

The price paid for this arrangement is that the canard may limit the maximum lift coefficient of the complete airplane. If the canard must stall before the main wing, then the airplane’s maximum usable lift may be set by the forward surface rather than by the main wing. In addition, the canard wake may affect the main-wing loading and local stall progression. These effects are not captured by treating the main wing and canard as independent surfaces.

Therefore, canard and tandem-wing aircraft are best viewed as part of the broader family of multi-lifting-surface configurations. Although they are not biplanes in the usual geometric sense, they require the same kind of coupled aerodynamic reasoning. The lift distribution, induced drag, trim, stability, and stall behavior must be analyzed as properties of the complete lifting system.

Flying Wings

Flying wings are tailless aircraft in which the wing serves as both the primary lifting surface and the main structural body of the airplane. In the purest form, the payload, fuel, propulsion system, landing gear, and flight control surfaces are all integrated into the wing, with little or no separate fuselage and no conventional horizontal or vertical tail. This arrangement can offer reduced parasitic drag, improved structural and volumetric efficiency, and a larger fraction of the aircraft devoted to useful lift and payload. However, the same integration that makes the flying wing attractive also makes it difficult to design because the wing itself must provide lift, trim, longitudinal stability, directional stability, and control.

History

The flying-wing idea is almost as old as powered flight. Early designers recognized that the fuselage and tail of a conventional airplane produced weight and drag but relatively little lift. One early example was the Dunne D.8, a swept, tailless aircraft developed before WWI. Its design employed swept, wing-like geometry to provide stability without a conventional tail. Other early tailless and all-wing experiments followed, including work by Hugo Junkers and later by the German Horten brothers, whose Horten Ho 229 remains one of the most famous early jet-powered flying-wing designs.

In the United States, Jack Northrop became one of the strongest advocates of the flying-wing concept. Northrop’s work progressed through a series of experimental and prototype aircraft, including the Northrop N-1M, the Northrop N-9M, the large piston-powered Northrop YB-35, and the jet-powered Northrop YB-49. These aircraft demonstrated both the promise and the difficulty of the configuration. They offered the potential for high aerodynamic efficiency and large internal volume, but they also posed challenges for stability, control, propulsion integration, and operational practicality.

Interest in flying wings continued because the configuration offers advantages that are difficult to obtain with conventional airplanes. A smooth, blended wing shape can reduce drag-producing intersections and eliminate much of the wetted area associated with a separate fuselage and tail. The wing’s broad internal volume can also be used for fuel, payload, or distributed propulsion. Later designs, especially the Northrop Grumman B-2 Spirit, showed that the flying-wing arrangement could also provide major advantages for a low-observable airplane, as a smooth, tailless planform can reduce radar reflections when properly shaped.

Aerodynamics of Flying Wings

The main aerodynamic challenge is that a flying wing must satisfy several requirements simultaneously. A conventional airplane can use its horizontal tail for trim and longitudinal stability and its vertical tail for directional stability and yaw control. A flying wing must obtain these functions from planform shape, sweep, twist, airfoil pitching moments, elevons, split drag rudders, differential thrust, or other control devices. As a result, the spanwise loading of a flying wing must satisfy a broader set of design requirements than minimum induced drag alone, including trim, stability, and control.

Flying wing designs benefit from reduced parasitic drag compared to conventional airplanes of similar size and weight. Their continuous wing surfaces and carefully shaped aerodynamic profiles eliminate many of the typical sources of drag and flow interference found on traditional configurations. However, while conventional wing design has long aimed for an elliptical spanwise lift distribution to minimize induced drag for a prescribed span and lift, flying wings require a broader design compromise.

In contrast to the elliptical spanwise lift distribution, which minimizes induced drag for a fixed span, flying wings may use a bell-shaped or approximately bell-shaped spanwise loading to help provide directional stability and yaw control, as illustrated in Figure 11. This distribution concentrates more lift toward the inboard sections of the wing, reducing the wing root bending moment for a given lift and span. However, this comes at the cost of increased induced drag for a fixed span. One way to offset this penalty is to use a somewhat longer wing with a higher aspect ratio, subject to allowable root bending moment, wing weight, stiffness, aeroelastic, and configuration constraints.

The bell-shaped spanwise loading produces upwash over the outer wing, which helps provide a flying wing with directional stability and yaw control.

Creating such a bell-shaped spanwise loading, which is achieved through suitable variations in wing chord, sweep, twist, airfoil section, and control-surface bias, establishes upwash regions over the outer wing panels. This upwash can slightly tilt the local aerodynamic force forward in those regions, producing a locally negative induced drag. It is this forward inclination of the local force over the outer wing, rather than a finite force at the mathematical wingtip itself, that helps provide directional stability and yaw control. In addition, it can lead to proverse yaw in response to aileron or flaperon inputs,^[2] a feature especially important for flying wings, which lack a conventional vertical tail and rudder.

Bell-Shaped Loading

The aerodynamic consequences of the bell-shaped spanwise loading require further exposition. A starting point is the extended spanwise loading solutions discussed by Prandtl in 1933, which stemmed from Munk’s original formulation of lifting-line theory.^[3] Prandtl concluded that if the wing span or aspect ratio is not a design constraint, other forms of the spanwise lift distribution may help optimize the aerodynamic and structural efficiency of a wing. However, he makes no mention of any application to tailless airplanes or flying wings. Prandtl introduced a more general spanwise circulation loading function given by

(98) $\begin{equation*} \Gamma (y) =\Gamma_{0} \left( 1-\left(\dfrac{y}{s}\right)^2 \right)^{\mu} \end{equation*}$

where $\mu$ = 1/2 gives the classic elliptical loading and $\mu$ = 3/2 gives a “bell-shaped” loading. For the same total lift and span, the bell-shaped loading has a higher peak circulation at the wing centerline than the elliptical loading. It then decreases more rapidly toward the wingtip, where lift over the outer portion of the wing is much lower, thereby producing the characteristic “bell shape,” as shown in Figure 12.

The two idealized spanwise circulation loadings of interest for further comparative analysis are the classical elliptical form, as a reference, with $\mu$ = 1/2, i.e.,

(99) $\begin{equation*} \Gamma_e(y) = \Gamma_{0,e} \left( 1-\left(\dfrac{y}{s}\right)^2 \right)^{1/2} = \Gamma_{0,e} \sqrt{1-\left(\dfrac{y}{s}\right)^2} \end{equation*}$

and the bell-shaped form with $\mu$ = 3/2, i.e.,

(100) $\begin{equation*} \Gamma_b(y) = \Gamma_{0,b}\left( 1-\left(\dfrac{y}{s}\right)^2\right)^{3/2} \end{equation*}$

where $\Gamma_{0,e}$ and $\Gamma_{0,b}$ are the respective loadings, or circulation values, at mid-span. A standard symmetric wing of semi-span ${s = b/2}$ is assumed.

Downwash

There is a non-uniform downwash over the wing with the bell-shaped spanwise loading, compared to the uniform downwash obtained with the elliptical loading, with an upwash being produced over the outer part of the wing. For the well-known elliptical spanwise loading, the downwash, $w$ , is constant across the span, i.e.,

(101) $\begin{equation*} w_e = \frac{\Gamma_{0,e}}{4 s} = \text{constant} \end{equation*}$

For an untwisted elliptical planform, uniform downwash results in a constant induced angle of attack and a constant local lift coefficient across the span, representing the minimum-induced-drag configuration for the wing.

The downwash corresponding to the bell-shaped loading can be obtained from the lifting-line method discussed previously, which expresses the circulation as

(102) $\begin{equation*} \Gamma(\theta) = 4 \, s \, V_\infty \sum_{n=1}^{\infty} A_n \sin n\theta \end{equation*}$

where the coefficients $A_n$ can be determined from the prescribed loading. The bell spanwise loading distribution is

(103) $\begin{equation*} \Gamma_b(y) = \Gamma_{0,b}\left( 1-\left(\dfrac{y}{s}\right)^2\right)^{3/2} \end{equation*}$

Using the standard lifting-line transformation $y = -s\cos\theta$ , where $0\le\theta\le\pi$ , gives $\Gamma_b(\theta) = \Gamma_{0,b} \sin^{3}\theta$ . The trigonometric identity $\sin^{3}\theta = \dfrac{1}{4}\! \left( 3 \sin\theta - \sin 3\theta \right)$ shows that only the first and third harmonics appear in the loading distribution, i.e.,

(104) $\begin{equation*} \Gamma_b(\theta) = 4 \, s \, V_\infty \left( A_1 \sin\theta + A_3 \sin 3\theta \right) \end{equation*}$

Matching coefficients gives

(105) $\begin{equation*} A_1 = \frac{3 \Gamma_{0,b}}{16 \, s \, V_\infty} \quad \text{and} \quad A_3 = - \frac{\Gamma_{0,b}}{16 \, s \, V_\infty} \end{equation*}$

For any general circulation distribution, the downwash is

(106) $\begin{equation*} w(\theta) = V_\infty \sum_{n=1}^{\infty} n \, A_n \frac{\sin n\theta}{\sin\theta} \end{equation*}$

and substituting for the bell-shaped loading gives

(107) $\begin{equation*} { w_b(\theta) = V_\infty \left( A_1 + 3 A_3 \frac{\sin 3\theta}{\sin\theta} \right) = \frac{3 \Gamma_{0,b}}{16 s} \left( 1 - \frac{\sin 3\theta}{\sin\theta} \right) } \end{equation*}$

The trigonometric identity ${\sin 3\theta = 3 \sin\theta - 4 \sin^{3}\theta}$ gives $\dfrac{\sin 3\theta}{\sin\theta} = 3 - 4 \sin^{2}\theta$ , so

(108) $\begin{equation*} w_b(\theta) = \frac{3 \Gamma_{0,b}}{16 s} \left( 4 \sin^{2}\theta - 2 \right) = \frac{3 \Gamma_{0,b}}{8 s} \left( 2 \sin^{2}\theta - 1 \right) \end{equation*}$

The geometric transformation gives $\sin^{2}\theta = 1 - (y/s)^{2}$ , so

(109) $\begin{equation*} w_b(y) = \frac{3 \Gamma_{0,b}}{8 s} \left(1 - 2 \left( \frac{y}{s} \right)^{2}\right) \end{equation*}$

This result means that the downwash with the bell-shaped spanwise loading is a maximum at the wing root and becomes upwash outboard of $|y|/s = \dfrac{1}{\sqrt{2}} = 0.707$ , as shown in Figure 13.

The bell-shaped spanwise lift loading produces higher inboard downwash but upwash outboard of 71% of the semispan.

Wing Lift

The lift on a wing, according to the Kutta-Joukowski theorem, is given by

(110) $\begin{equation*} L = \bigintsss_{-s}^s \varrho_\infty \, V_\infty \, \Gamma(y) \, dy \end{equation*}$

The standard lifting-line transformation $y = -s\cos\theta$ can be used, for which $dy = s \sin\theta\, d\theta$ . In this case, the total wing lift is

(111) $\begin{equation*} L = 2 \varrho_{\infty} \, V_{\infty} \, s\int_{0}^{\pi/2} \Gamma(\theta) \, \sin\theta\,d\theta \end{equation*}$

Substituting $\Gamma_e = \Gamma_{0,e} \, \sin\theta$ for the elliptical loading gives

(112) $\begin{equation*} L_e = 2\varrho_{\infty} \, V_{\infty} \, s\, \Gamma_{0,e}\int_{0}^{\pi/2}\sin^{2}\theta\, d\theta = \varrho_{\infty} \, V_{\infty} \, s \, \Gamma_{0,e} \left( \frac{\pi}{2} \right) \end{equation*}$

In terms of the $\theta$ coordinate, the bell-shaped spanwise circulation distribution is given by

(113) $\begin{equation*} \Gamma_b(\theta) = \Gamma_{0,b} \left( \sin^2\theta \right)^{3/2} = \Gamma_{0,b} \, \sin^3\theta \end{equation*}$

Substituting this distribution into the lift equation gives

(114) $\begin{equation*} L_b = 2\varrho_{\infty} \, V_{\infty} \, s \, \Gamma_{0,b} \int_{0}^{\pi/2} \sin^{4}\theta \, d\theta = \varrho_{\infty} \, V_{\infty} \, s \, \Gamma_{0,b} \left( \frac{3\pi}{8} \right) \end{equation*}$

If the elliptically loaded and bell-loaded wings produce the same lift, then $L_b = L_e$ , so their peak values of circulation are related by

(115) $\begin{equation*} \dfrac{ \Gamma_{0,b}}{\Gamma_{0,e}} = \dfrac{\pi/2}{3\pi/8} = \dfrac{4}{3} \end{equation*}$

Induced Drag Distribution

Because of the upwash generated by the bell-shaped spanwise loading on the outer wing panels, the local aerodynamic force can be slightly inclined forward, resulting in locally negative induced drag. The downwash has been previously derived in Eq. 109, so the form of the local induced drag coefficient follows directly. For the elliptical loading, using the equal-lift reference value $\Gamma_m = \Gamma_{0,e}$ , then

(116) $\begin{equation*} C_{d,i,e}(\eta)= \frac{\Gamma_m^{2}}{2 \, s \, V_\infty^{2}}\, \frac{\sqrt{1-\eta^{2}}}{c(y)} \end{equation*}$

where $\eta = y/s$ . For the bell loading, using $\Gamma_{0,b} = 4\Gamma_m/3$ , then

(117) $\begin{equation*} C_{d,i,b}(\eta)= \frac{4\,\Gamma_m^{2}}{3 \, s \, V_\infty^{2}}\, \frac{\left(1-\eta^{2}\right)^{3/2}\! \left(1-2\eta^{2}\right)} {c(y)} \end{equation*}$

where ${c}$ is the local wing chord. As shown in Figure 14, for which a rectangular planform with ${c}$ = constant is assumed, the induced drag becomes negative outboard of $|y|/s$ = 0.707. Notice that when using Eq. 115, then

(118) $\begin{equation*} \frac{C_{d,i,b}(0)}{C_{d,i,e}(0)} = \frac{4/3}{1/2} = \frac{8}{3} = 2.67 \end{equation*}$

Induced drag distribution on wings with elliptical and bell-shaped spanwise loadings.

It is the forward inclination of the local aerodynamic force over the outer wing panels, as shown at the sectional level in Figure 15, that can contribute to directional stability and provide favorable yaw control characteristics on a flying wing. However, the net drag on the wing is still positive for a given span and aspect ratio, with the bell-shaped form of wing loading producing higher overall induced drag compared to a wing with elliptical loading. As shown later, the corresponding spanwise loading factor is positive, with $\delta_b=1/3$ .

The region of upwash over the outer wing panels produces a forward inclination of the local aerodynamic force.

Lift Coefficient Distribution

For a rectangular planform where the chord ${c}$ and section lift-curve slope are constant, the local section lift coefficient is directly proportional to the bound circulation. Consequently, the spanwise lift-coefficient distributions corresponding to the two loadings are

(119) $\begin{equation*} C_{l,e}(y)=\frac{2\,\Gamma_{0,e}}{V_{\infty} \, c} \sqrt{1-\left(\frac{2y}{b}\right)^{2}} \quad \text{and} \quad C_{l,b}(y)=\frac{2\,\Gamma_{0,b}}{V_{\infty} \, c} \, \left( 1-\left(\frac{2y}{b}\right)^{2} \right)^{3/2} \end{equation*}$

Normalizing by the respective mid-span values gives the purely geometric forms

(120) $\begin{equation*} \frac{C_{l,e}(y)}{C_{l,e}(0)} = \sqrt{1 - \left(\frac{2y}{b}\right)^{2}} \quad \text{and} \quad \frac{C_{l,b}(y)}{C_{l,b}(0)} = \left(1 - \left(\frac{2y}{b}\right)^{2}\right)^{3/2} \end{equation*}$

or

(121) $\begin{equation*} \frac{C_{l,e}(\eta)}{C_{l,e}(0)} = \sqrt{1 - \eta^{2}} \quad \text{and} \quad \frac{C_{l,b}(\eta)}{C_{l,b}(0)} = (1 - \eta^{2})^{3/2} \end{equation*}$

where $\eta = y/s$ . Therefore, for the same rectangular planform, the bell-shaped loading has substantially lower outboard $C_l$ values than the elliptical loading. This distribution can give a flying wing favorable stall qualities because the lower outboard loading helps delay tip stall, allowing the flaperons to remain effective deeper into the stall progression. By comparison, the elliptically loaded rectangular wing has relatively higher outboard loading than the bell-loaded wing, so it is less favorable from the standpoint of delaying outboard stall.

Wing Twist Distribution

It is necessary to use planform shape, chord distribution, airfoil selection, and nonuniform wing twist to produce the bell-shaped form of spanwise loading, with significant nose-down twist, or washout, generally being needed over the outer span. The general form of the nonlinear spanwise twist distribution, $\theta_{\rm tw}(y)$ , needed to create the bell-shaped loading is of the form

(122) $\begin{equation*} \theta_{\rm tw}(y) \ \propto \ A \eta^{2} + B \left( 1- \left(1- \eta^2\right)^{3/2} \right) \end{equation*}$

where $\eta = y/s$ , and where $A$ and $B$ depend on the planform, chord distribution, lift-curve slope, and airfoil zero-lift angle of attack.

For the bell-shaped loading, the spanwise circulation and induced velocity are

(123) $\begin{equation*} \Gamma(\eta) = \Gamma_{0,b}\left(1-\eta^{2}\right)^{3/2}, \qquad w(\eta) = \frac{3\Gamma_{0,b}}{8s}\left(1-2\eta^{2}\right) \end{equation*}$

and the corresponding induced angle of attack is

(124) $\begin{equation*} \alpha_i(\eta) = \frac{w(\eta)}{V_\infty} \end{equation*}$

For a wing with local chord $c(\eta)$ , the section lift coefficient is related to the circulation by

(125) $\begin{equation*} C_l(\eta) = \frac{2\Gamma(\eta)}{V_\infty c(\eta)} = \frac{2\Gamma_{0,b}}{V_\infty c(\eta)} \left(1-\eta^{2}\right)^{3/2} \end{equation*}$

Using the linear section lift relation,

(126) $\begin{equation*} C_l(\eta) = C_{l_\alpha} \left( \alpha_{\rm geom}(\eta) - \alpha_i(\eta) - \alpha_0(\eta) \right) \end{equation*}$

the local geometric angle of attack can be written as

(127) $\begin{equation*} \alpha_{\rm geom}(\eta) = \frac{C_l(\eta)}{C_{l_\alpha}} + \alpha_i(\eta) + \alpha_0(\eta) \end{equation*}$

The twist distribution relative to the root is then

(128) $\begin{equation*} \theta_{\rm tw}(\eta) = \alpha_{\rm geom}(0) - \alpha_{\rm geom}(\eta) \end{equation*}$

With this convention, positive $\theta_{\rm tw}$ corresponds to washout relative to the root section. Therefore,

(129) $\begin{equation*} \theta_{\rm tw}(\eta) = \frac{C_l(0)-C_l(\eta)}{C_{l_\alpha}} + \bigg( \alpha_i(0)-\alpha_i(\eta)\bigg) + \bigg( \alpha_0(0)-\alpha_0(\eta)\bigg) \end{equation*}$

If the ideal thin-airfoil value $C_{l_\alpha}=2\pi$ is used, then the induced-angle contribution is

(130) $\begin{equation*} \alpha_i(0)-\alpha_i(\eta) = \frac{3\Gamma_{0,b}}{4sV_\infty}\eta^2 \end{equation*}$

For a rectangular wing of constant chord $c$ , the section lift coefficient is

(131) $\begin{equation*} C_l(\eta) = \frac{2\Gamma_{0,b}}{V_\infty c} \left(1-\eta^2\right)^{3/2} \end{equation*}$

and the required twist distribution becomes

(132) $\begin{equation*} \theta_{\rm tw}(\eta) = \frac{\Gamma_{0,b}}{\pi V_\infty c} \left( 1-\left(1-\eta^2\right)^{3/2} \right) + \frac{3\Gamma_{0,b}}{4sV_\infty}\eta^2 + \bigg( \alpha_0(0)-\alpha_0(\eta)\bigg) \end{equation*}$

For a rectangular wing, $A\!R=2s/c$ , and the relation between the peak circulation and the wing lift coefficient for the bell-shaped loading is

(133) $\begin{equation*} \frac{\Gamma_{0,b}}{V_\infty c} = \frac{8C_L}{3\pi} \end{equation*}$

Therefore, the required twist distribution for a rectangular wing is

(134) $\begin{equation*} \theta_{\rm tw}(\eta) = \frac{4 C_L}{\pi A\!R}\,\eta^{2} + \frac{8C_L}{3\pi^{2}} \left( 1-\left(1-\eta^{2}\right)^{3/2} \right) + \bigg( \alpha_0(0)-\alpha_0(\eta)\bigg) \end{equation*}$

where $C_L$ is the design lift coefficient of the wing, such as in cruise flight. This equation represents the geometric twist required to produce the bell-shaped spanwise loading for the constant-chord case, as illustrated in Figure 16. The final term accounts for spanwise variation in the zero-lift angle of attack, which may arise from airfoil camber, reflex, or spanwise airfoil changes. If the wing has taper, sweep, airfoil variation, elevon bias, or other nonuniform features, the same lifting-line approach can be used, but the actual twist distribution must be adjusted for those effects.

To produce the bell-shaped spanwise loading, significant nonlinear wing twist is needed.

The same approach can be extended directly to a tapered wing. For a linearly tapered wing panel, the local chord over one semi-span may be written as

(135) $\begin{equation*} c(\eta) = c_r\left( 1-(1-\lambda_c)\eta\right) \end{equation*}$

where $c_r$ is the root chord, $c_t$ is the tip chord, $\lambda_c=c_t/c_r$ is the taper ratio, and $0\leq \eta \leq 1$ . The section lift coefficient required to produce the bell-shaped circulation is then

(136) $\begin{equation*} C_l(\eta) = \frac{2\Gamma_{0,b}}{V_\infty c_r} \frac{\left(1-\eta^2\right)^{3/2}} {1-(1-\lambda_c)\eta} \end{equation*}$

so the corresponding twist distribution becomes

(137) $\begin{equation*} \theta_{\rm tw}(\eta) = \frac{\Gamma_{0,b}}{\pi V_\infty c_r} \left( 1 - \frac{\left(1-\eta^2\right)^{3/2}} {1-(1-\lambda_c)\eta} \right) + \frac{3\Gamma_{0,b}}{4sV_\infty}\eta^2 + \bigg( \alpha_0(0)-\alpha_0(\eta)\bigg) \end{equation*}$

This result shows that the taper changes the required twist because the same prescribed circulation distribution produces a different sectional lift-coefficient distribution when $c(\eta)$ varies along the span. Figure 17 shows representative results for $A\!R=10$ and $C_L=0.6$ , with no spanwise variation in $\alpha_0$ . The rectangular wing corresponds to $\lambda_c=1.0$ , while smaller values of $\lambda_c$ represent increasing taper. In this example, increasing taper reduces the magnitude of the required nose-down twist over the outer wing because the decreasing chord already contributes to shaping the sectional loading. In practice, taper, sweep, airfoil variation, and elevon bias must all be included in the final lifting-line or lifting-surface calculation.

Required nose-down twist to produce the bell-shaped spanwise loading for linearly tapered wings with $A\!R=10$ and $C_L=0.6$ . The rectangular wing corresponds to $\lambda_c=1.0$ .

Airfoils for Flying Wings

Flying-wing configurations require airfoils that satisfy not only lift generation but also the pitching-moment characteristics needed for trim and stability in the absence of a horizontal tail. For a conventional cambered airfoil, the pitching-moment coefficient about the aerodynamic center is usually negative, i.e., $C_{m_{ac}} < 0$ , which tends to produce a nose-down moment. A conventional airplane can balance this moment with a horizontal tail, but a flying wing must satisfy the pitching-moment trim condition using the wing itself.

Flying wings commonly use reflexed airfoils because the wing alone must satisfy the pitching-moment trim requirement. Reflexed airfoils incorporate an upward curvature near the trailing edge, reducing the magnitude of the negative sectional pitching moment and often bringing $C_{m_{ac}}$ closer to zero. However, a reflexed airfoil alone does not guarantee trim. The trim condition is an integrated aircraft-moment condition that involves the spanwise lift distribution, airfoil pitching moments, sweep, twist, elevon deflection, planform shape, and center-of-gravity location.

The trim condition may be written in the general form

(138) $\begin{equation*} M_{\rm cg} = \int_{-s}^{s} \left( \varrho_\infty V_\infty \Gamma(y)\big(x_{\rm ac}(y)-x_{\rm cg}\big) + q_\infty c^2(y) C_{m_{ac}}(y) \right) dy = 0 \end{equation*}$

where $\varrho_\infty V_\infty \Gamma(y)$ is the local lift per unit span, $x_{\rm ac}(y)$ is the local aerodynamic-center location, $x_{\rm cg}$ is the center-of-gravity location, and $C_{m_{ac}}(y)$ is the sectional pitching-moment coefficient about the local aerodynamic center. This expression shows why the spanwise loading, airfoil pitching moments, planform, sweep, twist, and center-of-gravity location are coupled in a tailless aircraft.

The use of reflex also has aerodynamic penalties. Reflex generally reduces the effective camber, shifts the zero-lift angle, and lowers the section lift coefficient obtained for a given geometric angle of attack. It may also reduce the maximum lift coefficient and alter the pitching-moment variation with lift. These effects are especially important for small UAVs, where Reynolds-number effects can strongly influence airfoil performance, separation behavior, and control-surface effectiveness.

In the context of lifting-line theory, the sectional zero-lift angle enters directly through

(139) $\begin{equation*} C_l = C_{l_\alpha} \left( \alpha_{\rm geom} - \alpha_i - \alpha_0 \right) \end{equation*}$

so that variations in $\alpha_0$ along the span influence the effective angle of attack and, therefore, the spanwise loading. A distribution of $\alpha_0$ is aerodynamically equivalent to a distribution of geometric twist, and both must be considered when designing a flying wing to achieve a desired loading, such as the bell-shaped distribution. Consequently, the aerodynamic design of flying wings typically involves a coupled choice of airfoil shape, geometric twist, control-surface bias, center-of-gravity location, and planform to achieve acceptable performance, trim, stability, and control.

Wing Bending Moments

Prandtl’s 1933 argument concerned the relationship among spanwise loading, induced drag, and structural bending, using a bending-related measure as a proxy for wing structural weight. A full structural-weight comparison would require integrating the bending moment distribution, and in some formulations, the square of the bending moment, over the span. For the present introductory comparison, however, it is useful first to examine the simpler root-bending moment produced by the two idealized loading distributions. This root-bending comparison clearly indicates why the bell-shaped loading can reduce structural bending for a given lift and span, although it should not be interpreted as a complete structural-weight optimization. Later, R.T. Jones^[4] made similar arguments with similar outcomes.

The wing root bending moment for one semi-span is obtained by integrating the spanwise loading multiplied by its moment arm. For the right semi-span, let $y=s\cos\theta$ , with $0\leq \theta \leq \pi/2$ , so that

(140) $\begin{equation*} M_{\rm root} \ \propto \ \int_0^s y\,\Gamma(y)\,dy = s^2 \int_0^{\pi/2} \cos\theta \, \Gamma(\theta) \, \sin\theta \, d\theta \end{equation*}$

For the elliptical loading, where $\Gamma_e(\theta) = \Gamma_{0,e} \sin\theta$ , then

(141) $\begin{equation*} M_e \ \propto \ s^2 \Gamma_{0,e} \int_0^{\pi/2} \cos\theta \, \sin^2\theta \, d\theta = \left( \frac{s^2}{3} \right) \Gamma_{0,e} \end{equation*}$

For the bell-shaped loading, where $\Gamma_b(\theta) = \Gamma_{0,b} \sin^3\theta$ , then

(142) $\begin{equation*} M_b \ \propto \ s^2 \Gamma_{0,b} \int_0^{\pi/2} \cos\theta \, \sin^4\theta \, d\theta = \left( \frac{s^2}{5} \right) \Gamma_{0,b} \end{equation*}$

To produce the same total lift, the peak circulation of the bell-shaped loading must be $\Gamma_{0,b} = \dfrac{4}{3} \Gamma_{0,e}$ , as shown previously in Eq. 115. Substituting this relationship gives

(143) $\begin{equation*} M_b = \frac{4}{3}\Gamma_{0,e}\left(\frac{s^2}{5}\right) = \left(\frac{4s^2}{15}\right)\Gamma_{0,e} \end{equation*}$

so the ratio of the bending moments is

(144) $\begin{equation*} \frac{M_b}{M_e} = \frac{4s^2/15}{s^2/3} = \frac{4}{5} = 0.80 \end{equation*}$

This result indicates that the bell-shaped loading yields a lower wing root bending moment, specifically a 20% reduction compared to the elliptical loading for the same total wing lift and span, as illustrated in Figure 18. This outcome has significant implications for the design of a wing with a bell-shaped spanwise loading, as it can reduce structural loading or permit an increase in span, but it should not be interpreted as a complete structural-weight calculation.

The bell-shaped spanwise lift loading reduces the wing root bending moment by 20% for the same total wing lift and span.

Total Induced Drag

The induced drag for a wing with the bell-shaped spanwise loading distribution can now be determined from the lifting-line harmonic coefficients. For any spanwise loading written as

(145) $\begin{equation*} \Gamma(\theta)=4\,s\,V_\infty \sum_{n=1}^{\infty} A_n \sin n\theta \end{equation*}$

the total lift depends only on the first harmonic coefficient, $A_1$ , whereas the induced drag depends on the harmonic contribution

(146) $\begin{equation*} \sum_{n=1}^{\infty} n A_n^2 \end{equation*}$

The usual induced-drag factor may then be written as

(147) $\begin{equation*} 1+\delta = \frac{\sum_{n=1}^{\infty} n A_n^2}{A_1^2} = 1+\sum_{n=2}^{\infty} n\left(\frac{A_n}{A_1}\right)^2 \end{equation*}$

For the elliptical loading, only $A_1$ is present, so $\delta_e=0$ . For the bell-shaped loading,

(148) $\begin{equation*} \Gamma_b(\theta)=\Gamma_{0,b}\sin^3\theta = \frac{\Gamma_{0,b}}{4} \left(3\sin\theta-\sin3\theta\right) \end{equation*}$

and therefore

(149) $\begin{equation*} A_1=\frac{3\Gamma_{0,b}}{16sV_\infty} \qquad \text{and} \qquad A_3=-\frac{\Gamma_{0,b}}{16sV_\infty} \end{equation*}$

so that

(150) $\begin{equation*} \frac{A_3}{A_1}=-\frac{1}{3} \end{equation*}$

For the same total lift and span, the elliptical and bell-shaped loadings have the same value of $A_1$ . Therefore, the induced-drag factor for the bell-shaped loading is

(151) $\begin{equation*} 1+\delta_b = \frac{A_1^2+3A_3^2}{A_1^2} = 1+3\left(\frac{1}{3}\right)^2 = \frac{4}{3} \end{equation*}$

Thus,

(152) $\begin{equation*} \frac{C_{D_{i,b}}}{C_{D_{i,e}}} = 1+\delta_b = \frac{4}{3} = 1.333 \end{equation*}$

Because, in general, the induced drag coefficient is

(153) $\begin{equation*} C_{D_i}=\frac{(1+\delta)C_L^2}{\pi\,A\!R} \end{equation*}$

and because $\delta_e=0$ for the elliptical loading, then

(154) $\begin{equation*} \delta_b=\frac{1}{3} \end{equation*}$

This result shows that the bell-shaped spanwise loading produces 33.3% more induced drag than the elliptical distribution for the same total lift, wing span, and wing area. This outcome is expected because the bell-shaped loading contains a nonzero third harmonic, which increases the induced drag relative to the elliptical loading.

Check Your Understanding #3 – Bell-shaped loading and flying-wing tradeoffs

A flying wing is designed to use a bell-shaped spanwise loading instead of an elliptical loading. For the same total lift, span, and wing area, assume that the bell-shaped loading has an induced-drag factor

$1+\delta_b = \frac{4}{3}$

and a root-bending-moment ratio

$\frac{M_b}{M_e}=0.80$

relative to the elliptically loaded wing.

For the same span and lift, what is the induced-drag ratio $C_{D_{i,b}}/C_{D_{i,e}}$ ?
If the bell-loaded wing span is increased by 25% while keeping the same wing area, what is the new aspect-ratio ratio $A\!R_b/A\!R_e$ ?
What is the new induced-drag ratio relative to the original elliptically loaded wing?
What happens to the root bending moment after this 25% span increase?

Show solution/hide solution.

For the same span, lift, and wing area, the induced-drag ratio is
$\frac{C_{D_{i,b}}}{C_{D_{i,e}}} = 1+\delta_b = \frac{4}{3} = 1.333$

Therefore, the bell-shaped loading has 33.3% more induced drag than the elliptical loading for the same span.
If the span ratio is
$\lambda = \frac{b_b}{b_e}=1.25$

and the wing area is unchanged, then the aspect ratio scales with the square of span, so

$\frac{A\!R_b}{A\!R_e} = \lambda^2 = (1.25)^2 = 1.5625$
The induced-drag ratio becomes
$\frac{C_{D_{i,b}}^{(\text{new})}}{C_{D_{i,e}}} = \frac{4/3}{1.5625} = 0.853$

Therefore, after increasing the span by 25%, the bell-loaded wing has about 14.7% less induced drag than the original elliptically loaded wing.
For a fixed loading shape and fixed total lift, the root bending moment scales approximately with span. Therefore,
$\frac{M_b^{(\text{new})}}{M_e} = 0.80 \times 1.25 = 1.00$

The 25% span increase brings the bell-loaded wing back to the same root bending moment as the original elliptically loaded wing.

Design Application

The primary reason for using a bell-shaped spanwise loading is that, for flying wings or other tailless airplane configurations, the upwash over the outer wing panels helps provide directional stability and can reduce or even reverse adverse yaw effects. Ideally, this goal should be achieved without incurring either an aerodynamic or a structural penalty. Therefore, a final step is to determine the span increase that a bell-loaded wing could have while retaining the same root bending moment as an elliptically loaded wing, with the same total lift and wing area.

From Eq. 140, the wing root bending moment contains the factor $s^2\Gamma_0$ . However, in this design comparison, the total wing lift is held fixed. Because the total lift scales as

(155) $\begin{equation*} L \ \propto \ s\Gamma_0 \end{equation*}$

holding $L$ fixed requires $\Gamma_0\propto 1/s$ . Therefore, for a fixed total lift and a fixed loading shape, the root bending moment scales as

(156) $\begin{equation*} M_{\rm root} \ \propto \ s \end{equation*}$

or equivalently $M_{\rm root}\propto b$ .

From the previous results, the root bending moment under bell-shaped loading is 80% of that under elliptical loading for the same span and total lift, i.e.,

(157) $\begin{equation*} \frac{M_b}{M_e}=0.80 \end{equation*}$

Therefore, the span of a bell-loaded wing that produces the same root bending moment as an elliptically loaded wing can be increased by

(158) $\begin{equation*} b_b=b_e\left(\frac{1}{0.80}\right)=1.25\,b_e \end{equation*}$

which corresponds to a 25% increase in wing span. If the wing area is held constant, then the aspect ratio scales with the square of the span, so

(159) $\begin{equation*} \frac{A\!R_b}{A\!R_e} = \left(\frac{b_b}{b_e}\right)^2 = (1.25)^2 = 1.5625 \end{equation*}$

Using Eq. 153, the corresponding induced-drag ratio becomes

(160) $\begin{equation*} \frac{C_{D_i,b}^{(\text{new})}}{C_{D_i,e}} = \frac{4/3}{1.5625} = 0.853 \end{equation*}$

This result shows that, in this idealized root-bending comparison, adopting a bell-shaped loading and increasing the span without increasing the root bending moment can reduce the induced drag by about 14.7%. The reduction occurs because the higher aspect ratio more than offsets the induced-drag penalty associated with the non-elliptical loading. In general, if

(161) $\begin{equation*} \lambda=\frac{b_b}{b_e} \end{equation*}$

then

(162) $\begin{equation*} \frac{C_{D_i,b}}{C_{D_i,e}} = \frac{1+\delta_b}{\lambda^2} = \frac{4/3}{\lambda^2} \end{equation*}$

as shown in Figure 19. Therefore, the use of a sufficiently high aspect ratio can offset the aerodynamic penalty of a less-than-ideal spanwise loading distribution by reducing induced drag.

Induced drag ratio of the bell-shaped loading to elliptical loading as a function of span ratio, based on $C_{D_i,b}/C_{D_i,e}=(4/3)/\lambda^2$ .

Induced drag ratio of the bell-shaped loading to elliptical loading as a function of span ratio, based on $C_{D_i,b}/C_{D_i,e}=(4/3)/\lambda^2$ .

This result is important in airplane design and is not limited to flying wings. When geometric span alone constrains the wing design, the elliptical spanwise loading remains the minimum-induced-drag solution. However, the main reason for using an alternative spanwise loading, in this case the bell-shaped loading, is the upwash it produces over the outer wing panels, which can incline the local aerodynamic force slightly forward relative to the local freestream direction, as previously described. This condition can provide directional stability on a tailless flying wing and can also produce desirable proverse yaw effects when flaperons or ailerons are applied. This characteristic is crucial for many flying-wing designs. It is the primary aerodynamic advantage of designing the wing to give a bell-shaped spanwise loading, despite the penalty of a higher induced-drag factor, $\delta_b=1/3$ . If the wing span is not a strict design constraint, then the induced-drag penalty from the bell-shaped loading, or from another non-elliptical loading, can be offset by using a wing of higher span and aspect ratio. However, increased span also affects structural weight, stiffness, aeroelastic behavior, packaging, ground handling, and other design constraints.

For a fixed loading shape and fixed total lift, the root bending moment is proportional to the product of lift and span. Therefore, a 25% increase in span can be achieved without increasing the root bending moment when using the bell-shaped loading, because the bell loading has only 80% of the root bending moment of the elliptical loading at the same lift and span. This increase in span reduces induced drag because of the higher aspect ratio. Further increasing the wing span beyond this value will reduce induced drag further, but only by increasing the root bending moment and, therefore, the structural demand.

A simple first-order structural estimate can be made by combining the bell loading’s lower root bending moment with the span increase. If the span ratio is $\lambda=b_b/b_e$ , then the root-bending-moment ratio scales approximately as

(163) $\begin{equation*} \frac{M_b}{M_e} = 0.80\,\lambda \end{equation*}$

For $\lambda=1.25$ , this ratio is unity, so the root bending moment is unchanged. If $\lambda>1.25$ , then the root bending moment increases. In addition, the geometric size and structural volume of the wing will generally increase with span, so the aerodynamic benefit of increasing span must eventually be balanced against the resulting structural and aeroelastic penalties. Therefore, when drag, weight, stiffness, and practical configuration constraints are considered together, the optimal design will generally require only enough additional span to make the aerodynamic benefit worthwhile.

Summary & Closure

Biplanes, triplanes, and flying wings show why the classical isolated-wing model must sometimes be extended. All are governed by circulation, downwash, induced angle of attack, and induced drag, but each configuration introduces additional constraints involving interference, structure, trim, stability, and control.

For biplanes and triplanes, the main aerodynamic issue is mutual interference between lifting surfaces. Gap, stagger, decalage, planform, twist, relative span, and lift sharing all affect the loading, lift-curve slope, induced drag, and stall behavior. These configurations were attractive in early aviation because they provided large wing area and compact structural depth, but their interference, wetted area, strut, wire, and junction drag penalties eventually favored cantilever monoplanes.

Flying wings present a different design problem because the wing must provide lift, trim, stability, and control without conventional tail surfaces. Therefore, the best spanwise loading is not determined solely by induced drag. The central lesson is that aerodynamic optima are conditional, and the best configuration depends on the full set of aerodynamic, structural, stability, control, aeroelastic, manufacturing, and mission constraints.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

Why did biplanes dominate early aviation even though their aerodynamic efficiency was generally lower than that of clean monoplanes?
How do gap, stagger, and decalage influence the aerodynamic behavior, trim, and stall characteristics of a biplane?
Munk’s stagger theorem states that stagger alone does not change induced drag under ideal lifting-line assumptions. Why, then, was stagger still important in practical biplane design?
Why did triplanes offer attractive performance advantages during WWI, and why did those advantages largely disappear as monoplane structures improved?
What is the difference between the ideal induced-drag comparison of a biplane and the total drag comparison of a practical biplane aircraft?
Why is the elliptical lift distribution not necessarily the best spanwise loading for a flying wing?
How does bell-shaped spanwise loading produce outboard upwash, and why can this be useful for yaw control on a tailless aircraft?
What is meant by locally negative induced drag, and why does this not imply that the total induced drag of the wing is negative?
How are airfoil pitching moment, wing twist, sweep, elevon deflection, and center-of-gravity location coupled in the trim of a flying wing?
Under what design constraints can a bell-shaped loading become preferable even though it has higher induced drag than elliptical loading for the same span?

Other Useful Online Resources

For additional resources on biplanes, triplanes, flying wings, and related aerodynamic concepts, follow up on some of these online resources:

A short video explaining why monoplanes eventually replaced biplanes in most aircraft designs.
A useful video that discusses the history and design trade-offs of biplanes.
A video on the history of triplanes and why three-wing configurations briefly became important during WWI.
A video on the Sopwith Triplane and its importance in WWI fighter development.
A video explaining why the Fokker Dr.I was aerodynamically effective but also limited by drag and speed.
The National Museum of the U.S. Air Force fact sheet on the Fokker Dr.I Triplane.
The San Diego Air & Space Museum page on its Fokker Dr.I reproduction and the aircraft’s association with Manfred von Richthofen, the “Red Baron.”
A U.S. Air Force article on the historical development of Northrop flying wings and the later B-2 Spirit.
A video on the history of Northrop flying wings before the B-2 Spirit.
A NASA fact sheet on the Prandtl-D flying-wing research aircraft and its use of bell-shaped lift distribution.
The Smithsonian National Air and Space Museum page on the Prandtl-D Ship 1 research model.

von Mises, R. (1959). Theory of Flight. Dover Publications, New York, pp. 244–249. ↵
Proverse yaw occurs because the increased lift and reduced drag on the down-going aileron, facilitated by the upwash, can reduce or even reverse the adverse yaw effects that would typically occur with a conventional airplane. ↵
Prandtl, L., "Über die Tragfähigkeit der Flügel (On the Lift of Wings)," Zeitschrift für Flugtechnik und Motorluftschiffahrt, 1933. See also: Hunsaker, D. F., and Phillips, W., "Ludwig Prandtl’s 1933 Paper Concerning Wings for Minimum Induced Drag, Translation and Commentary," AIAA 2020-0644. AIAA Scitech 2020 Forum. January 2020. ↵
Jones, R. T., "The Spanwise Distribution of Lift for Minimum Induced Drag of Wings Having a Given Lift and a Given Bending Moment," NACA Technical Note 2249, Dec. 1950. ↵

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Introduction to Aerospace Flight Vehicles Copyright © 2022–2026 by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.

Introduction

Biplanes

History

Aerodynamics of Biplanes

Classic Theory

Effect of Gap on Biplane Interference

General Theory

Munk’s Stagger Theorem

Local Lifting-Line Solution

Lift Sharing

Illustrative Model

Design Tradeoffs for Biplanes

Triplanes

History

Aerodynamics of Triplanes

Canards, Tandem Wings, & Other Coupled Lifting Surfaces

Flying Wings

History

Aerodynamics of Flying Wings

Bell-Shaped Loading

Downwash

Wing Lift

Induced Drag Distribution

Lift Coefficient Distribution

Wing Twist Distribution

Airfoils for Flying Wings

Wing Bending Moments

Total Induced Drag

Design Application

Summary & Closure

License

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