A box wing's fins earn their keep in the spar

A wake that closes on itself solved Prandtl’s best wing system of 1924: two wings, one above the other, joined at their tips by vertical fins, so that the wake behind them is a closed rectangle. At a gap of a fifth of the span it has two-thirds of the induced drag of an elliptic monoplane of the same span and lift. A lighter spar turns a box wing into a biplane then asked what that saving costs in structure. It held the root bending moment — the moment of the lift about the aircraft’s centreline, which is what sizes the spar, and the constraint under which Prandtl’s own loading of 1933 beats the ellipse — to a stated fraction of the elliptic monoplane’s, and found the fins’ share of the saving spent first: at a moment of eighty per cent, a box wing and a biplane of the same gap had the same drag to four decimal places.

That essay was explicit about what it had not done. The moment it held was the lift’s, summed over both wings, and how that moment is shared between the two spars “depends on how the tips are joined, which is structure and is not computed here”. A box wing’s fins are not only surfaces in the wake. They are members joining the ends of the two spars, and a structure joined that way does not carry a moment the way two separate spars do. This essay joins them, and asks whether the fins earn in the spar what they gave up in the wake.

Two ways to hold a moment #

At the root of a wing the lift’s moment about the centreline has to be held by something. A single spar holds it by bending: its upper flange in compression, its lower flange in tension, with the spar’s own depth — a few per cent of the chord, a hundredth of the span on an ordinary wing — as the lever arm between them. The flanges must carry the moment divided by that depth, and that is why the root moment sizes the spar: halve the depth and the flanges double.

Two spars a gap apart can do something one spar cannot. If they are joined, the upper spar can be put in compression along its length and the lower in tension, and that pair of forces is a couple whose lever arm is the gap — a fifth of the span on Prandtl’s box, twenty times the depth of either spar. A couple on a lever arm twenty times as long needs a twentieth of the flange force for the same moment. That is the whole reason a braced biplane was light: its interplane struts and wires made the two wings into one deep truss. Aerodynamically, two wings one above the other can sit anywhere along the stream without changing their total induced drag; structurally, where they are joined matters a great deal.

So the structural question has two parts. Statics fixes the total — the spars’ bending moments plus the couple must equal the lift’s moment, whatever the joint — but nothing in statics fixes how it is split. The split is decided by stiffness: by how easily each spar bends, how easily it stretches, and how stiffly the fins join them.

The frame #

The calculation takes half the box, since it is symmetric. Two horizontal spars half a span long are clamped at the centreline, one a gap above the other, and a vertical fin joins their tips with rigid joints. Each is an ordinary beam, with an axial stiffness and a bending stiffness. The spars are pairs of flanges a hundredth of the span apart. The fin’s stiffness is given as a multiple of a spar’s, and the natural value of that multiple is about one: a fin of the wing’s own section, turned on edge, bends sideways through its own thickness, just as a spar bends through the wing’s thickness. The spars carry the lift of the free box optimum at the same gap, taken from the Trefftz-plane solution of the essay before, and the frame is solved by the direct stiffness method.

The structural cost is counted as the structural moment: the moment a single spar of the same depth would carry for the same flange material. Bending counts in full; the couple counts at the spar’s depth over the gap. For a fin-less biplane, whose spars carry their own moments, the structural moment is simply the lift’s moment, and so it is for a box whose tip joints are ignored.

The couple, against the fin’s stiffness #

The first figure is the split. With fins a hundredth as stiff as the spars, the spars bend almost independently and the couple carries a few per cent. As the fins stiffen the couple grows, and with fins as stiff as the spars — a fin of the wing’s own section — it carries 29.5 per cent of the root moment at a gap of a fifth. The structural moment falls to 0.72 of the lift’s moment. Stiffer fins add little more: the curves flatten, and even rigid fins bring the couple only to 31 per cent.

The gap hardly matters, which is the first surprise. A larger gap gives the couple a longer lever arm but makes the fin longer and more flexible, and the two effects nearly cancel in the share the couple carries; what the larger gap buys is that the couple it does carry is cheaper, since its material goes as the depth over the gap.

Why a tip joint stops at a third #

The second figure explains the ceiling. Suppose the fins are rigid and the spars far stiffer in stretching than in bending, which they are: a spar a hundredth of the span deep is thousands of times stiffer along its length than across it. Then the two tips cannot move horizontally relative to each other, the rigid fin between them cannot rotate, and each spar becomes a cantilever whose tip is held level — a guided tip. A guided cantilever under uniform load has a root moment of $qL^2/3$ instead of $qL^2/2$, and a moment of $qL^2/6$ at its tip, bending it back the other way. The fin supplies that tip moment to both spars, and it can do so only by passing a horizontal force from one spar to the other: the couple. Two tips each needing $qL^2/6$ out of a total of $qL^2$ is a third.

The frame reproduces the third to five figures at every gap. For lift falling towards the tips — elliptic, or the box optimum’s — the guided tip moment is a smaller part of the root moment and the ceiling is lower, 31 per cent for the box. The gap does not appear, because a joint at the tip can do only one thing: stop the tips rotating. It cannot put shear into the spars along their length, which is what a truss does, and that is why a strutted biplane can carry nearly all its moment as a couple and a box wing joined only at the tips cannot carry more than a third.

A spar that bends back near its tip #

The third figure shows what the joint does along the spar. With the tips unjoined, the moment falls from its root value to nothing at the tip, as in any cantilever. Joined by a fin as stiff as the spar, the root carries 0.705 of what it did, the moment passes through zero a little under halfway out, at 0.43 of the half-span, and the outer half of the spar bends the other way, with 0.29 of the root’s unjoined moment at the tip. With a rigid fin the root carries 0.69 and the tip 0.31. The spar is working over its whole length, and the peak it has to be sized for has come down by nearly a third.

That has two consequences a designer would notice. The outer spar now needs material in both flanges for both signs of moment — a spar that was tapered to almost nothing at the tip cannot be, since its tip carries three-tenths of the root’s old moment. And the fin carries that moment too: it is a structural member, bending along its height, and its section and its joints are sized by the couple as much as by its own side force.

Counted in the spar, the box pays for its fins #

The fourth figure puts the frame back into the aerodynamic trade. The line is the monoplane: the least induced drag Prandtl’s 1933 problem allows at each root moment — the optimum that does not matter as long as the moment is free — rising steeply as the moment is held below the elliptic one — at 0.78 of the elliptic moment, the best monoplane pays 1.39 of the elliptic drag. The biplanes sit to the right of the box points, with more drag and moments near the elliptic value: their spars are separate, so their structural moment is their lift’s moment. A box whose tip joints are ignored sits just beside them, with less drag — 0.671 of the elliptic at a gap of a fifth — and a lift’s moment eight per cent above the elliptic, because the free box optimum carries its lift further out.

Counted as a frame, with fins as stiff as its spars, the same box moves left by nearly a quarter. At a gap of a fifth it needs a spar for 0.78 of the elliptic monoplane’s moment while paying 0.671 of its induced drag: less drag and less spar than the elliptic monoplane at once, where a monoplane with that spar would pay more than twice as much drag. At a gap of two-fifths, 0.80 of the moment for 0.544 of the drag. The essay before found that holding the lift’s moment spent the fins’ aerodynamic saving; counting the spar’s moment gives it back, with interest, because the constraint it held was a quarter too strict.

It follows that the free box optimum — the one with the lowest drag — is available to a box wing whose spar is sized like an elliptic monoplane’s, with a fifth of the spar to spare. No moment constraint binds at all until the spar is held below 0.78 of the monoplane’s.

What the spar’s own depth takes back #

The fifth figure varies the one number the others held fixed, the spar’s depth against the gap. The couple’s material goes as the depth over the gap, so the deeper the spar relative to the gap, the less a couple saves over bending: a spar a twentieth of the gap deep saves 28 per cent with fins as stiff as itself, one half the gap deep only 12. On a box wing with an ordinary section and a gap of a fifth of the span the spar is well to the left, and the saving is close to its full value; on a thick wing with a small gap it is much less, and the fins’ structural case weakens with it.

What the fins cost in the air #

The fins are paid for twice in the air, and only one of those payments was in the essay before. In the Trefftz plane they carry side force and shape the wake, which is how they lower the induced drag. On the aircraft they are also surfaces the air rubs along, and the second of a wing’s two drags — skin friction, in proportion to wetted area — is charged for them whether or not they carry any load. At a gap of a fifth of the span, two fins with the wings’ chord add a tenth to the wetted area of the two wings. At the lift coefficients of cruise, where induced drag is a minority of the total, a tenth more friction area is not small against a third off the induced drag, and the aerodynamic case for the fins is correspondingly narrower than the Trefftz plane alone suggests.

The structural case has no such charge against it. The fins are already there, already bending along their height, and the frame puts them to work with no added surface. Counted in the spar, the fins buy a quarter of the structure at a gap of a fifth; counted in the air, they buy a third of the induced drag and cost a tenth more friction. It is the first of those, not the second, that makes a box wing hard to argue against — which is the opposite of how the configuration is usually presented, as an aerodynamic idea with a structural price.

The same accounting applies to a wing that leaves the plane. A winglet is half a fin: it lowers the induced drag by spreading the wake out of the plane, and it adds friction and root moment at the tip, where a moment is dearest. A box’s fin is a winglet whose far end is held, and holding it is what turns a moment at the tip from a cost into a support.

What was checked #

The sixth figure is the ledger. A single clamped spar under uniform load gives the textbook root moment and tip deflection to two parts in $10^{10}$. Two spars joined by a rigid tip block, loaded unequally, were solved a second way, by the force method: cut the block from the upper spar, write the three unknown forces it passes, and solve the three compatibility conditions from closed-form cantilever flexibilities. The frame and the force method agree on both root moments and the couple to a part in a million. Statics holds at every fin stiffness — bending plus couple equal to the lift’s moment to $10^{-11}$ — which is the check that no load has leaked. And the guided limit, a third for a uniform load, comes out at every gap.

What the frame leaves out #

The fins’ own loads. The fins carry side force in the wake, small at the free optimum, and the frame here loads them only through the spars.

Torsion and the chordwise direction. The frame is plane. A real box wing’s rear wing is swept forward and its front wing back, and the joint at the tip then transfers torsion and chordwise bending as well; the spars are not in one vertical plane.

Flutter. A frame is stiffer in bending than two cantilevers, which helps, but joined wings have coupled modes that separate wings do not — the kind of coupling that makes flutter possible — and the structural moment is not the only thing a joined wing is sized for.

The loading held fixed. The couple’s share was computed for the free optimum’s lift. Rerunning the optimisation with the structural moment as its constraint would move the loading, and with it the share, slightly; since the free optimum already fits inside an elliptic monoplane’s spar, the rerun changes nothing until the spar is held much lower.

The convention the numbers depend on #

Drags and moments are ratios to the elliptic monoplane of the same span and lift. The lift’s moment is summed over both wings and taken about the centreline. The structural moment is bending plus the couple times the spar’s depth over the gap: the moment one spar of that depth would carry for the same flange material. Fin stiffness is the fin’s bending stiffness in the plane of the frame as a multiple of one spar’s, and the spars are two flanges a hundredth of the span apart unless stated.

Who worked it out #

Prandtl gave the box wing’s induced drag in 1924 and posed the monoplane’s moment-constrained optimum in 1933. The structural argument for joined wings — that joining the tips of two lifting surfaces lets them carry bending as a couple — is Wolkovitch’s, whose joined-wing papers of the 1980s made it the case for a swept-back front wing joined to a swept-forward rear one. Frediani and his colleagues at Pisa revived Prandtl’s box as the PrandtlPlane from the 1990s, and their designs are the reason the question has an audience.

Still open: joining the spars along the span #

The ceiling of a third comes from joining the spars only at their tips. A box wing can have more: a strut from the lower wing to the upper somewhere along the span, or a fin partway out, turns the frame into something closer to a truss, and a truss carries its moment almost entirely as a couple. Each strut is also a surface in the wake. The next calculation puts a strut at a stated fraction of the half-span, with its own wetted area and its own share of the wake in the Trefftz plane, and asks at what spanwise position a strut buys more in the spar than it costs in drag — whether a box with one strut on each side beats Prandtl’s plain box, measured the way this essay measures it.