The corner a soap film cannot finish

Two volumes, and both of them are torques solved Prandtl’s membrane analogy rather than describing it. The stress function of a twisted bar obeys the same equation as the height of a soap film stretched across a hole of the bar’s shape and blown up from beneath, so the film’s volume is the torque and its slope is the shear stress. For a square, a circle, an ellipse and a slit the relaxation reproduced every closed form there is, and its main finding about stress was a quiet one: the stress is largest at the middle of a side and zero at a corner, because a film pinned to two edges meeting at a right angle has nowhere to be steep.

That was a corner pointing outward. The essay ended by naming the other kind: the keyway, whose corners point inward, where “a re-entrant corner makes the film infinitely steep and the elastic stress concentration is unbounded — the practical reason shafts have generous radii”. This essay cuts the keyway and follows the film into its corner.

A keyway in a round shaft #

The shaft is 50 mm in diameter with a keyway 14 mm wide and 5.5 mm deep cut along it, the proportions a shaft of that size is given to take a standard key. At the bottom of the keyway two corners turn from the floor into the side walls, and from the material’s side each is a re-entrant right angle: the steel wraps three quarters of the way round it. The corners are rounded, in the figures that follow, to a fillet whose radius is varied from nothing to a few millimetres.

The film is found the same way as before, by relaxing Poisson’s equation on a grid over the section, with the true distance to the boundary used at every node beside it. Across the whole shaft a grid of a third of a millimetre is fine enough for the torsion constant, which it gives to four figures. It is not fine enough for the corner, whose fillet may be a few tenths of a millimetre, so the corner is solved a second time in a window a few millimetres across, on a grid a hundred times finer, with the film’s heights along the window’s edges taken from the whole-shaft solution. The two are the same calculation at two scales, and the peak stress is read only from the middle of the window, away from the edges the coarse solution fixed.

On the left the film is a dome, almost the paraboloid of the plain shaft, with a notch pressed into its top. The contours — lines of equal height — follow the circle round most of the shaft and then turn in to follow the keyway, crowding together as they go round its bottom corners. Crowded contours are a steep film, and a steep film is a high stress. On the right, at a hundred times the scale, the crowding has nowhere to stop until it meets the fillet.

Little stiffness, a lot of stress #

The keyway removes 11 per cent of the shaft’s torsional stiffness, and that number is almost reassuring. The stiffness is twice the volume under the film, and the keyway presses down only the top of the dome, where the film is flattest and contributes least. Removing material from where the film is flat costs little, which is why a keyway, a flat or a splined end barely changes how far a shaft twists.

The stress is another matter, and it is not in proportion to anything the stiffness says. The plain shaft’s surface stress under a torque $T$ is $2T/\pi R^3$. With the keyway, the stress at the bottom corners, rounded to 0.4 mm, is 2.81 times that. A designer who reduced the shaft’s stiffness by a tenth and raised its stress by a tenth would be designing for 1.11. The corner asks for 2.81.

Where the stress goes, round the surface #

Round the shaft’s surface the keyway does two things, and neither is what the corner does. Far from it, on the side opposite the keyway, the surface carries 1.11 times the plain stress — nearly the plain stress divided by the share of stiffness kept, 1.13, because a shaft twisted by the same torque with 88 per cent of its stiffness twists 13 per cent more, and far from the keyway the film is nearly the plain shaft’s, scaled by that twist. It is a little less steep than that because the whole film leans slightly away from the notch pressed into its top.

Near the keyway’s mouth the surface stress falls almost to nothing. The mouth’s edge, where the side of the keyway meets the shaft’s circumference, is a corner pointing outward, and a film pinned to two edges meeting at an outward corner is flat there. The steel beside the keyway’s mouth is carrying almost no shear at all. So the keyway has taken stress away from the surface near it and put it — with interest — into the corners at its bottom. A stress concentration is a redistribution: the torque is the same, the stress is carried somewhere else, and the somewhere else is the corner pointing the other way.

A corner with no stress to find #

What the corner’s stress is depends, alarmingly, on how it is computed.

With the corner rounded to half a millimetre, refining the grid does what refining a grid should: the answer settles, at 2.66, and the last two refinements agree to a per cent. With the corner sharp it does not. Each halving of the grid spacing raises the peak by about a quarter — 3.38, 4.19, 5.23, 6.55 — and there is no sign of it stopping.

It does not stop, because there is nothing to stop at. Near a re-entrant corner whose material subtends an angle $\alpha$, the solution of Poisson’s equation behaves like $\rho^{\pi/\alpha}$ at a distance $\rho$ from the corner, and the stress, its slope, like $\rho^{\pi/\alpha - 1}$. For three quarters of a turn, $\alpha = 3\pi/2$, that is $\rho^{-1/3}$: the stress at a sharp keyway corner is infinite, and a grid of spacing $h$, which can only see as close as $h$, reports $h^{-1/3}$. The computed exponent, −0.32, is that fact measured. A finer analysis of a sharp corner is not a more accurate one. It is an analysis that has moved closer to an infinity.

The same exponent, $-1/2$ rather than $-1/3$, is the one at the tip of a crack, where the material wraps all the way round. That is not a coincidence of mathematics; it is the reason a crack’s strength is set by a toughness rather than a stress — a number with the units of a stress times the square root of a length, which is exactly what an infinite stress that grows as $\rho^{-1/2}$ needs to be described by — and it is where the history of this subject went next. A re-entrant corner of three quarters of a turn is a milder singularity, and it would need a “toughness” with a cube root of a length in it; nobody uses one, because corners are rounded, and the toughness a crack is assessed with is itself a property that depends on how the crack front samples the steel.

What a fillet buys #

A fillet replaces the corner’s infinity with a length. The stress it leaves follows, for small radii, the same law the grid did — the radius to the power −1/3 — because the film near a small fillet looks, from far enough away, like the film near a sharp corner, and the fillet cuts the singular part off at its own radius. Halving a small fillet’s radius raises the corner’s stress by a quarter; doubling it lowers the stress by a fifth.

The law flattens as the fillet grows, because a fillet of a couple of millimetres in a keyway 5.5 mm deep is no longer small beside anything, and the corner is then less a corner than a round-bottomed slot. At 2.5 mm the stress is 1.88 times the plain shaft’s, and a keyway rounded that much would no longer take a square key.

That is the practical constraint the curve sits under. A key is a rectangular bar with small chamfers on its edges, and the fillet at the bottom of the keyway has to clear them; the standards that fix keyway dimensions fix the fillet with them, and for a shaft of this size it is a few tenths of a millimetre. At that radius the corner carries between two and a half and three times the stress the rest of the shaft was designed for — which is why keyways are where shafts crack.

How deep a keyway is worth cutting #

Deepening the keyway costs stiffness slowly and stress quickly. From 2 mm to 9 mm deep the shaft keeps between 97 and 78 per cent of its stiffness, and the corner’s stress rises from 1.67 to 3.70 times the plain value. The depth is set by the key — enough of it has to sit in the shaft to carry its shear without rolling out — and the figure is the price of that depth: every millimetre of it moves the corner further into the dome of the film, where the contours are already crowded, and the corner has to turn them more sharply.

Which way a corner points #

The keyway and the square of the earlier essay make the whole rule between them. A corner pointing outward — the square’s corners, the edges of the keyway’s mouth — is a place where the film is pinned on two sides that meet at less than a straight angle, and it is flat there: the stress is zero, and the material at that corner is doing nothing in torsion. A corner pointing inward is a place where the film is pinned on two sides that meet at more than a straight angle, and it is vertical there: the stress is infinite until something rounds the corner. The sign of the stress concentration at a corner is decided by which way the corner points, and its size by how sharply it turns.

Every way of connecting a shaft to what it drives is a choice among corners. A flat ground along the shaft — the D-profile of a small motor shaft — has only outward corners and nowhere for the stress to crowd; it costs more stiffness than a keyway of the same depth, because it removes a wider slice of the dome’s top. A spline has many inward corners, each shallow and generously rounded, and shares the torque among them. A keyway has two inward corners that are deep, sharp and loaded by the key as well as by the torsion. A shrink fit or a tapered locking bush has no corner at all, and transmits the torque by friction through a section that stays round. The designs differ in cost and in how they are assembled; in the film they differ only in which way their corners point and how much they are rounded.

The same film describes more than torsion. The velocity of a viscous fluid flowing slowly along a duct obeys the same equation over the duct’s cross-section, with zero at the walls, so a duct shaped like the keyed shaft has a nearly stagnant corner at each outward corner and an infinite wall shear at each inward one; the seepage of water through a dam’s cross-section does the same round the corners of a cut-off wall. A torsion engineer’s keyway, a hydraulic engineer’s duct and a geotechnical engineer’s cut-off have the same singular corner, and the same cure.

Griffith’s films #

The analogy was first used as an instrument, not a proof, and on exactly this problem. In 1917, at the Royal Aircraft Factory, A. A. Griffith and G. I. Taylor stretched soap films across holes cut to the shapes of shaft sections — keyways, splines, hollow sections — blew them up with a small pressure, and measured their heights and slopes optically to find the torsional stiffness and the stress in each. Soap films made the analogy quantitative: a film is a physical solution of Poisson’s equation over any shape that can be cut into a plate.

What the films showed at keyway corners was the difficulty this essay’s grid shows. The film’s slope at a sharp inward corner could not be measured, because it grew without limit as the measurement was taken closer to the corner, and a rounded corner’s stress depended on its radius in a way no strength-of-materials formula anticipated. Griffith’s next work was on why real materials are weaker than their atoms’ bonds predict, and he found the answer in the same place: at the tips of cracks and scratches, where the local stress is limited only by the radius at the tip. His theory of fracture, published in 1921, starts from that observation. A stress concentration that no stress can describe is the connection between the two pieces of work, and a keyway’s corner was one of the first places it was seen.

A keyed shaft, by hand #

The plain 50 mm shaft has a torsion constant of $\pi R^4/2 = 613{,}600$ mm⁴, and under a torque of 1 kN·m its surface stress is $2T/\pi R^3 = 2 \times 10^6/(\pi \times 15{,}625) = 40.7$ N/mm². The keyway leaves 0.884 of the torsion constant, 542,400 mm⁴, so under the same torque the keyed shaft twists $1/0.884 = 1.13$ times as far. The surface opposite the keyway carries $1.11 \times 40.7 = 45$ N/mm², a little under the 46 that the extra twist alone would give. The corner, rounded to 0.4 mm, carries $2.81 \times 40.7 = 114$ N/mm², and rounded to 0.1 mm, $4.27 \times 40.7 = 174$.

A crude estimate from the lost stiffness would put the whole keyed shaft at 46 N/mm² and miss the corner by a factor of two and a half. An estimate that treats the corner as sharp has no number at all to offer, and the grid shows why.

The analogy, and the shaft it idealises #

The shaft is long and the keyway runs its whole length, so every section is the same and the problem is two-dimensional. A real keyway ends somewhere — in a run-out where the cutter lifted, or in the rounded end of an end-milled slot — and at its end the stress is three-dimensional and higher again.

The key is absent. The film is for a shaft carrying torque through its own section with an empty keyway; a key transmitting torque bears on one wall of the keyway and puts a contact pressure and a shear into the corner beside it, which add to the torsional stress rather than replacing it.

The steel is elastic. A ductile shaft under a single overload yields at the corner and redistributes, and the corner’s stress concentration then matters for fatigue, where the detail rather than the steel decides, much more than for a single static torque.

What the pictures cannot show #

That the peak stresses are at a point. Every factor quoted is the largest slope the film reaches, at one place on the fillet, and the steel a few tenths of a millimetre away carries much less. In fatigue the material averages over a small volume and feels less than the peak; for a hole in a plate that averaging is gentle, and at a fillet whose radius is comparable to the averaging length it takes a large share of the concentration away. The factor is the elastic answer, and what a crack actually responds to is somewhat less.

Nor can they show the film’s third dimension. The contours are a plan of a surface whose height is the stress function, and the steepness that is the stress is read off as the crowding of lines, which the eye underrates by the square of the scale difference between the two panels.

Still open: the shaft with a hole through it #

A keyed shaft is simply connected: its film has one boundary, the outline, and is pinned to it at height zero. A shaft with a hole bored along it, off centre for a lubrication passage or a cable, has two boundaries, and the film over it has a flat island at the hole whose height is not zero and not known in advance — it is the extra unknown that makes the torsion of a hollow or multiply connected section a different problem, and the thick-walled form of the circuit a thin closing plate creates. Whether an off-centre hole weakens a shaft in the way a keyway does, crowding the film into its nearest wall, or whether its island’s height takes up the difference, is the question the film leaves when it is given a second edge to be pinned to.