More hangers, and the same steel

The hanger that cannot be made soft enough tried to take the fatigue range out of a network arch’s hangers by making them softer, and found that it barely could. Thirty-two crossed hangers on a 100 m span are the web of a truss whose chords are the arch and the tie, and a live load’s shear goes through the web whatever the web is made of: a tenth of the stiffness took 6 per cent off the worst range, and a fiftieth left four-fifths of it. The range belongs to the truss, not to the bar.

That essay ended on the other lever. Tveit’s network arches for longer spans do not use 32 hangers at one slope. They use many more, so that each hanger crosses several of its neighbours and a vertical section through the arch cuts five or six of them; the shear at any section is then shared among more diagonals. Whether sharing it among more reduces each hanger’s range in proportion — so that the area each needs falls with the number — or whether each range stays a fixed fraction of its own largest force, decides whether a network arch’s hanger steel is set by its geometry or by how many hangers it has.

The same arch, divided more finely #

The arch is the earlier essays’: a 100 m span with a 17 m rise, a tie and an arch rib, 60 kN/m of dead load, and 60 kN/m of live load run on from either end — every position of the patch, and the dead load alone, with every hanger that would go into compression taken out and the frame solved again until the set of slack hangers stops changing. Each hanger is a 3,000 mm² steel bar. The hangers lean alternately at 60° from horizontal, as before; what changes is how many there are, from 16 to 96, spaced evenly along the tie.

For each hanger, its largest and smallest force over all the load positions are its envelope, and the difference is its range — the number its fatigue detail is checked against, since a hanger is a threaded or forged bar whose stress rises and falls with every passing vehicle.

The hero figure shows the result hanger by hanger. On 32 hangers the ranges sit between about 400 and 680 kN, alternating as the hangers lean one way and the other. On 64 they sit between about 200 and 370; on 96, between about 120 and 240, with the hangers nearest the supports lower still. The pattern along the span is the same each time, and its scale falls as the hangers multiply.

More hangers across every section #

The mechanism that the earlier essays named is the shear in the web, and the next figure counts the web.

A vertical section through the arch cuts every hanger whose two ends lie on either side of it. With 32 hangers at 60° that is two to four, 2.8 on average over the middle half of the span; with 96 it is eight to ten, 8.5 on average. The count falls towards the supports, where the arch is low and a hanger at 60° spans less of the tie. Three times the hangers put three times as many across each section — the hangers are three times as close, and at the same slope each spans the same distance.

The live load’s shear at a section is carried by the tie, the arch and the diagonals that cross it, and the earlier essays found that in a network arch nearly all of it goes through the diagonals. Shared among three times as many, each carries a third. The hangers that cross a section share its shear, and the number that cross it is set by how close the hangers are.

The range falls as one over the number #

The next figure plots the worst and the mean range against the number of hangers.

The worst range falls from 1,252 kN on 16 hangers to 678 on 32, 369 on 64 and 241 on 96. The dotted line is 678 kN scaled as $32/n$, and the worst range follows it within ten per cent from 16 hangers to 96; the mean range follows it as closely. That answers the question the earlier essay left in the first of its two forms: sharing the shear among more crossings reduces each hanger’s range in proportion to the number of hangers. It does not stay a fixed fraction of the hanger’s own largest force in any way that would defeat the sharing; the largest force falls too.

The proportion is not exact, and the departure is in one direction. Doubling from 32 to 64 hangers takes the worst range to 54 per cent of what it was, not 50; tripling to 96 takes it to 36 per cent, not 33. The hangers near the supports, where the section count falls off, carry a little more than the average share, and the worst hanger is one of them.

And the same steel #

A hanger sized for its range needs an area in proportion to it, at whatever stress range its detail allows. The total area of all the hangers is then the number of hangers times the range, and the next figure plots that.

The total does not fall. Sized for the worst hanger’s range, sixteen hangers need 20.0 MN of range in total, 32 need 21.7, 64 need 23.6 and 96 need 23.2. Sized each for its own, 15.8 MN on 16 hangers and 17.7 on 96. Across a sixfold change in the number of hangers, the hanger steel sized for fatigue changes by less than a fifth, and it rises rather than falls. The hangers’ fatigue steel is set by the shear the arch carries, not by how many hangers it is divided among.

That is the second of the earlier essay’s two forms of the question, and both turn out to be true at once. Each hanger’s area falls with the number of hangers, because each carries a smaller share; the total area does not, because the shear being shared is the same. The arch has a live load to carry across its span, the web carries its shear, and a web made of more, thinner diagonals carrying the same shear needs the same total area of diagonal — as a truss with twice as many diagonals of half the area is the same truss for its web stress.

Steeper hangers are worse #

The slope is the other geometric choice, and the next figure varies it.

On 64 hangers, the worst range is 352 kN at 45° from horizontal, 369 at 60° and 511 at 70°. A steeper hanger spans less of the tie, so a section cuts fewer of them and each carries more shear; it also carries its share of shear less efficiently, because a diagonal’s vertical component is what resists shear and a steep diagonal carries the deck’s own weight more directly instead. Between 45° and 60° the range barely moves; beyond 60° it rises steeply, by 39 per cent at 70° on 64 hangers and 17 per cent on 32. The flatter hangers of Tveit’s designs are not a matter of appearance: they are on the flat part of this curve, where the slope costs nothing.

At the other end, a flatter hanger is longer, and it crosses more neighbours. Below about 45° the hangers near the supports start to reach past the end of the arch, and the arrangement has to be modified near the ends, which is part of why real network arches vary the slope along the span rather than holding one.

The tie stops bending first #

More hangers do buy something, and the last figure shows the first thing.

With 16 hangers the tie carries 1,253 kN·m and the arch 1,727 at their worst, because the hangers are far enough apart that the tie spans between them as a beam and the arch bends between its nodes. With 32 the tie’s moment has fallen to 316 kN·m and the arch’s to 588. Beyond that, both fall slowly: 176 and 432 kN·m on 96 hangers. Cross the hangers and the bending goes found that crossing the hangers turns the arch and tie into chords of a truss; this figure shows that the truss is essentially complete at about 32 hangers on this span, a hanger every three metres, and that more hangers after that make the chords’ local bending between nodes smaller without changing the global action.

So the two effects of more hangers separate cleanly. The tie and arch stop bending by about 32 hangers, after which their bending falls only as their spans between nodes shorten. The hangers’ ranges go on falling as one over the number for as many hangers as there are, and the total hanger steel goes on being the same.

Why the web keeps the shear however it is divided #

It would be natural to expect a denser web to be stiffer, and a stiffer web to attract more of the shear from the tie, so that the hangers’ total share rose with their number. The totals rise a little — 20.0 MN of range on 16 hangers, 23.6 on 64 — and that is the reason. But the effect is small, because the web had nearly all the shear already. The stiffest path takes the load when two paths share a displacement, and in a network arch the web of crossed diagonals is so much stiffer in shear than the tie is in bending that the earlier essays found nearly all the shear in the web already on 32 hangers. Three times the hangers make the web stiffer still and take the tie’s small share smaller. There is little more shear to attract.

The contrast is with vertical hangers, which the truss with no diagonals describes: a web with no diagonals carries shear only by bending the chords between the hangers, so its hangers carry the deck’s weight and almost none of the live load’s shear, and their ranges are small — 194 kN on the same arch — while the tie bends. More vertical hangers would shorten the tie’s spans and reduce its bending, but would not move shear into the hangers, because a vertical member cannot carry it. Crossed hangers take the shear, and then share it among however many there are.

Each hanger’s range is set by its influence line, the force it carries as a unit load moves along the deck, and the worst place to stand is where that line is largest. For a diagonal in a truss’s web, the line changes sign at the panel the diagonal crosses: a load on one side puts it in tension and a load on the other in compression, so its range is the sum of the two lobes. With more hangers each diagonal crosses a shorter stretch of the tie, and both lobes shrink in proportion, which is the one-over-the-number law seen hanger by hanger. The hangers that carry the shear twice found the two lobes as the patch ran on from either end; here they shrink together as the hangers multiply, and the shape along the span stays the same.

What more hangers buy #

If the total hanger steel is fixed by the arch, the case for more hangers has to be made on other grounds, and there are three.

The first is the detail. A hanger’s fatigue strength is set by its connection, and for a threaded bar EN 1993-1-9 reduces it for diameters above 30 mm by $(30/d)^{0.25}$. A 3,000 mm² bar is about 62 mm in diameter and keeps 83 per cent of its detail’s strength; three times as many hangers of a third the area are about 36 mm and keep 96 per cent. The same total steel then lasts longer, or a little less of it lasts as long — not because sharing reduced the range in total but because the detail decides and the steel does not, and a smaller detail is a better one.

The second is the slack hangers. Every arrangement here has some hangers that go slack under part of the live load — six on 32 hangers, ten on 96 — and a slack hanger’s range is its whole largest force. A slack hanger also changes the truss while it is slack, and its neighbours take its share. With more hangers, the loss of one to slackness, or to an accident, or to replacement, is a smaller part of the web, and the arch can be designed to carry its load with any one hanger missing at a much smaller penalty.

The third is the deck. A tie supported every metre carries its own deck loads between hanger points as a short beam, and its local bending is small; a tie supported every six metres needs a deeper section for local loads alone. The tie moments in the last figure include that local bending, and their slow fall beyond 32 hangers is the tie becoming a member sized by its axial force alone. The deck that keeps the hangers in the truss found how much of a network arch’s behaviour depends on the tie being stiff enough to keep its hangers working; a dense web asks less of it.

The web, by hand #

The shear in a network arch’s web at a section, under a patch of live load from one end to that section, is about what a simply supported beam would carry there: at the quarter point, with the patch on the first quarter, $q a^2/2L = 60 \times 25^2/200 = 188$ kN. Shared among the hangers that cross the section, about 2.8 on 32 hangers, each carries $188/2.8 = 67$ kN of shear, and at 60° from horizontal its axial force is $67/\sin 60° = 77$ kN. A full reversal from one patch to the other doubles that, about 155 kN of range for the average hanger at the quarter point.

That is a third of the 528 kN mean range the envelope finds, and the difference is instructive: the envelope’s range is not from one patch to its mirror at one section, but from the worst patch for each hanger to the patch that slackens it — including the dead load’s own pretension coming off as the hanger goes slack, which a pure shear calculation leaves out. But the scaling in the hand estimate is the scaling the envelopes found. With 96 hangers the same shear is shared among 8.5, and each carries a third as much; the number of hangers times each one’s share is the shear, whatever the number.

One slope, one load, one plane #

One slope and even spacing. Tveit’s hangers vary in slope along the span, and the spacing on the tie is sometimes uneven to keep the arch nodes evenly spaced. A varying slope puts steeper hangers near the crown, where the section count is high anyway, and flatter ones near the supports, where it is low, which evens out the ranges along the span without changing the total.

One load pattern. The live load here is a uniform patch running on from either end. A real bridge’s fatigue load is a set of vehicles of different weights and axle spacings, and the range for each hanger is a spectrum, not a single number. The scaling with hanger count is a property of the web’s sharing and should hold for any load; the totals would change with the load model.

A plane frame. Real network arches have two arch planes braced together, and a load off the centreline loads one plane more than the other; each plane’s web then shares its own plane’s shear, exactly as here.

Still open: hangers that are not all bars #

Every hanger here is a steel bar, stiff in tension and slack in compression. Some network arches use cables or strands instead, which have their own fatigue category — far better than a threaded bar’s, worse at their anchorages — and a sag under their own weight that makes them softer at low force. A cable hanger near slack is therefore softer than a bar of the same area at the same force, and it sheds shear to its neighbours exactly when it is closest to going slack. Whether a web of cables evens out its ranges by softening the hangers that are about to slacken, or concentrates them on the hangers that stay taut, is the question that the hanger’s material puts to the sharing this essay found.