Structural Engineering & Statics
where the load goes
Series

Torsional constant — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · sections
  2. 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.

    part 2 · sections
  3. The film over an angle's corner. Contours of Prandtl's stress function over the corner of an equal angle 100 × 100 × 10 mm with a 12.0 mm root radius and 6.0 mm toe radii: left as rolled, right with sharp corners. Along each leg the contours run parallel to its faces; at the corner they bulge into the metal the fillet adds, round the outside of the L. The rolled angle's J is 6.80 cm⁴ and the sharp one's 6.18 cm⁴. The dot marks the steepest slope: on the fillet in the rolled angle, in the re-entrant corner in the sharp one, where it grows without limit as the grid is refined.

    The corner with one fillet

    A rolled I-section's web meets its flanges in two T junctions, four fillets, and its torsion constant is 7 to 12 per cent above Σbt³/3 because of them. An angle meets itself in one L, and a channel's web meets its flanges in two. Solved over the real outline, an L junction's fillet adds about two-thirds of what a T junction's does with the same plates. The sum then errs on either side: low by 8% for an I-section with a channel's plates, high by 0.8% for the channel itself, and anywhere from 5% low to 2% high for angles, depending on which length of leg is counted. The fillet's peak stress is still the inscribed circle's, to a few per cent.

    part 3 · sections

All series