Structural Engineering & Statics
where the load goes
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Stressed ribbon — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The abutment force is the sag turned upside down. A 100 m ribbon carrying 35 kN/m at a sag of 2.0 per cent of its span. H = wL²/8f, so the horizontal force at each abutment is 21875 kN — 6.25 times the entire weight of the deck, and five times what a suspension bridge of the same span and weight at a tenth would have needed. The curve is a reciprocal and it has no flat part: halving the sag doubles the force, at any sag. What stops a designer flattening it further is not the ribbon, which is in tension and cannot buckle. It is what the ground at each end will take, and at 6.25 deck-weights that is usually rock or a very large anchor block.

    The deck that is its own cable

    Every other cable structure hangs something from the cable. A stressed ribbon hangs nothing — the walking surface is the catenary, laid at a fiftieth of the span rather than a tenth, because a footbridge has to be walkable. That one decision hands the abutments six and a quarter times the entire weight of the bridge.

    part 1 · structures
  2. A load on one span moves the other. The change in level along two 100 m spans of stressed ribbon at a sag of 2.00 per cent, carrying 35 kN/m, when 20 kN/m of live load is added to the left-hand span, with a rigid pier (dashed) and with a pier of 200,000 kN/m (solid). With a rigid pier the loaded span sags a further 195 mm and the other does not move. With a pier of 200,000 kN/m the pier top leans 25 mm towards the load, the loaded span sags 358 mm and the unloaded span rises 191 mm; the pier carries 4,953 kN of the 9,440 kN a rigid pier would.

    The pier that carries only the difference

    A stressed ribbon continued over a pier pulls on it from both sides, and under its own weight the two pulls cancel. Load one span and they do not: the pier is asked for the difference, nearly ten meganewtons for a crowd on one of two hundred-metre spans. Whether it gives it depends on its stiffness against a number the ribbon supplies itself, and whatever the pier does not carry, the ribbon turns into movement — a third of a metre of extra sag in the loaded span and a fifth of a metre of rise in the empty one.

    part 2 · structures
  3. One span hung, the other still bare tendon. The level along two 100 m spans at a sag of 2.00 per cent, 35 kN/m finished and 3.0 kN/m of bare tendon, the tendons at 1100 N/mm² when finished, with the deck hung on the left span only (solid) and as finished (dashed), on a pier of 200,000 kN/m. The hung span sags 2.04 m and pulls 21,442 kN; the bare tendon beside it sags 0.20 m and already pulls 18,363 kN, because it was cut 455 mm shorter than the span so as to carry its share when finished. The pier is asked for the difference, 3,080 kN, and leans 15 mm towards the hung span — against 4,953 kN under 20 kN/m of crowd on one finished span.

    The tendon that pulls before the deck arrives

    A stressed ribbon is hung one span at a time, and the obvious fear is the stage at which one span carries its deck and the next does not: the pier between them asked for a whole span's thrust. It is not asked for that, because tendons cut to the finished length are already stretched across the empty span and pulling. For two 100 m spans at a fiftieth, the rigid pier's stage force is 4,109 kN against a crowd's 9,440 — unless the tendons were sized generously, when the stage overtakes the crowd.

    part 3 · structures
  4. The pier reads the difference between the two spans' temperatures. Contours of the pier force between two 100 m spans at a sag of 2.00 per cent with the deck hung on one, the tendons at 1,100 N/mm² when finished, on a pier of 200,000 kN/m, every half meganewton, over the warming of the hung span's tendons (across) and of the bare tendon's (up) since they were cut. With neither warmer the pier carries 3,080 kN; warming the bare tendon 30 °C alone gives 4,126 kN, the hung span alone 2,342 kN, and both together 3,392 kN. The contours are straight and parallel: the pier gains 35 kN for each degree the bare tendon warms and loses 25 for each degree the hung span does, so a warming both share moves it only 30 per cent as far as the same warming on the bare tendon alone, and what the pier feels is mostly how much warmer one span's strand is than the other's.

    The strand in the sun beside the strand in the shade

    Halfway through hanging a stressed ribbon, one span is a sagging cable full of segments and the next is bare tendon stretched nearly straight, and the pier between them carries the difference of their pulls. A warm day takes tension off both — and at the flat sag a ribbon is laid at, the hung span is so nearly all stretch that it loses three quarters of what the bare one does. What moves the pier is not the weather but the shade: strand in the sun beside strand under concrete, 35 kN on the pier for every degree between them. No sun slackens the bare tendon. The hour does change something else, which is the level the joints are cast at.

    part 4 · structures
  5. The flatter the ribbon, the more the casting temperature matters. For two 100 m spans of a stressed ribbon under 35 kN/m on a pier of 200,000 kN/m, one span's joints cast 20 °C warmer than the other's: the force left on the pier as a share of the force a crowd of 20 kN/m on one span puts there, against the ribbon's sag as a percentage of its span. At a sag of 1 per cent the casting leaves 82 per cent of a crowd (1,902 kN against 2,316); at 2 per cent, 14 per cent; at 3, 3 per cent; at 4, 1 per cent.

    The span cast in the afternoon

    A stressed ribbon's joints are cast on whatever day the programme reaches them, and a span cast on a warm afternoon sets around strand that has lengthened in the sun. It cools that night as a concrete band eleven times stiffer than the strand, and the band does not stretch back as the strand would have. On a ribbon with a sag of 2 per cent, a span cast 20 °C warmer than its neighbour ends 72 mm higher than it was set and leaves 713 kN on the pier between them — a seventh of what a crowd on one span puts there. On a flatter ribbon the same afternoon costs more than the crowd. The difference between the two spans is the strand's flexibility against the band's, and only the elastic part of it differs; the part that comes from the sag is the same for both.

    part 5 · structures

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