The hanger that cannot be made soft enough

The hangers that carry the shear twice ran a live load across a network arch from either end and found that its crossed hangers swing through nearly all of their force: the worst of 32, on a 100 m span with a 17 m rise, from 49 kN to 727, a range of 678. Vertical hangers on the same arch swing through 194. The crossed hangers are the web of a truss whose chords are the arch and the tie, and a patch of load’s shear passes through them as through a truss’s diagonals, from one sign to the other as the load runs on and off. That is why crossing the hangers takes the bending away, and why it puts a fatigue range into every hanger.

The essay ended on a way out that the design rules do not consider. A hanger that stretches more takes less of the shear, and a tie that bends a little more carries it instead. Whether a network arch with deliberately compliant hangers — longer, of a higher-strength bar, or with a spring at the anchorage — moves enough of the shear back into the tie to bring the hangers’ ranges inside their details’ fatigue strength, without giving back the bending the crossing was meant to remove, was the question.

Softer hangers swing through almost as much #

The arch is the earlier essays’: 100 m span, 17 m rise, 32 hangers crossing at 60°, each a 3,000 mm² steel bar with an axial stiffness of 630 MN; 60 kN/m of dead load and 60 kN/m of live load run on from either end, the slack hangers taken out at every stage. Each hanger’s axial stiffness is then scaled by a factor — a spring in series at the anchorage below one, a larger bar above it — and the envelopes found again.

The range barely moves. Halving the hangers’ stiffness takes 7 kN off the worst range, 1 per cent. A tenth of the stiffness takes 43 kN off, 6 per cent. Even a fiftieth — a spring so soft that each hanger stretches by tens of centimetres under its own dead load — leaves the worst range at 557 kN, four fifths of what it was, and three times the vertical hanger’s. Making the hangers stiffer does nothing at all: four times the stiffness, 684 kN.

The mean range falls a little faster than the worst, from 528 to 478 kN at a tenth and 400 at a fiftieth, because the hangers near the springings, which carry the least shear, give up proportionally more of it. The worst hangers are near the quarter points, where the web’s shear is largest and the arch’s own slope helps least, and they are the ones the softening reaches last.

What the softer hangers give back is bending #

Where the range does fall, the bending comes back, and faster. A tenth of the stiffness puts a third more moment into the tie for 6 per cent less range; a fiftieth multiplies the tie’s moment by 3.7 for 18 per cent less range. The tie and the arch are the truss’s chords, and a chord carries shear only by bending; every kilonewton of shear the web gives up arrives in the chords as moment, at the lever arm of a panel. Softening the web is a way of turning the truss back into a Vierendeel frame, and a Vierendeel frame is what the crossing was designed to replace.

The range belongs to the truss #

Drawn against each other, the range and the bending show what softening can and cannot reach. The crossed arch can be walked along a short line, from 678 kN and 316 kN·m towards 557 and 1,154, paying more than three times the bending for a fifth of the range. The vertical arch is nowhere near the line. It has a third of the range and eleven times the bending, and the line that leads from the stiff network towards it would have to bend down sharply somewhere beyond a fiftieth of a bar’s stiffness — where the hangers have stopped being hangers.

So there is no compliant network arch that has a vertical arch’s ranges and a network arch’s bending, or anything near either. The range belongs to the truss. As long as the crossed hangers are stiff enough to be the web — and at a fiftieth of a bar’s stiffness they still are — they carry the shear a web carries, and that shear swings sign as the load passes. The only arrangement with small ranges is the one whose hangers carry no shear at all.

Why the web keeps the shear #

The softening does so little because of how much stiffer the web is than the alternative. A live load on half the span is a uniform load of half its intensity and an antisymmetric load of 30 kN/m up on one half and down on the other. The uniform part goes to the arch as thrust and barely touches the hangers’ ranges. The antisymmetric part is what the hangers cycle with, and it has to cross the span from one half to the other as shear, either through the web or through the chords’ bending.

Through the chords it would go as two beams, the arch and the tie, each bending over a half-span of 50 m. Their combined bending stiffness, about $360{,}000\ \text{kN·m}^2$, carrying 30 kN/m across 50 m as a simply supported beam would, deflects by $5 \times 30 \times 50^4/(384 \times 360{,}000) = 6.8$ m. The chords alone are hopeless at it. That is the bending a truss with no diagonals has to supply, and on vertical hangers it is what the 3,505 kN·m in the tie is made of: the chords carrying a little of the shear at a very large deflection, with the arch’s thrust line doing the rest.

Through the web the same shear goes as axial force in the two or three hangers a vertical section cuts. A diagonal at $\theta$ to the horizontal carrying a share $V$ of the shear is stretched by $V L_d/(EA\sin\theta)$, which moves its panel’s ends apart vertically by that over $\sin\theta$, so a web of diagonals has a shear stiffness of $EA\sin^2\theta\cos\theta$ for each one a section cuts: $630{,}000 \times 0.75 \times 0.5 = 236{,}000$ kN, and about 600,000 for the two or three of them. Carrying 30 kN/m across a half-span of 50 m in shear, the web deflects by $q\ell^2/8GA = 30 \times 50^2/(8 \times 600{,}000) = 16$ mm.

Sixteen millimetres against 6.8 m: the web is more than four hundred times stiffer than the chords at carrying the antisymmetric shear. Made ten times softer it is still forty times stiffer, and the chords take a fortieth of the shear; a fiftieth as stiff, it is still nine times stiffer, and the chords take about a tenth of it, of the order of the 18 per cent the worst range fell by. Only at about a four-hundredth of a plain bar’s stiffness would the chords take half, and by then a hanger under its dead load would be stretched by metres.

So the shear stays in the web for the same reason the stiffest path takes the load: not because the web is stiff, but because the alternative is so much softer. The hangers’ range follows the shear, and the shear follows the ratio, and no practical spring moves the ratio far enough to matter.

Soft hangers stay taut #

The design rules for network arches are written around slackness: a crossed hanger that goes slack under a heavy partial load takes its triangle out of the truss, and the deck that keeps the hangers in the truss found how much bending that gives back. Softening helps there.

At the bar’s own stiffness six hangers go slack at some stage of the live load; at a tenth of it, none. The hangers whose least force came nearest to nothing swing a little less far as they soften, and that little is enough to keep them in tension. But the worst range at a tenth of the stiffness is still 635 kN, and on a 3,000 mm² bar that is 212 N/mm² every time the heaviest traffic passes. Taut is not the problem. A hanger can be in tension at every stage and still be the most heavily cycled member on the bridge, and the rule that keeps it taut does nothing about the range.

The envelope barely moves #

Hanger by hanger, a twentieth of the stiffness shaves both ends of each envelope a little and changes its shape not at all. The hangers near the quarter points keep their long bars, near-zero to near-maximum; the hangers at the ends keep their short ones. A spring at every anchorage that lets the hangers stretch twenty times as far has bought a tenth of the worst range and given back twice the tie’s bending.

The spring itself is not a small object. A hanger near the quarter point is about 14.5 m long, so a 3,000 mm² bar has an axial stiffness of about 43 kN per millimetre; a spring that leaves it a twentieth of that has to have about 2.3 kN per millimetre, and under the worst hanger’s dead-load force of 259 kN it is compressed by over a hundred millimetres and swings by more than two hundred and fifty with every heavy vehicle. That is the travel of a railway buffer, at both ends of all 32 hangers, for 10 per cent off the worst range.

Sized for the range #

The hanger that cannot be made soft enough has to be made large enough.

Because the force range hardly depends on the hanger’s stiffness, the stress range is very nearly the force range over the area, and the area can be chosen for it directly. The worst hanger’s 678 kN range is 226 N/mm² on 3,000 mm²; a threaded bar, whose thread puts it in detail category 50, needs about 13,800 mm² to bring the range to its category, and about 18,900 to bring it under its constant-amplitude limit, below which a cycle of that size does no damage. Its strength needs 1,640 mm² at 460 N/mm². Sized for its range, the hanger is eight times the bar its strength asks for.

And the larger bar costs the range nothing, because stiffness does not move it: four times the stiffness, 684 kN against 678. That is the answer to whether a hanger should be sized for its force, its range or its stiffness. Its force is the smallest of the three demands; its stiffness is not a lever at all; its range decides, and the area that satisfies it brings its own stiffness, which the truss neither needs nor minds.

A bar of 13,800 mm² is a solid rod 133 mm across, and a thread on a rod that size is weaker still than category 50, because fatigue strength falls with size. That is why network arch hangers are not threaded rods: they are flat plates or round bars with forged or machined ends, whose details are rated well above a thread’s, and where the area the range asks for is smaller in proportion.

What does reduce the range #

If stiffness does not, three things do, and none of them is a property of the hanger.

More hangers. The earlier essay found the worst range falling nearly in proportion to their number, 1,252 kN on 16 hangers and 467 on 48, because the shear at a section is shared among more diagonals. The fraction of each hanger’s largest force that it swings through stays near nine tenths, so more hangers make each smaller without making any of them less of a fatigue member; but the area each needs for its range falls with the count, and so does the weight of the detail at each end.

A better detail. The range is fixed by the structure; the stress range the detail sees is fixed by the area; what the detail survives is its category. A forged eye or a machined end on a round bar is rated far above a thread, and a hanger plate with its end shaped and ground well above a fillet-welded attachment. Choosing the detail is choosing the multiplier on everything above.

Less of the live load counting. Most cycles do not count: a detail’s cut-off removes the small ones, and a crossed hanger’s long influence line makes its smaller cycles bigger than a vertical hanger’s — every vehicle on the span moves it — so the spectrum the detail sees is harsher than the envelope alone suggests. The envelope is the largest range; the damage is the count of all of them, and that is a matter of the traffic, not of the hanger.

The spring, by hand #

A hanger near the quarter point runs from the tie up to the arch about 12.5 m above it at 60°, so it is about 14.5 m long. Its axial stiffness as a plain bar is $EA/L = 210{,}000 \times 3{,}000/14{,}500 = 43$ kN/mm. To leave it a twentieth of that, a spring in series must satisfy $1/k = 20/43 - 1/43$, so $k = 43/19 = 2.3$ kN/mm.

The worst hanger carries 259 kN under the dead load alone, so the spring is compressed by $259/2.3 = 113$ mm before any traffic arrives, and swings by $607/2.3 = 264$ mm between the worst hanger’s least and largest forces. A tenth of the stiffness, which buys 6 per cent, needs a spring of 4.8 kN/mm and a little under half those movements.

Elastic hangers and a patch of load #

The live load is a patch run on from either end, as in the earlier essay. A real fatigue load is a train of axles, which produces several cycles per passage on a hanger with a long influence line; the envelope is the largest range among them, and the counting of the smaller ones is the loops a crack grows on.

The springs are linear and the hangers elastic. A spring with a preload, or one that stiffens as it closes, behaves differently near its limits; a hanger that is pretensioned to keep it taut has the same range, since the pretension moves the envelope and not its width — the conclusion the dead load already drew for the hangers’ own weight.

The arch and the tie keep their stiffness. A stiffer tie takes more of the shear as the hangers soften, and the line in the trade-off figure would bend sooner; but a tie stiff enough to carry the shear by bending is a tie sized for the moments the vertical arch has, which is the structure the network was chosen to avoid.

A web is a web #

The crossed hanger’s range is not a property of the hanger but of its place in the structure. It is a diagonal of a truss whose shear changes sign as the load crosses, and a diagonal carries its share of that shear in proportion to the truss’s stiffness against shear, which its own axial stiffness hardly enters until it is made too soft to be a diagonal. That is the same lesson the stiffer path that takes the load teaches in reverse: the load goes to the stiffest path, and a web made softer is still enormously stiffer, as a path for shear, than a chord’s bending.

So the choice the network arch offers is not between ranges and bending along some continuum a designer can tune. It is between two structures. The crossed hangers make a truss, with chords that hardly bend and a web that cycles through nearly all its force; vertical hangers make a frame, with hangers that carry only their panel’s weight and chords that bend under every partial load. A network arch is chosen for its chords, and its hangers are then sized for their range.

Still open: hangers that cross twice #

Every hanger here crosses its neighbours at one slope, and each section of the span is cut by two or three of them. Tveit’s designs for longer spans use more hangers at varying slopes, so that each hanger crosses many others and a vertical section cuts five or six; the shear at any section is then shared among more diagonals, each carrying a smaller part of it. Whether sharing the shear among more crossings reduces each hanger’s range in proportion — so that the area each needs falls with the number crossing a section — or whether the range of each stays a fixed fraction of its own largest force, as the count of hangers found, decides whether the fatigue size of a network arch’s hangers is set by its geometry or by how many of them there are.