Structural Engineering & Statics
where the load goes
Series

Membrane analogy — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A soap film over a hole, and its volume is the torsion constant. The stress function for Saint-Venant torsion of a square of 100 mm, relaxed on a 97 by 97 grid until it stopped moving — 385 sweeps. Its contours are the lines the shear stress runs along, its slope is the magnitude of that stress, and twice its volume is the torsion constant: 1.4053e+7 mm⁴ against a closed form of 1.4058e+7, an error of -0.035 per cent. The steepest slope is 67.515 at the boundary, at the point on it nearest the centre — which is why the peak shear in a solid section is at the middle of the longest side and never at a corner, where the film comes down to zero from two directions and its slope vanishes.

    Two volumes, and both of them are torques

    Torsion of a solid section that is not a circle has no elementary answer, and for thirty years after Saint-Venant posed it the only way to get one was to blow a soap film over a hole cut in a plate and measure it. The film is not an illustration of the solution. It is the solution, and so is a heap of sand poured on the same hole.

    part 1 · sections
  2. The film over a keyed shaft, and the corner it climbs. Contours of Prandtl's stress function — the height of a soap film blown up over a hole of the section's shape — for a 50 mm shaft with a keyway 14 mm wide and 5.5 mm deep, its bottom corners rounded to 0.40 mm; the shear stress is the slope of the film, so it is highest where the contours crowd. Left, the whole shaft: the contours follow the circle and bend round the keyway, and the keyway costs 11 per cent of the plain shaft's torsional stiffness. Right, a window 3.0 mm across around the keyway's corner, solved on its own fine grid with its edges taken from the whole film: the contours crowd into the fillet, where the stress is 2.81 times the plain shaft's surface stress under the same torque.

    The corner a soap film cannot finish

    Cut a keyway into a round shaft and its torsional stiffness falls by only 11 per cent. The stress does not fall; it moves. It leaves the surface beside the keyway's mouth almost unloaded and crowds into the keyway's bottom corners, where a soap film stretched over the section would have to stand vertical. Computed on finer and finer grids, a sharp corner's stress never settles — it grows by the cube root of two every time the grid is halved — and a fillet of a few tenths of a millimetre is what turns an infinite answer into three times the plain shaft's.

    part 2 · sections
  3. The film over a bored shaft has an island that floats. Contours of Prandtl's stress function — a soap film blown over the section — for a 50 mm shaft with a 10 mm hole 15.0 mm off its axis, at the same pressure; the shear stress is the film's slope. Left, the bored shaft: the film is pinned to the outline and to a flat island over the hole, shaded, which floats at 55 per cent of the solid shaft's peak height, where the film's pull round its edge balances the pressure on it. The shaft keeps 93 per cent of the solid shaft's stiffness, and its largest stress, at the dot, is 1.70 times the solid shaft's surface stress under the same torque. Right, the same shaft slit along its length from the hole to the surface: the island is pinned at the outline's height, the film sags into it, and the shaft keeps 67 per cent, its largest stress 2.81 times.

    The island the film floats

    Bore a hole along a shaft, off centre for a lubrication passage or a cable, and the soap film that solves its torsion has a second edge: a flat island over the hole whose height nobody knows in advance. Let it float where the film's pull balances the pressure on it, and the shaft keeps nearly all its stiffness and doubles the stress beside the hole, as a hole in any shear field does. Pin it at the outline's height instead and the same shaft has been slit to its hole, and loses a third of its stiffness. The island's height is the whole difference between a bore and a cut.

    part 3 · sections
  4. Two islands in one film: level with each other, or not. Contours of Prandtl's stress function for a 50 mm shaft bored with two 8 mm holes, their edges 2.0 mm apart; the shear stress is the film's slope, and each hole is an island, shaded, floating where the film's pull round it balances the pressure on it. Left, the holes side by side at 12.0 mm from the axis: by symmetry both islands float at the same height, 219, the film between them stays nearly flat, and the steel there carries almost nothing. The shaft keeps 95 per cent of the solid shaft's stiffness, and its largest stress, at the dot, is 1.13 times the solid shaft's surface stress. Right, one hole inside the other on one radius: the inner island floats at 237 and the outer at 156, so the film between them is steep and the steel between the holes carries up to 1.96 times; the shaft's largest stress, 2.03 times, is at the outer hole's outer edge, with the steel between the holes close behind.

    Two islands in one film

    Bore a hole along a shaft and its torsion film acquires an island that floats at a height of its own. Bore two and there are two islands, and the steel between them is a strip of film stretched from one to the other. Side by side at one radius, the islands float level, the strip is flat, and two holes cost twice one hole's stiffness and none of its stress. One inside the other on a radius, the islands float at different heights, the strip is steep, and the steel between the holes becomes the shaft's weak point.

    part 4 · sections
  5. A passage opposite a corner, and opposite a flat. The section of a 50 mm shaft with a hexagonal bore whose flats are 8 mm from the axis and whose corners are rounded to 1.5 mm, with a passage 5 mm across centred 13 mm from the axis, opposite a corner of the bore (left) and opposite a flat (right), and the largest shear stress each leaves round the passage and round the bore, as a multiple of the plain round shaft's surface stress under the same torque. Opposite a corner the passage peaks at 1.25 and the bore at 1.15; opposite a flat, 1.19 and 0.54. With no passage the bore's corners carry 0.54.

    The bore with corners and the passage beside it

    A hollow shaft's bore is not always round: a hexagonal socket or a square drive turns the island in Prandtl's soap film into a polygon, and every corner of the bore is a corner of the steel that the film cannot finish. Its corners carry twice the stress of its flats. A passage drilled through the wall beside it can sit opposite a corner, where it doubles that stress again, or opposite a flat, where the bore does not notice it at all.

    part 5 · sections

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