Bending moment — where it appears
Named by 10 essays across one field — each of them below, with the objects they name alongside it.
The loading nobody used
Elliptic loading is the least-drag answer to a question no aeroplane asks. Constrain the moment the lift makes about the wing root instead of the span, and a different curve comes out — five quarters of the span for sixty-four seventy-fifths of the drag, both exact — with an upwash over the outer wing and a yaw that turns the right way.
A wing that leaves the plane
A winglet is not a fence and it does not block anything escaping round the tip. The induced drag of any system of lifting surfaces depends on one cross-section of its wake and on nothing else whatever, and a wake that reaches upwards is cheaper for the same reason a wake that reaches sideways is.
A lighter spar turns a box wing into a biplane
Prandtl's best wing system has two-thirds of a monoplane's induced drag at a gap of a fifth of the span, and part of that saving is carried by circulation turning the corner into its fins. Ask the box to bend its root less and it pays about half what a monoplane pays — but a biplane with no fins pays nearly as little, and by the time the spar is a fifth lighter the fins carry almost nothing and the box has become the biplane it was built from.
A box wing's fins earn their keep in the spar
Constrain a box wing's root bending moment and its fins stop saving drag: it becomes a biplane. But that constraint counted the lift's moment, not the spars', and the fins join the spars into one frame. Joined at the tips, the two spars hand part of the moment to a couple across the gap — tension in one, compression in the other — which costs far less material than bending. With fins as stiff as the spars that is three-tenths of the moment; with rigid fins it stops at a third, whatever the gap, because a tip joint can only guide a tip. Counted in the spar, the box has less drag and a lighter spar than the elliptic monoplane at once.
A bending moment never chooses a span
Hold a wing's lift and its bending moment and ask for the least induced drag, and the answer does not stop: the span grows for ever and the tips push down. Both celebrated answers — Prandtl's bell at √(3/2) of the span for 8/9 of the drag, and the root-moment wing at 4/3 for 27/32 — are the span at which the least-drag loading's tips stop lifting. The structure sets how much is saved; a condition nobody states sets where the saving ends.
One wing root feels the rolling gust its lift cannot
The reverse-flow theorem says the gust weight of any load is the loading the wing would carry at an incidence shaped like that load's arm. For the lift that is the familiar uniform-incidence loading. For the bending moment at one root it is the loading of a wing washed in towards one tip only — a weight that is not symmetric, and so passes the rolling half of a gust that the lift and the two roots together cancel. That half is a sliver of the variance and adds up to a fifth to how often the root's moment crosses its mean.
Sweep slows a wing root's gust cycles and leaves their excess over the lift
On a straight wing every station meets a gust at the same instant, and one wing root's bending moment crosses its mean nearly a fifth more often than the lift, because it reads the rolling half of the gust. Sweep the wing back and the outer stations meet the gust later, by their distance aft. That delay averages the gust along the flight path as a second, softer chord would: at 35° every load crosses its mean about a tenth less often, the root slightly more than the lift since its weight sits further out, and the variance hardly changes at all. The rolling excess survives almost whole.
A spar that must lift itself stops the span at a quarter of the weight
Held only to a lift and a bending moment, the least-drag wing never stops growing. An aeroplane's wing stops because its spar has weight, and the weight has to be lifted. Count it, and the best span is the one at which the spar is exactly a quarter of the lift when the wing's area is fixed, and a third when its chord is — whatever the loading. The best loading then sheds some of its tip load, but not all the way to Prandtl's bell; with the chord held and nothing else charged, it is the bell exactly, and the more of the drag the wing's own area makes, the nearer it comes back to the ellipse.
A stiffness floor keeps the span and gives the tips their load back
A spar sized by strength alone thins to nothing at the tips, and charged as lift it stops the best span where it weighs a quarter of everything the wing carries, with the tips a little unloaded. A real outer wing is sized by stiffness, and stiffness sets a floor under the spar's weight that no reduction of the tip load can lower. Put that floor in and the tips get their load back — the best loading slides from a sixth of the way to the bell towards the ellipse — the quarter does not survive, rising to two fifths as the floor spreads over the outer two thirds of the span, and the span itself barely moves. A weight per metre cannot stop a span.
A strut inside a box wing is invisible to the wake, and halves the spar
Prandtl's best box wing has, everywhere inside it, air moving straight down: no sideways flow at all. A strut joining its two wings anywhere along the span therefore meets no flow it could turn, carries no load, and leaves the induced drag exactly where it was — the wake cannot see it. The spar can. A strut two fifths of the way out, as stiff as a spar and rigidly joined, halves the flange material of both spars, though it is pushed and pulled by almost nothing: the two wings lift equally and bend together, so the strut works through its joints, and a pinned one does nothing. Its only aerodynamic price is friction, and a slim strut pays it.
Named alongside it
The objects these essays reach for when they reach for this one.
Model limitInduced dragOptimisationSpan loadingThe Trefftz planeConstraintLifting lineSpan efficiencyBox wingCirculationDownwashFatigue