Structural Engineering & Statics
where the load goes
Series

Shear centre — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The shear centre of a channel. A channel of 80 by 200, with the shear flow in its flanges drawn. Those flows form a couple, so the load has to be applied 31.7 outside the web to leave the section untwisted — a point in the air, outside the material entirely.

    The point that is not in the section

    A channel loaded down its web twists. To stop it, the load must be applied through a point outside the steel entirely — in the air beside the section, where nothing can be attached.

    part 1 · sections
  2. The shear centre of a channel. A channel of 100 by 250, with the shear flow in its flanges drawn. Those flows form a couple, so the load has to be applied 38.0 outside the web to leave the section untwisted — a point in the air, outside the material entirely.

    The eccentricity a purlin cannot avoid

    A channel's shear centre is outside the material, so a load applied anywhere on the section misses it. The distance is fixed by the proportions rather than by the detailing, it is 38 mm on an ordinary purlin, and the torque it produces is not an error anybody made.

    part 2 · sections
  3. The shear centre waits for a thick plate. Where the shear centre of a channel 300 mm deep with 100 mm flanges, all 10 mm thick sits, measured from the web towards the toes, against the thickness of a plate closing the toes as a fraction of the channel's own, on a logarithmic scale. Open, it is 33.3 mm outside the web. Closed by a plate as thick as the rest it is at the middle, 50.0 mm inside the web, towards the toes. With a plate a thousandth as thick it has moved 0.3 per cent of the way; a hundredth, 3 per cent; a tenth, 23 per cent. It is half way at 0.28 of the thickness and crosses the web at 0.22.

    The plate that closes the circuit

    Close a channel with a plate across its toes and its shear centre is expected to come in from the air beside the web to the middle of the box. It does — once the plate is about a quarter as thick as the channel. A plate a hundredth as thick moves it three per cent of the way while multiplying the torsion constant by eight, so the load keeps its lever, the section stops twisting, and the whole torque goes round the circuit as a shear flow through the thinnest plate in it.

    part 3 · sections
  4. The interior web pulls the shear centre toward itself. The shear centre of a two-cell box 600 mm wide and 300 mm deep on its centrelines, 10 mm walls, its interior web 200 mm from the left, its distance from the left against the interior web's thickness as a multiple of the other walls', on a logarithmic scale. The single box's middle is 300 mm (dashed) and the web is at 200 (dotted). A web a hundredth as thick as the walls moves it 0.6 per cent of the way; a tenth, 5; as thick as the walls, 27; half way at 5.4 times; a hundred times, 92.

    The web that moves the centre and not the twist

    A thin plate closing a channel into a box changes its torsion at once and its shear centre hardly at all. An interior web dividing a box into two cells does the reverse. Off the middle of a 600 mm box, a web as thick as the walls takes 38 per cent of the vertical shear — more than either outer web — and pulls the shear centre 27 mm toward itself, while the torsion constant changes by less than one per cent. The two walls are the same kind of wall; what differs is which flow they are part of.

    part 4 · sections
  5. An inclined web takes the shear centre off its line. Where the shear centre of a trapezoidal box 1,500 mm deep, 3,000 mm across the top and 2,000 mm across the bottom (15 mm top plate, 12 mm elsewhere) lies as an interior web from the top flange's middle is leaned so that its foot moves along the bottom flange (solid, dots at each position), against where it would lie with a vertical web at the inclined web's mid-depth position (dashed); the cross marks the box with a vertical web at the middle, 791 mm up. A vertical web moves it sideways only, within 2 mm of that height. The inclined web lifts it as it leans: with its foot 500 mm left of the middle it is −66 mm across and 800 mm up; 500 mm right, 66 across and 800 up.

    The web that leans and lifts the centre

    In a rectangular box an interior web pulls the shear centre sideways toward itself and leaves its height alone, because a vertical web cannot carry horizontal shear. Most steel box girders are trapezoids, and their webs lean. Leaning the outer webs puts the shear centre below the centroid instead of above it; leaning an interior web lifts the shear centre as well as pulling it, by up to 35 mm in a 1.5 m box; and a vee of interior webs meeting at the bottom flange drops it by as much as 200. A web that leans carries horizontal shear, and whatever carries horizontal shear decides where the shear centre is in height.

    part 5 · sections
  6. The tub girder's shear centre at three stages of its building. A steel trough 1,500 mm deep, 3,000 mm between its web tops and 2,000 mm across its 12 mm bottom flange, with 12 mm webs and a 400 × 20 mm flange on each web, drawn to scale. Open, as it is lifted, its shear centre lies 706 mm below the bottom flange, outside the steel entirely. Closed across the top by a bracing truss equivalent to a plate 0.50 mm thick, which carries shear and no bending stress, it is still 538 mm below. With a 250 mm concrete deck cast and hardened across the web tops it is 1,127 mm above the bottom flange, inside the box. The steel's centroid, 673 mm up, does not move until the deck is cast.

    The box that is a trough until its deck is cast

    A composite tub girder is a box only once its concrete deck has hardened. Before that it is an open steel trough — two leaning webs and a bottom flange — whose shear centre lies seven hundred millimetres below its bottom flange, in the air, and whose resistance to twisting is the thickness of its plates cubed. A light truss across the top is what holds it together while the deck is poured, and it is a plate for shear and nothing else: an angle of ordinary size is worth half a millimetre of steel. That half-millimetre multiplies the trough's torsion constant two thousandfold, and leaves its shear centre almost exactly where it was. The bracing closes the box for twisting; it does not close it for the shear centre.

    part 6 · sections
  7. The diagonals over the support are the girder's shortest-lived members. The years a diagonal's connection of the top bracing of a 1.5 m deep composite tub girder under a 250 mm deck, continuous over two 40 m spans, 480 kN lorries passing 2.5 m off its centre line lasts at two million such lorries a year, along a span, for fatigue categories 50, 71 and 90, logarithmic and drawn at 1,000 years where the range is below the category's cut-off and the life is without limit; dotted, 100 years. Where the deck is whole every category lasts without limit. At the worst diagonal, 34.5 m along, category 50 lasts 2 years, category 71 lasts 8 years, category 90 lasts 25 years.

    The bracing nobody took out

    The light truss that closes a steel tub girder while its deck is poured is usually left in once the deck has hardened, because taking it out from under a finished slab is more trouble than it is worth. Over most of a span that is harmless: the deck lifts the neutral axis to the top flanges and takes ninety-nine per cent of the shear round the top, and a passing lorry puts five newtons per square millimetre in a diagonal. Over an interior support, where the deck is cracked to its bars, the same diagonals sit six hundred millimetres from the axis and are handed a share of the torque, and each heavy lorry puts forty in them — the shortest-lived connections in the girder, on members nobody designed for anything after the day the concrete went off.

    part 7 · sections

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