Structural Engineering & Statics
where the load goes
Concept

Saint-venant torsion — where it appears

Torsion carried by shear stresses circulating in each cross-section of a member free to warp, at a stiffness GJ proportional to the rate of twist. It is the whole of an open section's resistance only when nothing restrains the warping.

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 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.

Open sectionBimomentBredts formulaClosed sectionDecay lengthEquilibriumFree bodyMembrane analogyRolled sectionRoot filletSecond momentSection shape

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