The hangers that carry the shear twice

Cross the hangers and the bending goes set a tied arch’s hangers on a slant, crossing each other, and showed that the tie, which on vertical hangers bends hard under a load on half the span, hardly bends at all: the arch, the tie and the crossed hangers make a truss, and a truss carries a partial load by axial forces in its web. The deck that keeps the hangers in the truss followed the price of that — under a heavy partial load some crossed hangers would have to push, and a hanger cannot, so they go slack and give some of the bending back.

Both essays looked at one load: the deck’s weight and a live load on half the span. A bridge carries a live load that moves. Every train and every lorry runs on from one end and off at the other, and each hanger’s force rises and falls as it passes. What a hanger has to survive is not one force but the range between its largest and its smallest, millions of times. This essay asks what crossing the hangers does to that range.

The envelope of a crossed hanger #

The arch is the earlier essays’: 100 m span, 17 m rise, 32 hangers crossing at 60° to the horizontal, 60 kN/m of dead load on the tie. A live load of 60 kN/m runs on from the left end to every point of the span, then from the right end to every point, and each hanger’s force is computed at each stage — with slack hangers taken out, as before. The largest and smallest force each hanger sees, over all those stages and the dead load alone, is its envelope.

The bars are long. The dead load alone puts between 150 and 280 kN in most hangers. As the live load runs on, each hanger’s force rises to between 450 and 750 kN at some stage and falls to near nothing at another: hanger 11, a fifth of the way along, swings from 49 kN to 727, a range of 678. Six hangers, the end pairs and a few near the crown, go slack at some stage and swing from nothing.

The worst range is 90% of the largest force in any hanger. If every hanger is sized to the same stress for its largest force, the stress in the worst one swings through nine-tenths of that stress with every vehicle.

The same arch on vertical hangers #

The comparison is the same arch with its hangers vertical.

A vertical hanger carries the deck’s weight over its own panel, nothing else to first order, and the live load adds its own weight over the same panel when it is there. Its force runs from the dead load’s share, 194 kN, to the dead and live load’s, 388; its range is the live load’s share, 194 kN, half its largest force, because the live load here equals the dead load. The bars are short and all the same.

The tie of the vertical arch is paying for this. Under a load on half the span it bends hard, which is what the crossed hangers were introduced to stop. But its hangers see only the load directly above them.

Why the crossed hangers swing #

A crossed hanger is a diagonal in a truss, and a diagonal carries shear.

Under the dead load the shear in a symmetric arch is carried mostly by the arch’s own curvature — the arch is close to funicular for a uniform load, and the hangers carry the deck’s weight to it in roughly equal shares. A live load on part of the span is not uniform, and the truss carries its antisymmetric part as a beam would: by shear, which passes through the web. The shear at a section changes sign as the load’s front passes it, so a diagonal that is stretched by the shear when the load is on one side is relieved when the load is on the other. On top of the dead load’s share each crossed hanger carries a shear term that swings both ways.

So each crossed hanger carries the shear of a moving load as a web member does, and its influence line has both signs. The vertical hanger’s influence line is a single spike; the crossed hanger’s is a long wave across the span, and the range is the difference between the wave’s peak and its trough. That is what the essay’s title means: the crossed hangers carry the deck’s weight to the arch, and they carry the live load’s shear through the truss, and the second job is the one that swings.

On average the crossed hangers’ ranges are 2.9 times the vertical ones’. The alternation in the plot is the two families of hangers, leaning one way and the other: under a load front at a given point one family is stretched and the other relieved, and their ranges differ by the angle each makes with the shear it carries. The end hangers, close to the bearings where the shear is carried by the arch’s springing, have the smallest ranges.

Two families, and the hangers at the ends #

Reading the envelope hanger by hanger separates the two families. Hangers 3, 5, 7, 9 and 11, leaning one way, carry 260 to 280 kN of dead load and swing through 610 to 680 kN; hangers 4, 6 and 8, leaning the other, carry 150 to 170 kN and swing through about 420 to 430. The difference is which way each family leans against the shear a load from the nearer end produces: one family is stretched by it and also carries more of the deck’s weight; the other is relieved by it.

The end hangers are the exception the envelope makes plainest. Hanger 1, nearest the bearing, carries only 29 kN of dead load, because so close to the bearing the arch’s springing and the tie carry the load there between them; it swings from nothing to 285 kN, slack at one stage and taut at another. Hanger 2 beside it is the only one in the arch whose smallest force is its dead load: 260 kN, rising to 521, the range of a vertical hanger. Towards the crown, hangers 10, 12 and 16 dip to zero or close to it at some stage: the shear there changes sign as each load front passes, and a hanger that a load from one end relieves almost entirely is loaded by the load from the other.

Dead load moves the force and not the range #

The obvious response is to make the deck heavier: more dead load keeps the hangers in tension, which is what the earlier essay’s rules are written to do.

It does keep them taut. It does not change their range. On a hanger that stays taut the structure is linear, and the dead load adds the same force to every state the live load creates; the largest and the smallest both rise by it, and their difference does not. The worst range is 678 kN at 60 kN/m of dead load and 678 kN at 160. The largest force rises from 756 to 1,231 kN.

A heavier deck therefore makes the fatigue picture worse in the only way it can change it: the hanger has to be larger for its strength, so the stress range falls in proportion to the area — but the area is set by the larger force, and the mean stress the range sits on rises. What the dead load buys is that the hanger stops going slack, which matters for a different reason: a hanger that goes slack and snaps taut again is hit, and its anchorage sees an impact that no static range contains.

Below 60 kN/m the range is shorter — 570 kN at 20 kN/m — because many hangers go slack and their force cannot fall below zero; the bottom of their swing is cut off. That is a reduction in range bought with slackness, which the earlier essay found also gives the tie its bending back.

The slope that suits strength is not the one that suits fatigue #

The hangers’ slope is the network arch’s main design variable, and it moves the force and the range differently.

The largest force is least near 60°, 756 kN, which is close to the slope network arches are usually given: steep enough that the hangers cross several times and share the load, flat enough that they reach along the span. The range is least near 50°, at 627 kN, and it rises steeply above 60°: at 70° the hangers are nearly vertical in their reach but still cross, most of them go slack at some stage, and the worst range is the whole of the largest force.

So the slope that suits the hangers’ strength is not the one that suits their fatigue, and the two are a few degrees apart in a direction that a designer optimising for steel weight would not move.

More hangers, smaller swings, the same share #

Adding hangers is the other variable.

The range falls nearly in proportion to the number of hangers: 1,252 kN with 16, 678 with 32, 467 with 48. Each hanger’s share of a patch’s shear falls as more of them share it. But the largest force falls in the same proportion, so the range stays between 87% and 91% of the largest force whatever the count. More hangers make each one smaller; they do not change the fact that each one’s force swings through nearly all of itself. A crossed hanger is a fatigue member by its geometry, not by its size.

What nine-tenths of the stress means #

A range is a force; fatigue is checked as a stress, and the two are joined by the hanger’s area, which is chosen for strength.

Suppose every hanger is sized so that its largest service force puts it at the same stress — a bar of high-strength steel at, say, 250 N/mm² under the most onerous load. A vertical hanger’s stress then swings through half of that, 125 N/mm², with every heavy vehicle. The worst crossed hanger’s swings through nine-tenths, 225 N/mm². Fatigue strength is quoted as the stress range a detail survives for two million cycles, and a welded attachment of the kind a hanger is fixed by is rated at a few tens of newtons per square millimetre on that scale. Neither hanger survives as sized, but the crossed one is further out by nearly a factor of two in stress, which on the cube law of steel fatigue is a factor of nearly six in life.

Not every vehicle is the heaviest, and most of the cycles a structure sees do not count: below a detail’s cut-off a cycle does no damage at all, and a spectrum of traffic reduces to an equivalent range nearer the heaviest vehicles than the average. The crossed hanger’s long influence line makes that worse rather than better, because every vehicle on the span moves its force, where a vertical hanger feels only the vehicles over its own panel. So a crossed hanger is sized up for its range, below the stress its strength would allow, or it is given an anchorage detail of a much higher category — a forged or machined end in place of a welded plate — or it is designed to be found in time, with an inspection interval set by how fast a crack in it would grow.

The influence line is the whole difference #

The reason the two arrangements differ so much is visible in a single picture, not drawn here but implicit in the envelopes: a hanger’s influence line, its force against the position of a unit load.

A vertical hanger’s influence line is a narrow spike over its own panel, with small tails where the tie’s bending passes some of a nearby load to it. A crossed hanger’s is a wave that runs most of the length of the span, positive over one stretch and negative over another, because a load anywhere changes the shear the web carries. The worst place to stand for a member with a two-signed influence line is a pair of places, one for each sign, and a moving load visits both on every passage. The range is the area of the wave’s positive lobe plus the area of its negative one, times the load’s intensity — and for a crossed hanger both lobes are large.

The ranges, read from a free body #

The vertical hanger’s range can be checked with nothing but its panel. A hanger 3.125 m from its neighbours carries $60 \times 3.125 = 187.5$ kN of dead load and the same again of live, so its force runs from 188 to 375 kN — the figure’s 194 and 388 are those numbers with the arch’s slight non-funicularity added — and its range is the live load’s share exactly.

The crossed hanger’s range needs the truss. A live load on half the span is a uniform load of half its intensity plus an antisymmetric one of $\pm 30$ kN/m, and on a simply supported 100 m span the antisymmetric part has a shear of $30 \times 100/4 = 750$ kN at the ends and at mid-span, changing sign at the quarter points. In a tied arch part of that shear is carried by the arch’s own slope — near the springings, almost all of it — and the rest by the web, where the few crossed hangers a vertical section cuts share it, each carrying its part divided by $\sin 60° = 0.87$. As the load runs on and off, the shear at a section swings from one sign to the other, so a hanger crossing it sees twice its share: with a few hundred kilonewtons of web shear shared among two or three diagonals, a range of several hundred. The figure’s 500 to 680 kN are of that order, and the vertical hanger’s 194 is not, because a vertical hanger carries no shear at all.

What the force range leaves out #

A force range is the input to a fatigue check, not its answer, and the step from one to the other has its own assumptions.

The detail decides. A hanger is a bar or a cable, and its fatigue life is set at its ends: the welded plate, the forged eye, the socket. The detail decides and the steel does not, and a welded hanger anchorage has a fatigue strength a small fraction of the bar’s yield stress. A range of nine-tenths of the design stress is far beyond what any welded detail survives for millions of cycles.

The live load here is a patch. A real fatigue load is a vehicle or a train of axles, which produces a sharper influence-line response and several cycles per passage on a long hanger. The patch gives the envelope; counting the cycles needs the train.

Hangers vibrate. A slender hanger has a natural frequency that a passing train or a vortex can excite, adding bending at its ends that no static analysis contains — and network hangers are often clamped together where they cross to stop it.

The tie and the arch are elastic. Their stiffness shares the load with the hangers; a flexible tie gives the hangers more of the shear and a stiff one less.

Still open: hangers that are made for the range #

The design rules for network arch hangers are written around keeping them taut, and the ranges here say that taut is not the problem: a taut crossed hanger’s range is set by the live load and the geometry, and the dead load cannot touch it. What can is the hanger’s own stiffness against the tie’s: 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, is the question of whether the hanger should be sized for its force, its range, or its stiffness.