Structural Engineering & Statics
where the load goes
Series

Network arch — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Cross the hangers and the chords stop bending. The same tied arch, the same sixteen hangers, the same load on half the span — hung vertically and hung as a network. Vertical hangers make the two chords a Vierendeel frame, which has no truss action at all, so a partial load is carried by bending: 3316 kNm in the tie and 6234 in the arch. Inclined hangers can carry the shear between the chords axially, and the same load gives 686 and 831 — factors of 4.8 and 7.5. The thrust is identical in both, because that is decided by the span and the rise and nothing else.

    Cross the hangers and the bending goes

    A tied arch with vertical hangers is a Vierendeel frame with a curved top chord — it has no truss action at all, so a load on half the span is carried by bending. Incline the hangers so they cross and the same two chords become a truss.

    part 1 · structures
  2. A hanger that would push is a hanger that is not there. The force in each hanger of a tied arch of 100 m span and 17 m rise on 32 crossed hangers, carrying 60 kN/m of dead load over the span and 120 kN/m of live load over its left half, against the hanger's position along the tie. Solved as though every hanger could push (pale bars), 4 come out in compression, the worst at 185 kN. Solved with those hangers taken out until no remaining one is compressed (dark bars), 8 are slack — more than the linear solution has in compression, because each hanger that goes passes its share to its neighbours and some of them follow.

    The deck that keeps the hangers in the truss

    Crossing the hangers of a tied arch turns it from a frame that bends into a truss that does not, as long as every hanger is pulling. A hanger cannot push, and under a load on half the span some of them would have to. Solved as it really is — with those hangers taken out, and the ones that follow them — the arch keeps the linear answer exactly until the live load on half its span is about equal to its own dead weight, and past that the tie's moment doubles and then quadruples. The arch's own weight is the pretension nobody had to apply, and a heavier deck is a stiffer network.

    part 2 · structures
  3. A crossed hanger swings through nearly all of its force. The largest and smallest force in each crossed hanger of a tied arch of 100 m span and 17 m rise on 32 hangers, with 60 kN/m of dead load and a live load of 60 kN/m running on from either end, each bar running from the one to the other. The worst range is 678 kN, from 49 to 727, in hanger 11; the largest force anywhere is 756 kN. 6 of the 32 go slack at some stage, and the worst range is 90% of the largest force anywhere: the live load's shear passes through the crossed hangers as through a truss's web, and swings them from nearly nothing to nearly their most.

    The hangers that carry the shear twice

    Crossing a tied arch's hangers turns them into the web of a truss, which is why the tie stops bending. It also means a live load's shear passes through them, so as a load runs on from either end each crossed hanger swings from nearly nothing to nearly its most: on a 100 m arch the worst swings through 678 kN, 90% of the largest force in any hanger, where a vertical hanger swings through 194, half of its own. More dead load raises the force and leaves the range alone. The arrangement that is best for strength is the one that hands its hangers to fatigue.

    part 3 · structures
  4. The range belongs to the truss. The worst hanger force range against the tie's largest moment in a tied arch of 100 m span and 17 m rise on 32 crossed hangers of 3,000 mm² bar, 60 kN/m of dead load and 60 kN/m of live load run on from either end, as the hangers' stiffness runs from four times a plain bar's to a fiftieth of it (dots), and for the same arch on vertical hangers (the lone dot). Softening the crossed hangers walks the bridge from 678 kN of range and 316 kN·m of tie moment to 557 and 1,154; vertical hangers sit at 194 and 3,505. The line bends nowhere near the vertical arch's point: a soft network is still a truss.

    The hanger that cannot be made soft enough

    A network arch's crossed hangers swing through nearly all their force as traffic passes, because they are the web of a truss. Make them softer — a spring at each anchorage — and the tie ought to take back some of the shear and the hangers' ranges ought to fall. They hardly do. A tenth of the stiffness takes 6 per cent off the worst range and adds a third to the tie's bending; a fiftieth takes 18 per cent off and multiplies the tie's bending by 3.7. The range belongs to the truss and leaves only when the truss does, which vertical hangers show at 194 kN against 678 and eleven times the bending. So the hanger is sized for its range, eight times the area its strength needs, and the stiffness that comes with it costs the range nothing.

    part 4 · structures
  5. More hangers, each swinging through less. Each hanger's force range, plotted at its mid-point along the span, for a tied arch of 100 m span and 17 m rise on 32, 64, 96 crossed hangers of 3,000 mm² bar at 60°, 60 kN/m of dead load and 60 kN/m of live load run on from either end. 32 hangers: the worst 678 kN, the mean 528; 64 hangers: the worst 369 kN, the mean 273; 96 hangers: the worst 241 kN, the mean 184.

    More hangers, and the same steel

    A network arch's crossed hangers are the web of a truss, and a live load's shear passes through them, so every hanger swings through nearly all of its force. The obvious cure is more hangers, each crossing more of its neighbours, so that every section's shear is shared among more diagonals. It works hanger by hanger — three times the hangers, a third of the range — and not at all in total: the hangers' fatigue steel, sized for their ranges, is set by the shear the arch carries and barely changes with how many hangers it is divided among. What more hangers buy is elsewhere.

    part 5 · structures

All series