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# Encyclopedia

In aerodynamics, wing loading is the loaded weight of the aircraft divided by the area of the wing.[1] The faster an aircraft flies, the more lift is produced by each unit area of wing, so a smaller wing can carry the same weight in level flight, operating at a higher wing loading. Correspondingly, the landing and take-off speeds will be higher. The high wing loading also decreases maneuverability. The same constraints apply to birds and bats.

A highly loaded wing on a Lockheed F-104 Starfighter.

## Units

Aircraft "weights" are always given as masses, i.e. in units like kilograms or pounds. Wing loadings are nearly always given in either lb/ft2 or kg/m2. Occasionally force units replace mass, so then the wing loading is in N/m2. To get from lb/ft2 to kg/m2, multiply by 4.88; to get from kg/m2 to N/m2, multiply by 9.81. This last number is the standard value of the gravitational field in m/s2; the real field, g varies slightly with place and altitude but by less than about 1%.

Aircraft Fun 160[2] ASK 21 Nieuport 17 Ikarus C42 Cessna 152 Vans RV-4 Spitfire V DC-3 B-17 Bf 109 B-36 Eurofighter F-104 A380 B747
(kg/m2)
6.3 33 38 38 51 67 120 123 190 210 272 311 514 663 740
Role hang glider glider fighter microlight private sports fighter airliner bomber fighter bomber fighter fighter airliner airliner
Date 2007 1990 1916 1997 1978 1980 1940 1936 1938 1937 1949 1998 1958 2007 1970

The table, which shows wing loadings is intended to give a feeling for the range of wing loadings used by fixed wing aircraft. Maximum weights have been used. There will be variations amongst variants of any particular type. The dates are approximate, indicating period of introduction.

The critical limit for bird flight is about 5 lb/ft2 (25 kg/m2)[3]. An analysis of bird flight which looked at 138 species ranging in mass from 1x10-2 to 10 kg, from small passerines to swans and cranes found wing loadings from about 1 to 20 kg/m2[4]. The wing loadings of some of the lightest aircraft fall comfortably within this range. One typical hang-glider (see table) has a maximum wing loading of 6.3 kg/m2, and an ultralight rigid glider[5] 8.3 kg/m2.

## Effect on performance

Wing loading is a useful measure of the general maneuvring performance of an aircraft. Wings generate lift owing to the motion of air over the wing surface. Larger wings move more air, so an aircraft with a large wing area relative to its mass (i.e., low wing loading) will have more lift at any given speed. Therefore, an aircraft with lower wing loading will be able to take-off and land at a lower speed (or be able to take off with a greater load). It will also be able to turn faster.

### Effect on take-off and landing speeds

Quantitatively, the lift force L on a wing of area A, travelling at speed v is given by

$\textstyle\frac{L}{A}=\tfrac{1}{2}v^2\rho C_L$ ,

where ρ is the density of air and CL is the lift coefficient. The latter is a dimensionless number of order unity which depends on the wing cross-sectional profile and the angle of attack. At take-off or in steady flight, neither climbing or diving, the lift force and the weight are equal. With L/A = Mg/A =WSg , where M is the aircraft mass, WS = M/A the wing loading and g the acceleration due to gravity, that equation gives the speed v through

$\textstyle v^2=\frac {2gW_S} {\rho C_L}$ .

As a consequence, aircraft with the same CL at take-off under the same atmospheric conditions will have take off speeds proportional to $\scriptstyle\sqrt {W_S}$. So if an aircraft's wing area is increased by 10% and nothing else changed, the take-off speed will fall by about 5%. Likewise, if an aircraft designed to take off at 150 mph grows in weight during development by 40%, its take-off speed increases to $\scriptstyle150 \sqrt{1.4} = 177$ mph.

Some flyers rely on their muscle power to gain speed for take-off over land or water. Ground nesting and water birds have to be able to run or paddle at their take-off speed and the same is so for a hang glider pilot, though he or she may get an assist from a downhill run. For all these a low WS is critical, whereas passerines and cliff dwelling birds can get airborne with higher wing loadings.

### Effect on climb rate and cruise performance

Wing loading has an effect on an aircraft's climb rate. A lighter loaded wing will have a superior rate of climb compared to a heavier loaded wing as less airspeed is required to generate the additional lift to increase altitude. A lightly loaded wing has a more efficient cruising performance because less thrust is required to maintain lift for level flight.

The second equation given above applies again to the cruise in level flight, though ρ and particularly CL will be smaller than at take-off, CL because of a lower angle of incidence and the retraction of flaps or slots; the speed needed for level flight is lower for smaller WS.

The wing loading is important in determining how rapidly the climb is established. If the pilot increases the speed to vc the aircraft will begin to rise with vertical acceleration ac because the lift force is now greater than the weight. Newton's second law tells us this acceleration is given by

$\textstyle Ma_c=\tfrac{1}{2}v_c^2\rho C_LA -Mg$

or

$\textstyle a_c=\frac{1}{2W_S}v_c^2\rho C_L -g,$

so the initial upward acceleration is inversely proportional to WS. Once the climb is established the acceleration falls to zero as the sum of the upward components of lift plus engine thrust minus drag becomes numerically equal to the weight.

### Effect on turning performance

To turn, an aircraft must roll in the direction of the turn, increasing the aircraft's bank angle. Turning flight lowers the wing's lift component against gravity and hence causes a descent. To compensate, the lift force must be increased by increasing the angle of attack by use of up elevator deflection which increases drag. Turning can be described as 'climbing around a circle' (wing lift is diverted to turning the aircraft) so the increase in wing angle of attack creates even more drag. The tighter the turn radius attempted, the more drag induced, this requires that power (thrust) be added to overcome the drag. The maximum rate of turn possible for a given aircraft design is limited by its wing size and available engine power: the maximum turn the aircraft can achieve and hold is its sustained turn performance. As the bank angle increases so does the g-force applied to the aircraft, this has the effect of increasing the wing loading and also the stalling speed. This effect is also experienced during level pitching manouevers. [6]

Like any body in circular motion, an aircraft that is fast and strong enough to maintain level flight at speed v in a circle of radius R accelerates towards the centre at $\scriptstyle\frac{v^2} {R}$. That acceleration is caused by the inward horizontal component of the lift, $\scriptstyle L sin\theta$, where θ is the banking angle. Then from Newton's second law ,

$\textstyle\frac{Mv^2}{R}=L\sin\theta=\frac{1}{2}v^2\rho C_L A\sin\theta.$

Tidying up gives

$\textstyle R=\frac{2W_s}{\rho C_L\sin\theta}.$

Gliders designed to exploit thermals need a small turning circle in order to stay within the rising air column, and the same is true for soaring birds. Other birds, for example those that catch insects on the wing also need high manouevrability. All need low wing loadings.

### Effect on stability

Wing loading also affects gust response, the degree to which the aircraft is affected by turbulence and variations in air density. A small wing has less area on which a gust can act, both of which serve to smooth the ride. For high-speed, low-level flight (such as a fast low-level bombing run in an attack aircraft), a small, thin, highly loaded wing is preferable: aircraft with a low wing loading are often subject to a rough, punishing ride in this flight regime. The F-15E Strike Eagle has been criticized for its ride quality, despite its wing loading of 650 kg/m2 (excluding fuselage contributions to the effective area), as have most delta wing aircraft (such as the Dassault Mirage III, for which WS = 387 kg/m2) which tend to have large wings and low wing loadings.

Quantitatively, if a gust produces an upward pressure of G (in N/m2, say) on an aircraft of mass M, the upward acceleration a will, by Newton's second law be given by

$\textstyle a=\frac {GA} {M}=\frac {G} {W_S}$,

### Water ballast use in gliders

Modern gliders often use water ballast carried in the wings to increase wing loading when soaring conditions are strong. By increasing the wing loading the lift-to-drag ratio is increased at higher airspeeds. The ballast can be dumped overboard when conditions weaken [8]. (see Gliding competitions)

## Design considerations

### Fuselage lift

The F-15E Strike Eagle has a large relatively lightly loaded wing

A blended wing-fuselage design such as that found on the F-16 Fighting Falcon or MiG-29 Fulcrum helps to reduce wing loading; in such a design the fuselage generates aerodynamic lift, thus improving wing loading while maintaining high performance.

### Variable-sweep wing

Aircraft like the F-14 Tomcat and the Panavia Tornado employ variable-sweep wings. As their wing area varies in flight so does the wing loading (although this is not the only benefit). In the forward position takeoff and landing performance is greatly improved.[9]

### Fowler flaps

The use of Fowler flaps increases the wing area, decreasing the wing loading which allows slower landing approach speeds.

## References

### Bibliography

• Meunier, K. Korrelation und Umkonstruktionen in den Größenbeziehungen zwischen Vogelflügel und Vogelkörper-Biologia Generalis 1951: p403-443. [Article in German]
• Thom, Trevor. The Air Pilot's Manual 4-The Aeroplane-Technical. 1988. Shrewsbury, Shropshire, England. Airlife Publishing Ltd. ISBN 1-85310-017-X
• Spick, Mike. Jet Fighter Performance-Korea to Vietnam. 1986. Osceola, Wisconsin. Motorbooks International. ISBN 0-7110-1582-1