Design and Analysis of Optimally-Loaded Lifting Systems

Ilan Kroo
Associate Professor
Dept. of Aeronautics and Astronautics
Stanford University
Stanford, California

This paper is an updated version of the AIAA paper 84-2507, published in Oct. 1984.
This paper is not copyright and is in the public domain. For information on related computed codes contact:


This paper describes a method for computing lift distributions of nonplanar lifting systems with minimum total drag and maximum lift subject to lift, structural weight, and trim constraints. Lifting surfaces are represented by a discrete vortex Weissinger model with lift-dependent section profile drag; induced drag calculations are based on Trefftz plane downwash. The method is applied to planar wings, for which the more realistic structural model produces results that differ significantly from Prandtl's classical result. The effectiveness of winglets in reducing wing drag at fixed weight is then examined. Finally, the method is used to compare canard, aft-tail, and three-surface configurations with both prescribed and optimal levels of longitudinal static stability.


AICaerodynamic influence coefficient matrix
ARaspect ratio, b^2/ S
bwing span
clocal chord
Cdsection drag coefficient
Cd0zero lift drag coefficient
Cd1lift-dependent drag coefficient (linear term)
Cd2lift-dependent drag coefficient (quadratic term)
Clsection lift coefficient
Cm0section zero lift moment coefficient
D0parasite drag
D1linear lift-dependent drag influences
D2quadratic lift dependent drag influences
DICdrag influence coefficient matrix
Jobjective function
l"spanwise" coordinate
LICmatrix of lift influences
Mpitching moment
M0pitching moment at zero lift
MICpitching moment influence coefficient matrix
Mbwing bending moment
nunit normal vector
qdynamic pressure
rdistance from vortex to control point
tairfoil thickness
U\inftyfreestream velocity
\vec{V}local velocity
WICweight influence coefficient matrix
x, y, zstreamwise, transverse, and vertical coordinates
\alphaanlge of attack
\lambdaLagrange multiplier
\Lambdawing sweep
\thetadihedral angle


CGcenter of gravity
x,y,zcomponents in x, y, z directions
indinduced (vortex) component
L,M,Wlift, moment, weight
Trefftzvalue in far wake
nnormal component
refreference value


Simple analytic results derived from three dimensional wing theory have provided useful guidelines for the design of "real" wings. Prandtl's solution for minimum induced drag with fixed span and total lift [1] constitutes a good first approximation to the ideal distribution of lift over an isolated wing. Prandtl later included structural weight effects by minimizing induced drag with a constant integrated value of wing bending moment [2]. Jones computed similar results using wing root bending moment as a constraint [3]. The minimum induced drag of non-planar wings and systems of wings is addressed by Munk [4] and by von Karman and Burgers [5]with a simple, approximate solution for two surfaces given by Butler [6] and Kroo [7]. Similar solutions for wings with winglets and for wing-tail combinations (with weight constraints) are discussed in Refs. 8-10. However, as the geometry becomes more complex, and as additional constraints are imposed, these analytical approaches become less useful.

A variety of computer programs have therefore been developed which permit calculation of minimum induced drag for rather general arrangements of lifting surfaces [11-12]. This paper describes such a program, used to compute optimal load distributions for systems of non-planar lifting surfaces. It is distinguished from currently available programs as it combines generality in both geometry and lift distributions with lift, trim, and/or structural weight constraints. The incorporation of section profile drag and maximum lift calculations make it a useful tool for comparing a variety of configuration concepts.

The following sections include a description of the method and several example results illustrating the effectiveness of winglets and the relative performance of canard, aft-tail, and three-surface designs.

Overview of the Method

The program computes the optimal distribution of lift on a system of lifting surfaces. This distribution is that which minimizes the total drag of the lifting system subject to any combination of the following constraints:

  1. Fixed total lift
  2. Trim with specified static margin
  3. Fixed structural weight

The constraints are specified by comparison with a reference wing of given span and area with rectangular planform and elliptic loading. The geometry of the lifting system is specified by defining a number of linearly-tapered elements with arbitrary sweep, taper, dihedral, and location. Results of the analysis include the relative drag of the system (normalized by the drag of the reference wing), the relative weight, the trimmed static margin (which is the optimal static margin unless constrained), the distribution of section lift coefficient, and the surface incidence distribution required to achieve these values.


The system of lifting surfaces is represented by a set of discrete horseshoe vortices. The relation between the strength of each vortex and the lift, induced drag, pitching moment, profile drag, and structural weight is computed by the methods discussed below. A system of linear equations is then constructed, the solution to which yields the vortex strengths producing minimum total drag subject to the imposed constraints. This representation provides very rapid analysis and lower storage requirements than vortex lattice methods with several chordwise panels.

Induced Drag

The induced drag of the vortex system is related to the distribution of "normal-wash" in the Trefftz plane (the velocity induced by the vortex system far downstream in a direction perpendicular to the bound vortex and the free-stream).

For the discrete vortex system:

The induced velocity in the Trefftz plane is related to the vortex strengths by the Biot-Savart law so that:

where Vnij is the velocity in the Trefftz plane at station i produced by unit vorticity at station j in a direction normal to the ith bound vortex and the freestream. This quadratic variation of induced drag may, therefore, be represented by:

where Gamma is a vector with components given by the vortex strength at each station and [DIC] is a matrix of coefficients from equation (1).


The total lift is assumed to vary linearly with the strengths of the vortices :

where $\theta$ is the dihedral angle. For the discrete vortex system:

Pitching moment (trim)

The pitching moment about the center of gravity is given by:

The difficulty here is the computation of the center of gravity location, given the distance from the center of gravity to the neutral point (as fixed by the desired static margin). The neutral point is computed with a non-planar, multi-element analysis program in which the wing is represented by skewed horseshoe vortices. Control points are placed at 3/4 chord points and total lift and moment are computed at unit angle of attack. The influence coefficients are saved and used later to compute the twist distribution required to achieve the desired loading.

Viscous Drag

It is important to include viscous drag in the determination of optimal load distribution. Since profile drag varies with lift coefficient, the inclusion of this term precludes extremely high section lift coefficients which might otherwise appear on planforms with small tip chords. The profile drag coefficient is assumed to vary quadratically with the local section lift coefficient:

Since Cl is related to the circulation, gamma, by:

with ci the local chord, the total viscous drag may be written:


The structural weight of the lifting systems is constrained by comparing systems with equal total area and with equal volumes of material required to support normal stresses due to bending. Although this excludes differences in the weight of material resisting shear loads, the error is negligible for wings of moderate or high aspect ratio. The weight of the fully-stressed bending material is proportional to the local bending moments so we may write:

The second sum includes all of the elements which are structurally connected to the ith element. The program decides which elements are connected, based on the input geometry. It should be noted that the bending material weight depends on the absolute value of the local bending moment. Should a solution be computed which produces a negative moment at certain stations, the program recomputes the WIC coefficients and the optimal load distribution.

Solution for minimum drag

The equations relating vortex strengths to the goal function and constraint equations are:

The problem is to minimize drag subject to the desired constraints. This is accomplished by defining a goal function with constraint equations appended with Lagrange multipliers:

The optimality conditions may be expressed by the equation:

Substitution of the previous expressions for drag, lift, moment, and weight in terms of the circulation at each station yields the following system:

When constraints are not imposed on lift, weight, or trim, the corresponding rows and columns of the matrix equation are deleted.

Solution for CLmax

The maximum lift coefficient is computed in a straightforward manner. Each panel is assigned a circulation based on the maximum section Cl and section chord. The panel or set of panels which produce the greatest change in moment for a given change in lift are identified and lift is removed as required to assure trim. If trim cannot be obtained without exceeding maximum section lift limits, the next most effective set of panels is identified, and the process repeated. At present no attempt has been made to model the section pitching moment changes associated with the deflection of high-lift devices (see Conclusions).

Computing the Twist Distribution

After the optimal circulation distribution has been determined (for minimum drag or maximum CL), the surface twist distribution is determined from the influence coefficient matrix used in the neutral point calculation. The neutral point was calculated by computing the circulation strengths associated with an untwisted surface at unit incidence:

[AIC] {gamma } = { 1 }

Since the circulation strengths are now known, a simple matrix multiplication determines the section incidences:

{alpha} = [AIC] { gamma}

Best Geometry

The preceding approach is used to compute the optimal circulation distributions for a specified geometry. In many cases it is the designers task to determine the geometry. The program described here may be used to investigate several specified designs, however, the method's extremely rapid execution makes it ideally suited for use with numerical optimization procedures. In the examples which follow, several parameters describing the configuration are varied and results are plotted for each case. For the two and three-surface designs, however, a simple 1-D optimizer was used to compute optimal span so that each plotted point represents several solutions.


Planar Wings

Figure 1 illustrates the relative drag of planar wings of various spans. The dashed curve represents Prandtl's solution for minimum induced drag with constant integrated bending moment. The potential savings of 1/9th of the induced drag with a span of SQRT(3/2) times that of the elliptical wing is attractive, but not predicted by the present method which suggests a savings of only 3% with a span of 105% of the elliptically-loaded reference wing.

The discrepancy arises from the fact that Prandtl's solution is not sensitive to the size of the wing structural box -- only to the lift distribution. The wings compared here have equal areas (so that variations in parasite drag and area-dependent structural weight do not complicate the comparison). Thus, as span is increased, the chord is reduced, and with specified section thickness- to-chord ratio, the depth of the structural box is reduced. Wings with very large spans are therefore penalized to a greater extent than predicted by considerations of bending moment alone. When profile drag effects are included, a savings of only 2% at CL=.5 with a 5% increase in span appears attainable.

Figure 1.

Similar results are plotted in figure 2 for wings with various amounts of linear taper. More highly tapered wings are structurally more efficient and permit larger spans for a given structural weight. However, the limitation placed on Cl by profile drag terms results in lower tip loads than otherwise desired. Thus the effectiveness of tapering the wing reaches diminishing returns near taper ratios of .1 to .2. If the airfoil thickness ratio were permitted to vary over the span the optimal taper ratio would increase. Off-design considerations and Reynolds number effects may further increase the desired taper ratio.

Figure 2.


In figure 3 the relative drag of wings with winglets is compared. Results are presented for winglets with a span of 10% and 20% of the wing semi-span with a taper ratio of 1.0. The expected shift in optimal span (7% lower optimal span with a 10% winglet) is accompanied by a shift in the minimum drag, with planar wings 1% to 2% better than wings with 10% winglets. This is attributable to the reduction of projected area and the corresponding increase in profile drag and structural weight.

Figure 3.

One would expect that winglets would be more effective if less area were moved from the wing to the winglet. Results for taper ratio of .2 (tip chord of winglet/root chord of wing) in figure 4 show that this is indeed the case. 10% winglets appear nearly as effective as planar wings with 5% greater span while 20% winglets show only a small penalty.

Figure 4.

Figure 5 illustrates the effect of winglet dihedral showing that no fundamental gains in performance may be achieved by canting the winglets inward or outward.

Figure 5.

It appears, therefore that either small winglets or tip extensions may be used to produce similar small performance gains. Additional factors must determine whether winglets or tip extensions are more appropriate for a given design. These include:

  1. CLmax: The projected area of the wings with 10% winglets in figure 4 is only 86% of the planar wing with equal wetted area.
  2. Interference: Boundary layer interference may pose greater problems for winglets -- especially those which are canted inward.
  3. Stability: The effects of winglets on lateral / directional stability may be adverse or favorable depending on the configuration.
  4. Structural constraints: Although the computed structural weight of each of the cases is constant, the material distributions differ markedly. When minimum gauge constraints are present the more uniform material distribution associated with winglets may carry an additional advantage.

Canard and Aft-Tail Designs

The relative drag of two-surface designs is depicted in figures 6 and 7. Drag is plotted vs the ratio of canard (or tail) span to wing span for three values of the parameter: canard (or tail) aspect ratio / wing aspect ratio. At each point the span of the wing has been varied and the minimum drag with optimal span, subject to lift, weight, and trim constraints, is plotted.

Figure 6.

Figure 7.

In these examples the total area is fixed at 150 sq ft; the reference wing has a 30 ft span with a design CL of .5. The center of gravity is 1 ft forward of the neutral point with a longitudinal separation between the two surfaces of 10 ft. A 1 ft vertical separation was used in figure 6 and a larger (5 ft) gap assumed for the analysis of figure 7.

Figure 6 illustrates the presence of two regions in this design space with superior performance. A minimum in the drag curve is achieved by canard designs with canard-to-wing span ratios between .2 and .4. A second minimum appears on the right side of the figure for aft-tail designs with tail-to-wing span ratios of .2 to .4 as well.

Because of the unfavorable interference of the canard on the wing, asymmetries appear in these curves. The best aft-tail designs achieve 2% to 3% lower drag than canard designs, and although in each case relatively high aspect ratio tail or canard surfaces are preferred, the drag is insensitive to the aspect ratio of aft-tail. Canard designs suffer large penalties in drag with low aspect ratio canard surfaces.

For a given wing span, tandem designs (bt/bw = 1) achieve the lowest induced drag. With a given structural weight, however, these designs are penalized severely, as indicated by the peaks at the center of figure 6. As the vertical separation between the surfaces is increased, the structural penalty is essentially unchanged, while the aerodynamic advantage is amplified. Figure 7 shows results for configurations similar to those in figure 6 except that the vertical gap has been increased from 1 ft to 5 ft. The major effect of this modification is a reduction in the drag of configurations with large span ratios. The peaks near the center of figure 6 are greatly reduced in figure 7. Tandem designs (with equal size surfaces) achieve drag ratios of about 1.0 with the larger vertical gap, compared with 1.12 for the smaller gap and .94 for the best aft tail design. The increased vertical gap also benefits canard designs with span ratios greater than .2 and aft-tail designs with bt/bw>.4. The increased gap is actually a disadvantage for small span aft-tails since wing/tail interference is favorable for tails carrying a download. The asymmetry between canard and aft-tail designs disappears when the trim constraint is relaxed. Results from the multiple surface optimization program without a trim constraint are plotted in figure 8. (Note that imposing no trim constraint is equivalent to setting the static margin to an optimum value.) The symmetry of figure 8 is expected from consideration of Munk's stagger theorem. Since the induced drag of a system of lifting surfaces is independent of their streamwise position (as of course are wing and tail structural weight and parasite drag) it follows that, for minimum drag, the load carried by a canard surface will be equal to that carried by an aft-tail of equal size. One parameter that is not independent of the streamwise position of wing and tail is the neutral point. Thus, this symmetry only applies when the static margin is not constrained.

Figure 8.

Relaxing the trim constraint reduces the drag throughout the design space, but its greatest impact appears on designs with small canards or tails -- and to a greater extent on canard designs than aft-tail designs. In fact, the configuration with least drag is a neutrally stable tailless design. The difference between the relative drag of this configuration (.87) and that of the best design which trims with a 12 inch stability margin (.94) shows the significant penalty associated with the requirement for stability and controllability.

Figure 9 shows the static margin at which the airplane trims with optimal load distributions. In order to achieve the drag values shown in figure 8, most canard designs must be statically unstable while most aft-tail designs require small positive static margins.

Figure 9.

These analyses have been performed for configurations with a given total area. However, it is often desirable to reduce wing area, and with it, the profile drag of the lifting surfaces. Stalling speed or field length constraints often limit the extent to which this is possible and thus, the maximum trimmed lift coefficient is an important measure of aircraft performance. The maximum trimmed CL of the configurations discussed previously (normalized by the maximum section lift coefficient) is shown in figure 10. As with total drag, the aft-tail configurations retain a small advantage over canard designs. Again, the maximum attainable lift coefficient is insensitive to tail aspect ratio while canard designs' CLmax varies strongly with aspect ratio. The highest CL is achieved by configurations with tail spans of 20% to 40% of the wing span. These are, conveniently, the same designs with lowest drag. The situation is less favorable for canard designs. Although small canards of high aspect ratio produce least drag, large canards of small aspect ratio achieve the highest CLmax. Moreover, the sensitivity of CLmax and drag to canard aspect ratio leads to greater compromises in each of these areas than would be required for an aft-tail design.

Figure 10.

The differences between aft-tail and canard configurations' maximum lift capability is again related to the trim constraint. There exists one position of the center of gravity for which each surface carries maximum lift. This optimal static margin is shown in figure 11. Nearly neutral stability is required for canard designs while static instabilities from 0 to 20% are necessary for aft-tail designs.

Figure 11.

The relative importance of these two performance indicies (CLmax and drag) depend on the aircraft's design mission. A design intended for high speed flight with a strict stalling speed constraint would be strongly affected by the maximum lift capability of the design while the comparisons of relative drag with fixed area applies more directly to an aircraft constrained by climb rate requirements.

Thus, the optimal design is influenced by the intended mission -- especially for canard designs with their greater sensitivity to aspect ratio changes and the large difference between the design with highest CLmax and the design with least drag. A typical compromise might consist of a canard design with equal wing and canard aspect ratios with bt/bw = .5. This design would achieve a CLmax of 72.5% that of an aft tail design with the same wing and tail areas and with bt/bw = .4. The drag of canard and wing would be 107% that of the wing/tail combination. Savings in propulsion system integration, fuselage layout, control system simplicity, etc., could conceivably lead one to favor the canard configuration, but in this example the initial aerodynamic compromise is large.

Three-Surface Designs

When design constraints make an aft-tail configuration unattractive (due to tail length limitations or stalling characteristics, for example) it is possible to combine some of the attributes of canard and conventional designs in a three-surface configuration.

The number of design variables increases dramatically as the number of surfaces is increased from two to three. This reduces the extent to which a few plots may cover the range of possible designs; figure 12, however, illustrates some interesting trends. In these cases the canard and tail are given the same span and aspect ratio. The tail is placed 10 ft behind the wing (as in the analysis of figure 6) and the canard is located 10 ft in front of the wing. The minimum drag occurs at canard and tail spans of 20% of the wing -- smaller than in the two-surface case. The minimum drag is slightly lower than wing/canard designs and slightly higher than wing/tail designs. The sensitivity to changes in canard/tail aspect ratio is similarly somewhere between the canard and aft-tail results of figure 6.

Figure 12.

The maximum trimmed lift coefficient is shown in figure 13. Again, the three- surface design resolves some of the primary difficulties with canard designs. The maximum lift coefficient is achieved with small canard and tail surfaces.

Figure 13.

Thus, the three-surface design provides distinct aerodynamic advantages over the wing/canard combination and, in this case, may closely approach the performance of a wing/tail system. No distinct performance advantage of the three-surface design as compared with the conventional design arises in this case, however.


The method described here permits rapid evaluation of the performance of configurations with multiple lifting surfaces at low speeds. Calculation of minimum total drag and maximum lift based on section properties yields an approximate, but useful comparison of wings with and without winglets and a variety of wing/tail/canard combinations. Although the results presented as examples of the application of this program are not comprehensive, certain conclusions are suggested.

Winglets do not provide performance advantages over planar wings from the standpoint of drag with given weight. However, they do not produce large penalties, so that if they are used to augment directional stability or if minimum gauge considerations are important, winglets may be advantageous.

Wing/aft-tail combinations achieve generally lower drag than wing/canard systems of equal weight and area. If the section CLmax is constant over all sections, aft-tail configurations exhibit greater maximum lift capability than canards of moderate aspect ratio. Relaxing static stability results in canard and aft-tail designs with very similar performance.

Three-surface configurations with small canard and tail surfaces do not experience the penalties associated with canard designs; however, unless restricted by limited tail length or other configuration dependent constraints, no obvious performance advantages apply to three-surface designs.

Several planned additions to the program would permit more accurate comparisons of performance. These include:


[1] Prandtl, L., Tiejens, O., Applied Hydro- and Aeromechanics, Dover Publ., New York, 1934.

[2] Prandtl, L., "Uber Tragflugel des Kleinsten Induzierten Widerstandes," Zeitshrift fur Flugtechnik und Motorluftschffahrt 24 Jg. 1933. (Reprinted in Tollmein, Schlichting, and Gortler, ed., Gesammelte Abhandlungen, Springer - Verlag, 1961.)

[3] Jones, R., "The Spanwise Distribution of Lift for Minimum Induced Drag of Wings Having Given Lift and Root Bending Moment," NACA TN-2249, 1950.

[4] Munk, M., "Minimum Induced Drag of Airfoils," NACA Rpt. 121, 1921.

[5] von Karman, T., Burgers, J.M., "General Aerodynamic Theory -- Perfect Fluids," Vol. II of Aerodynamic Theory, W. F. Durand, ed., Dover edition, 1963.

[6] Butler, G.F., "Effect of Downwash on the Induced Drag of Canard - Wing Combinaitions," J. of Aircraft, May 1982.

[7] Kroo, I., "Minimum Induced Drag of Canard Configurations," J. of Aircraft, Sept. 1982.

[8] Jones, R., Lasinski, T., "Effect of Winglets on the Induced Drag of Ideal Wing Shapes," NASA TM 81230, 1980.

[9] Kroo, I., McGeer, T.,"Optimization of Canard Configurations," ICAS-82-6.8.1, Aug. 1982

[10] McGeer, T., Kroo, I.,"A Fundamental Comparison of Canard and Conventional Configurations," J. of Aircraft, Nov. 1983.

[11] Lamar, J., "A Vortex Lattice Method for the Mean Camber Shapes of Trimmed Non-Coplanar Planforms with Minimum Vortex Drag," NASA TN-D-8090, 1976.

[12] Blackwell, J., "Numerical Method To Calculate the Induced Drag or Optimal Span Loading for Arbitrary Non-Planar Aircraft," NASA SP-405, May 1976.