The island the film floats

A soap film blown over a hole of a section’s shape solves the section’s torsion: the slope of the film is the shear stress, and twice the volume under it is the torque. A keyway cut into a round shaft is a change to the hole’s outline, and the film answers it at once, crowding into the keyway’s inside corners and spending some of its volume to do so. The film has one edge, it is pinned to that edge at height zero, and everything else follows.

A shaft with a hole bored along it is different, and not because the hole is a different shape. A lubrication passage drilled the length of a spindle, a cable run through a hollow axle, a bore off centre for a sensor: each leaves the section with two edges, the outline and the hole. The film over the section is pinned to the outline at height zero as before. Over the hole it has to meet something, and what it meets is a flat island — a weightless plate the size of the hole — whose height is not fixed by anything drawn. The question the keyway left was whether a bored hole crowds the film into its nearest wall as the keyway does, or whether the island’s height takes up the difference.

Where the island floats #

The island is part of the same membrane, under the same pressure, and it has to be in equilibrium. The pressure pushes it up over its whole area; the film pulls on its edge, down if the film round it slopes away and up if it slopes towards it. It floats where the two balance: the film’s slope integrated round the island’s edge equals twice the island’s area, in the units where the film’s equation is $\nabla^2\phi = -2$.

That is not an extra assumption brought in to fix the height. It is the condition that the shaft’s cross-section warps into a single surface round the hole — that going once round the hole and coming back to the start returns to the same axial displacement. A thin closed tube obeys exactly the same condition, and in a thin tube it is Bredt’s formula: the shear flow round the circuit is uniform, and twice the enclosed area times it is the torque. The bored shaft is a thick-walled tube with its hole off centre, and its island is the thick-walled version of the circuit a closing plate creates.

The film here is relaxed on a grid of 161 points across the shaft, each node next to an edge using its measured distance to that edge, and the island’s height is one more unknown, updated at every sweep from the flux round its edge. The check that it has been done correctly is the concentric hole, which is the hollow shaft: torsional constant $\pi(R^4 - a^4)/2$ and island height $(R^2 - a^2)/2$. For a 50 mm shaft with a 10 mm hole the film gives both within 0.3 per cent.

The left-hand film barely notices the hole. The contours that would have been circles round the shaft’s axis bend a little to pass round the island, and the island sits nearly at the height the solid film had there. That is the bored shaft: a section that has lost 4 per cent of its area and 7 per cent of its torsional stiffness.

The right-hand film is the same outline with the island pinned at height zero. A film pinned at zero round the hole is a film that sees the hole as part of the outside, and the section it describes is the shaft with a slit from the hole to the surface: a cut of no width that joins the hole’s edge to the outline, so that the two edges become one and there is no circuit. The film sags into the hole, the volume it loses is a third of the solid shaft’s, and the stress at the hole’s inner side reaches 2.81 times the solid surface stress.

The circuit is worth a third of the shaft #

The island’s height is the whole difference between a bore and a cut. Both shafts have exactly the same steel. One keeps 93 per cent of its stiffness and the other 67, and nothing distinguishes them but whether the material round the hole closes on itself.

Read the two curves from left to right. The bored shaft starts at 100 per cent with the hole on the axis — a 10 mm hole removes material where the film is flat and carries almost nothing — and loses stiffness slowly as the hole moves out, to 97 per cent at 10 mm off the axis and 93 at 15. Only in the last few millimetres, as the hole comes within a few millimetres of the surface, does it fall faster, to 89 per cent with 1.5 mm of steel left over the hole. A hole that close is a keyway’s worth of lost stiffness; anywhere else it is less.

The slit shaft runs the other way. Concentric, it is a thick ring cut through, an open section whose film can rise only as far as the ring’s thickness lets it, and it keeps 43 per cent. As the hole moves out the slit gets shorter and more of the shaft is solid again, so it recovers to 76 per cent. The gap between the two curves is what the closed circuit is worth, and it is largest exactly where the hole is safest — on the axis, where a slit costs the shaft 57 per cent and a bore costs it nothing. That is the thick-walled form of the factor a slit costs a thin tube: six hundred there, because a thin tube’s whole stiffness is its circuit, and a little over two here, because most of a thick shaft’s stiffness is in its solid material and only the rest is in the circuit.

The comparison with the area is the other thing the figure says. The hole removes 4 per cent of the area wherever it is, and the stiffness lost ranges from nothing to 11 per cent. Torsional stiffness is not a property of the area but of where the area is, weighted by how far the film would have risen over it; a hole on the axis removes the part of the film that was flat, and a hole near the surface removes the part that was steep.

One unknown per hole, as in a cell #

The island’s height is the same kind of unknown that every closed section carries, and seeing it that way explains why it cannot be guessed. A thin box with two cells has two circulating shear flows, and they are not fixed by equilibrium alone: the torque fixes only their weighted sum, and the second equation comes from asking that both cells twist at the same rate. A section with one hole through solid material has one such unknown, the island’s height, and one such equation, the balance of the island. Equilibrium of the solid steel round the hole says nothing about how much flow goes round the hole and how much turns back before reaching it; compatibility — the warping closing on itself — is what decides.

That is why the slit and the bore give such different answers from the same steel. A slit removes the compatibility condition and lets the two faces of the cut slide past each other along the shaft. The flow that would have circulated round the hole has nowhere to go but back, and turning it back costs the shaft its circuit. In a web that carries three actions at once the same distinction separates the flow a box carries round its cells from the flow its open parts carry across their thickness, and the closed part is always the stiffer, by a factor that grows as the square of the section’s size over its wall’s thickness.

It is also why a designer usually meets this result without the film. A hollow shaft is the commonest bored shaft there is, and its formula, $\pi(R^4 - a^4)/2$, already contains the island at its right height: concentric, the island’s balance gives $(R^2 - a^2)/2$ exactly, and a hollow shaft’s formula is what the floating island produces with no hole offset. What the film adds is the off-centre case, where no formula is printed in the handbooks a machine designer reaches for, and where the stiffness answer stays close to the hollow shaft’s long after the stress answer has moved away from it.

The stress answer matters more for a shaft than for most members, because a shaft turns. Every revolution under a bending load reverses the stress at the hole’s edge, and a shaft under steady torque with a fluctuating part sees the hole’s doubled stress as the range a fatigue check counts. The film gives the elastic peak; a fatigue assessment takes it as the stress concentration it then reduces by the material’s notch sensitivity.

A hole in a shear field #

Where the stress goes is the more important half, because the stiffness lost is small and the stress concentration is not.

The pattern round the hole is one that has nothing to do with shafts. In the solid shaft, near the hole’s position, the shear stress runs round the axis, so at the hole it runs in one direction — across the line from the shaft’s axis to the hole’s centre. A round hole in a field of shear running one way doubles the stress at the two points of its edge where the edge runs along the field, and leaves none at the two points where it runs across it. That is the anti-plane counterpart of the factor of three that a round hole puts into a plate in tension, and it is two, not three, because a shear field has no stress pressing across the hole for the edge to redistribute.

In the shaft the field is not uniform; it grows linearly from the axis. So the doubling applies to whatever the solid shaft’s stress would have been at each point of the hole’s edge. At the outer side, 20 mm from the axis, the solid shaft would carry 0.8 of its surface stress; doubled, 1.6. The film gives 1.54 a grid cell inside the edge, and the dashed curve follows the film round the outer half of the hole, where the stress is highest. On the inner half the film sits above the rule, because there the solid shaft’s stress falls towards the axis faster than any uniform field assumes, and the film round the island’s inner edge has to climb harder to reach it.

The rule turns a field calculation into one line. The largest stress in a shaft with a bored hole is about 2(e + a)/R times its solid surface stress — twice the solid stress at the hole’s outer edge, e + a from the axis — as long as that is more than one, and as long as the hole is clear of the surface. A small hole near the axis does nothing; a 10 mm hole 15 mm out gives 1.6 by the rule and 1.70 from the film, which resolves the edge more finely than the grid of the figure above.

When the steel over the hole runs out #

The rule assumes the hole sits in a field the shaft would have had anyway. When the hole comes close to the surface, the steel between them is a thin ligament carrying the whole shaft’s flow past the hole, and the field round the hole is no longer the shaft’s.

The curves leave the rule at about the same place on each: where the ligament — the steel between the hole’s outer edge and the surface — becomes thinner than the hole’s radius. For the 10 mm hole that is 5 mm of steel over it, a hole centred 15 mm out, where the film gives 1.70 against the rule’s 1.60; past that the stress rises much faster than the rule, to 2.15 with 3 mm of steel and 2.86 with 1.5 mm. The ligament is then carrying flow that would have spread over a hole’s width of shaft, and it carries it in a strip whose width is the ligament’s, so the stress goes roughly as the hole’s size over the ligament’s thickness rather than as the hole’s distance from the axis.

That puts the keyway in its place. The keyway in the same shaft, 14 mm wide and 5.5 mm deep with 0.4 mm fillets in its corners, has a peak stress of 2.81 times the solid surface stress at those fillets. A bored hole is better than a keyway anywhere until it comes within a millimetre or two of the surface, and worse than one after that. A 10 mm hole crosses the keyway at about 18.5 mm off the axis, 1.5 mm under the surface; a 16 mm hole crosses it at about 14 mm, with 3 mm of steel left. The larger the hole, the sooner it becomes the worst feature on the shaft.

Two practical rules follow from that and from the stiffness curve. A passage drilled along a shaft costs almost nothing in stiffness anywhere and nothing in stress near the axis, so if it can go near the axis it should. And if it has to go out towards the surface, the steel over it should be at least the hole’s own radius, which is where the doubling rule stops being enough; inside that, the hole has to be treated as a notch and checked as one.

The island near the surface #

What happens to the island as the hole moves out is a different view of the same transition.

A hole well inside the shaft floats its island a few per cent below where the solid film was — the film round the hole is pinned almost where it would have been anyway, and the field round it is the shaft’s. Near the surface the island sinks, to 37 per cent below with 1.5 mm of steel over it, because the thin ligament over the hole has too little width to hold the film up: the film climbs from zero at the surface to the island’s height across 1.5 mm, which is a steep slope, and the island has to sit lower for that slope to balance the pressure on it. That steep slope is the stress in the ligament. The island’s sinking and the ligament’s stress are one fact seen twice.

The volume under the island is the part of the torque the circuit carries — the flow that goes round the hole rather than being turned back by it — and it is small: 8 per cent of the torque with the hole on the axis, 2.6 per cent near the surface. That seems to contradict the slit shaft’s loss of a third of its stiffness. It does not, because pinning the island at zero changes the whole film, not just the part over the hole: a film with no island to climb to must fall back to zero round the hole, which takes away the volume over a whole annulus of the shaft rather than over the hole alone.

What the film leaves out #

A long shaft. Every number is Saint-Venant torsion, far from the ends. Where the hole stops — a blind bore, a cross-drilling meeting the axial one — the section changes along the length and the field is three-dimensional.

A smooth bore. The hole is a perfect circle. A drilled passage has a surface finish, and the doubling it carries is a doubling of whatever the surface’s own roughness already concentrates.

Elastic material. A shaft that yields first at the hole’s edge redistributes, and the sand heap that gives a section’s plastic torque has an island too — a flat-topped plateau over the hole at a height of its own.

One hole. Two passages, or a hole and a keyway together, interact through the film, and two islands float at heights that each depend on the other.

Still open: two islands that share a film #

A shaft with two bored holes, or a hollow axle with an off-centre passage drilled through its wall, has a film with two islands, each floating at a height set by its own balance and each affected by the other through the film between them. Whether two holes side by side at the same radius behave like one hole twice as wide — which the doubling rule would treat very differently from two holes, since the stress at a hole’s edge depends on its own size only through its distance from the axis — or whether the steel between them becomes a second ligament with its own concentration, is the question a second island puts to the film, and it is the one that decides how close together two passages may be drilled.