Structural Engineering & Statics
where the load goes
Concept

Membrane analogy — where it appears

Prandtl's observation that the stress function of a twisted bar obeys the same equation as a soap film blown up over a hole of the section's shape. The film's slope is the shear stress and twice the volume under it is the torsion constant.

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 What the fillets are worth across the section tables. For seven standard rolled sections, the torsion constant the film gives over the rolled outline, beside the published one (UB 203×133×25 5.96 against 5.96 cm⁴; UB 305×165×40 14.74 against 14.70 cm⁴; UB 457×191×67 37.14 against 37.10 cm⁴; UB 610×229×101 76.97 against 77.00 cm⁴; UC 152×152×23 4.63 against 4.63 cm⁴; UC 254×254×73 57.68 against 57.60 cm⁴; UC 305×305×97 91.22 against 91.20 cm⁴). Solid bars: what the fillets add to J over the same plates with sharp corners, 9 to 13 per cent. Second bars: how far the fillet's peak stress exceeds the flange face's, 45 to 65 per cent. Σbt³/3 is 7 to 12 per cent below the published constant.

The corner the rolling mill rounds

The torsion constant of an I-section is taught as Σbt³/3, the sum over three thin plates, and a rolled section is not three plates. Where its web meets each flange the mill leaves a fillet, a curve of steel about one per cent of the section's area, and the soap film whose volume is the torsion constant rises into it. Solved over the real outline, the film reproduces the published constant of seven standard sections to a fifth of a per cent and says the fillets are worth nine to thirteen per cent of it. It also puts the section's largest torsional stress on the fillet, half as much again as the flange formula every design check uses.

sections · Torsional constant

Named alongside it

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

Stress concentrationTorsional constantFilletOpen sectionRolled sectionRoot filletSaint-venant torsionShaftShear stressSingularityStress functionTorsion

All concepts