Two islands in one film

The island the film floats bored one hole along a round shaft and followed what Prandtl’s soap film does over the section. The film is pinned to the shaft’s outline, and over the hole it meets a flat island that is free to float: it rises until the film’s pull round its edge balances the pressure on it. The island’s height is the extra unknown a hole adds, the warping condition that keeps the shaft from tearing round the hole, and it decided everything that essay found — how much stiffness the hole costs, where the stress peaks, and why the same hole slit to the surface costs ten times as much.

That essay ended on a shaft with two holes. Two holes put two islands in the film, each floating at its own height, and between them a strip of film — the steel between the holes — stretched from one island to the other. Whether that strip is flat or steep is decided by how high each island floats, and each island’s height is decided by where its hole is. The question the earlier essay left was whether two holes side by side behave like one hole twice as wide, or whether the steel between them becomes a second ligament with its own concentration. The answer turns out to depend not on how close the holes are but on which way they lie.

A film with two islands #

The shaft is 50 mm in diameter and the holes 8 mm, each with its centre 12 mm from the axis — far enough out that a hole matters, far enough in that the steel outside it is a generous 9 mm. The film is solved over the section as the earlier essays solved it, by relaxation on a fine grid with the boundary handled cell by cell, and now with two islands, each floating where the film’s pull round its own edge balances the pressure on it. The torsional stiffness is twice the volume under the film, the islands included, and the shear stress at any point is the film’s slope there.

Each island obeys its own balance, and the two balances are coupled only through the film between them. That is the whole of what a second hole adds that a bigger first hole does not. One hole twice as wide has one island; two holes have two, and the film between them can do something the steel inside one big hole could not.

Side by side, the islands float level #

Put the two holes at the same distance from the axis, either side of a radius. By symmetry their islands must float at the same height, and they do: 219, a little below the 227 one such hole would float at alone. Between two islands at the same height the film has nowhere to go. It sags a little under its pressure and rises again, but its slope — which is the stress — is small. At the midpoint between two holes whose edges are 2 mm apart the steel carries 3 per cent of the solid shaft’s surface stress.

So the steel between two holes at one radius is not a ligament in the sense that a thin wall is. A thin wall between a hole and the surface carries the whole shear flow that has to pass between them, because the film must fall from the island’s height to zero across it. A strip of steel between two islands at the same height has no fall to make.

That makes the pair behave like neither of the things it was expected to. It is not one hole twice as wide: an 8 mm hole’s worth of steel sits between the two holes doing nothing useful, but it is not missing either, and the film over it holds the two islands apart in a way a single wide hole’s island could not. And it is not two independent holes either, because each island floats a little lower for the other’s company.

What two holes cost #

Turn the pair round the axis, from nearly touching to opposite each other, and almost nothing changes. The shaft keeps 96 per cent of its stiffness with the holes nearly touching and 95 with them opposite — very nearly twice the 3 per cent one hole costs, whatever the angle between them. Its peak stress stays between 1.13 and 1.18 times the solid shaft’s surface stress, which is the same as one hole’s 1.17. Two holes at one radius cost twice one hole’s stiffness and none of its stress.

The alternatives cost more of both. One hole with the pair’s total area, 11.3 mm across, keeps 94 per cent and peaks at 1.37. One hole twice as wide, 16 mm, keeps 87 per cent and peaks at 1.83. A designer who needs a given flow area along a shaft — coolant, oil, a cable — and has the choice between one large passage and two small ones gets a stiffer, less stressed shaft from the pair. The stress difference is the larger of the two, and it has a direct reading in the film: a hole’s peak is at its outer edge, where the film must fall from the island to the surface, and a large hole brings its outer edge — and so that fall — closer to the surface than two small ones at the same radius do.

The contrast with a hole in a plate pulled in tension is worth a sentence. There, the concentration factor does not depend on the hole’s size at all — a small hole and a large one in a wide plate both triple the stress beside them. In a twisted shaft the hole’s size matters through the steel it leaves outside it, because the stress a shaft carries rises toward its surface, and the film’s fall from island to outline has to happen across whatever steel is left.

A ring of holes #

The side-by-side result extends to any number of holes at one radius, which is the arrangement of the lightening holes in a hub or the bolt holes round a flange. Three 8 mm holes equally spaced at 12 mm cost 7.2 per cent of the stiffness, four cost 9.2, six cost 11.8: a little less than in proportion to their number, because each island floats a little lower as more are added — from 227 for one hole to 200 for six — and an island that floats lower holds the film down less round it. The peak stress does not move: 1.14 to 1.17 times the solid shaft’s surface stress for any number from one to six, and from four holes onward it is at the shaft’s surface between the holes rather than at a hole’s edge.

A ring of holes at one radius costs stiffness in proportion to its count and stress not at all, which is a better bargain than any single hole of the same area could offer. The holes’ islands float level by symmetry, and the steel between each pair of them is quiet for the same reason it is between two.

One inside the other #

Now put the same two holes on one radius, one inside the other, with the same 2 mm between their edges: centres at 7 and 17 mm from the axis. The islands no longer float level. A hole near the axis sits where the film is high, and its island floats high to match: alone, the inner hole would float at 278 and the outer at 145. Together they pull toward each other, to 237 and 156, but they cannot meet, and between them is 2 mm of steel across which the film must fall 81. The stress there, the film’s slope, is 1.71 times the solid shaft’s surface stress at the midpoint and nearly 2 at the outer hole’s inner edge. The shaft’s peak, 2.03, is just outside, at the outer hole’s outer edge.

Close the gap and the steel between the holes takes over the shaft. With half a millimetre between them it carries 3.5 times the solid shaft’s surface stress, and that is the shaft’s peak. With 5 mm it carries 1.13, and the peak has moved outward, to the steel between the outer hole and the surface. One inside the other, the steel between two holes is a second ligament, with a concentration of its own that grows without limit as it thins, which is exactly what the earlier essay’s question feared and exactly what the side-by-side arrangement avoids.

The two arrangements put the same holes the same distance apart, and this one is the slit’s milder cousin: a slit pins the island to the outline’s height across zero steel, and a hole close behind another pins it nearly to its neighbour’s across very little. What differs is the difference in height between their islands: nothing side by side, 81 one inside the other. The steel between two holes carries a shear flow in proportion to that difference over its own width, as a thin wall between two cells of a box girder carries the difference between the two cells’ circulations — the multicell rule, arrived at here for a solid shaft with holes rather than a tube with webs.

Turning the pair #

Between the two cases the transition is smooth. Turn the pair from lying along a radius to lying across it and the difference between the islands’ heights falls away, the stress in the steel between them falls with it, from 1.71 to 0.04, and the shaft’s peak falls from 2.03 to 1.22. The figure is the practical rule for placing two passages in a shaft: put them at the same radius, not one behind the other, and the steel between them can be thin without being stressed.

A hollow axle with a passage in its wall #

The commonest shaft with two holes is a hollow one with a second hole drilled through its wall — an oil passage along a crankshaft journal, a cable route along a hollow axle, a drain along a spindle. Here the two islands float at different heights by construction. The bore’s island sits at the hollow shaft’s height, the passage’s somewhere between that and zero, and the passage has steel on both sides of it, each carrying the fall between its island and its neighbour’s.

The peak stress is least, 1.49 times the solid shaft’s surface stress, with the passage’s centre 15 mm from the axis: 2 mm of steel between it and the bore, 7 mm between it and the surface. Placed close to the surface the passage peaks at 2.72; close to the bore, at 1.90. The safest place is nearer the bore than the surface, not midway, because the fall across the outer steel is from the passage’s island to zero — the whole height — while the fall across the inner steel is only the difference between two islands that are both high. Thin steel between two high islands carries less than thick steel between one island and the surface.

That is not the rule a designer would guess. The instinct is to put a passage midway through the wall, equidistant from both surfaces, as if each surface threatened the steel equally. The film says the outer surface threatens it more, because the film is pinned at zero there, and a passage pushed inward toward a bore that is itself floating high sits in a region where the film is nearly level already.

By hand, a sanity check on the islands #

The single hole’s island has a closed form only when the hole is at the centre, where it is the hollow shaft: the island floats at $(R^2 - a^2)/2$ and the stiffness is $\pi(R^4 - a^4)/2$, which the solution reproduces for the 20 mm bore to within a few parts in a thousand. Off-centre there is no closed form, but there is a bound: an island floats near the height the solid shaft’s film would have at the hole’s centre, $(R^2 - r^2)/2$, a little below it because the hole relieves the film around it. At 12 mm from the axis that height is 240; the 8 mm hole’s island floats at 227, and the two side-by-side islands at 219. At 7 mm the solid film’s height is 288 and the inner hole’s island floats at 278; at 17 mm, 168 and 145. The difference between the stacked pair’s islands, 81 after they have pulled toward each other, is a little under the 120 the solid film’s heights at their two centres differ by — which is the fall the steel between them would have carried if neither island could move.

Where a hole in a beam differs #

The same question — what the steel between two holes carries — has a different answer in a beam’s web, where holes are cut for services. There the material between two openings is a post carrying shear across the gap between the chords above and below, and its stress depends on the shear passing it, not on whether its neighbours are at the same height. The torsion film has no shear diagram to read: it has only the islands’ heights, and the steel between two holes in a twisted shaft carries load only if those heights differ. A rule of thumb carried from webs to shafts — keep holes well apart — is right for holes on one radius and needlessly cautious for holes side by side.

Where the film stops #

The shaft is long and uniformly twisted. Every result is Saint-Venant torsion, far from the shaft’s ends and from wherever the holes begin and stop. A passage that ends inside a shaft, or turns, has a three-dimensional stress field round its end that no section analysis sees, and that is often where a crack in a drilled shaft starts.

The steel is elastic. The concentrations here are elastic ones; a ductile shaft redistributes once the steel between two stacked holes yields, as a beam that has yielded sheds load from its first hinge to the rest of itself, and its torsional strength is less affected than its elastic peak suggests. In plastic torsion the film becomes a sand heap of constant slope, and the islands still float, but the steel between two stacked holes can carry no more than the yield stress, so the difference in their heights is capped by the gap times that slope. Fatigue, which is what limits most drilled shafts, reads the elastic peak.

The holes are perfect circles and the grid is finite. The film is solved on a grid about a third of a millimetre fine, and the half-millimetre gap is resolved by only a few cells across; its 3.5 is the least certain number here, and the direction of everything around it is not in doubt.

Finish, residual stress and passages that cross #

They cannot show a hole’s surface finish, which a drilled passage leaves rough and which decides how much of the elastic peak becomes a fatigue crack. They cannot show the residual stresses that drilling, rolling or quenching leave round a passage. And they cannot show a passage that crosses another, as oil passages in crankshafts do, where two islands meet and the film has no way to describe what the steel round the crossing does.

Still open: the bore that is not round #

A hollow shaft’s bore is not always round: a splined bore, a hexagonal socket, a keyed hole for a drive. Its island then floats at a height set by a balance round an edge that has corners, and the film between it and a passage in the wall falls across steel whose thickness varies round the bore. Whether a passage near a non-round bore should be placed opposite a flat or opposite a corner — and whether the corner a soap film cannot finish, on the outside of a keyed shaft, has a counterpart on the inside of a keyed bore that a passage must keep away from — is the question a bore with corners puts to a film with two islands.