Structural Engineering & Statics
where the load goes
Concept

Closed section — where it appears

A thin-walled section whose wall forms a closed loop, such as a tube or a box. Shear can circulate around the loop, so it resists twisting through the area it encloses and is far stiffer in torsion than an open section with the same wall.

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

One slit, and the torsional stiffness falls by a factor of hundreds. A 200 by 200 box of 8 mm wall, drawn closed and then slit along its length. Closed, the torque runs round the wall as a shear flow and the torsion constant is 5.66×10⁷ mm⁴; slit, the loop is broken and only each wall's own thickness resists, giving 1.31×10⁵ mm⁴. The ratio is 432 to one, so the same torque twists the slit section 432 times as far and raises a peak shear stress 36 times as high. Nothing about the material changed.

The slit that costs a factor of six hundred

Bending stiffness cares where the material is, and changes by a factor of two or three between sensible sections of the same area. Torsional stiffness cares whether the material forms a closed loop, and the penalty for not doing so is an order of magnitude squared.

sections · Torsional constant 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.

sections · Membrane analogy

Named alongside it

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

Open sectionTorsionWarpingBimomentBredts formulaDecay lengthEquilibriumFree bodyMembrane analogySaint-venant torsionSecond momentSection shape

All concepts