Structural Engineering & Statics
where the load goes
Concept

Fillet — where it appears

The radius that rounds a re-entrant corner, at a weld toe, the root of a keyway or the corner of a bore. A sharp corner concentrates stress without limit in an elastic analysis; the fillet makes the peak finite, and doubling it buys back only part of the excess.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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.

sections · Membrane analogy 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.

sections · Membrane analogy

Named alongside it

The objects these essays reach for when they reach for this one.

Membrane analogyStress concentrationTorsionTorsional constantClosed sectionShaftShear stressSingularityStress function

All concepts