Wing loading

The lift force L on a wing of area A, traveling at true airspeed v is given by

where ρ is the density of air and C_L is the lift coefficient. The lift coefficient is a dimensionless number which depends on the wing cross-sectional profile and the angle of attack.[9] At take-off or in steady flight, neither climbing nor diving, the lift force and the weight are equal. With L/A = Mg/A =W_Sg, where M is the aircraft mass, W_S = M/A the wing loading (in mass/area units, i.e. lb/ft² or kg/m², not force/area) and g the acceleration due to gravity, that equation gives the speed v through[10]

${\displaystyle \textstyle v^{2}={\frac {2gW_{S}}{\rho C_{L}}}}$ .

As a consequence, aircraft with the same C_L at takeoff under the same atmospheric conditions will have takeoff speeds proportional to ${\displaystyle \scriptstyle {\sqrt {W_{S}}}}$. So if an aircraft's wing area is increased by 10% and nothing else is changed, the takeoff 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 takeoff speed increases to ${\displaystyle \scriptstyle 150{\sqrt {1.4}}}$ = 177 mph.

Some flyers rely on their muscle power to gain speed for takeoff over land or water. Ground nesting and water birds have to be able to run or paddle at their takeoff speed before they can take off. The same is true for a hang glider pilot, though they may get assistance from a downhill run. For all these, a low W_S is critical, whereas passerines and cliff dwelling birds can get airborne with higher wing loadings.

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 having the effect of increasing the wing loading and also the stalling speed. This effect is also experienced during level pitching maneuvers.[11]

Load factor varying with altitude at 50 or 100 lb/sq ft

As stalling is due to wing loading and maximum lift coefficient at a given altitude and speed, this limits the turning radius due to maximum load factor. At Mach 0.85 and 0.7 lift coefficient, a wing loading of 50 lb/sq ft (240 kg/m²) can reach a structural limit of 7.33 g up to 15,000 feet (4,600 m) and then decreases to 2.3 g at 40,000 feet (12,000 m). With a wing loading of 100 lb/sq ft (490 kg/m²) the load factor is twice smaller and barely reaches 1g at 40,000 feet.[12]

Aircraft with low wing loadings tend to have superior sustained turn performance because they can generate more lift for a given quantity of engine thrust. The immediate bank angle an aircraft can achieve before drag seriously bleeds off airspeed is known as its instantaneous turn performance. An aircraft with a small, highly loaded wing may have superior instantaneous turn performance, but poor sustained turn performance: it reacts quickly to control input, but its ability to sustain a tight turn is limited. A classic example is the F-104 Starfighter, which has a very small wing and high 723 kg/m² (148 lb/sq ft) wing loading.

At the opposite end of the spectrum was the large Convair B-36: its large wings resulted in a low 269 kg/m² (55 lb/sq ft) wing loading that could make it sustain tighter turns at high altitude than contemporary jet fighters, while the slightly later Hawker Hunter had a similar wing loading of 344 kg/m² (70 lb/sq ft). The Boeing 367-80 airliner prototype could be rolled at low altitudes with a wing loading of 387 kg/m² (79 lb/sq ft) at maximum weight.

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 center at ${\displaystyle \scriptstyle {\frac {v^{2}}{R}}}$. That acceleration is caused by the inward horizontal component of the lift, ${\displaystyle \scriptstyle Lsin\theta }$, where ${\displaystyle \theta }$ is the banking angle. Then from Newton's second law,

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

Solving for R gives

${\displaystyle \textstyle R={\frac {2Ws}{\rho C_{L}\sin \theta }}.}$

The smaller the wing loading, the tighter the turn.

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 maneuverability. All need low wing loadings.

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 a wing loading of 650 kilograms per square metre (130 lb/sq ft) (excluding fuselage contributions to the effective area), whereas most delta wing aircraft (such as the Dassault Mirage III, for which W_S = 387 kg/m²) 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

${\displaystyle \textstyle a={\frac {GA}{M}}={\frac {G}{W_{S}}}}$,

decreasing with wing loading.

A further complication with wing loading is that it is difficult to substantially alter the wing area of an existing aircraft design (although modest improvements are possible). As aircraft are developed they are prone to "weight growth"—the addition of equipment and features that substantially increase the operating mass of the aircraft. An aircraft whose wing loading is moderate in its original design may end up with very high wing loading as new equipment is added. Although engines can be replaced or upgraded for additional thrust, the effects on turning and takeoff performance resulting from higher wing loading are not so easily reconciled.

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 average speed achieved across country can be increased to take advantage of strong thermals. With a higher wing loading, a given lift-to-drag ratio is achieved at a higher airspeed than with a lower wing loading, and this allows a faster average speed across country. The ballast can be ejected overboard when conditions weaken to maximize glider cross-country speed in gliding competitions.

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

A blended wing-fuselage design such as that found on the General Dynamics F-16 Fighting Falcon or Mikoyan 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.

Aircraft like the Grumman 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). When the wing is in the forward position takeoff and landing performance is greatly improved.[13]

Like all aircraft flaps, Fowler flaps increase the camber and hence C_L, lowering the landing speed. They also increase wing area, decreasing the wing loading, which further lowers the landing speed.[14]

• Anderson, John D. Jnr. (1999). Aircraft Performance and Design. Cambridge: WCB/McGraw-Hill. ISBN 0-07-116010-8.
• Spick, Mike (1986). Jet Fighter Performance-Korea to Vietnam. Osceola, Wisconsin: Motorbooks International. ISBN 0-7110-1582-1.