The stay shaken along its own length

Every excitation a stay cable is checked against pushes it sideways. Wind across it, rain running down it, a vortex shed from its lee: each is a force across the cable, and each is dangerous when it arrives at one of the frequencies read off the stay to find its tension. That essay also named, in four paragraphs and without a number, the one excitation that pushes nothing sideways. The deck a stay is anchored to moves, the anchorage moves with it, and a movement along the stay’s own line stretches the stay and lets it go again. The tension swings, and a member whose tension swings is a member whose stiffness swings.

The stay here is the same one: 120 m long, 3,500 kN, 60 kg a metre, inclined at 25°, with a fundamental of 1.01 Hz. A damper near its anchorage can give it about one per cent of critical damping and no more, and a line of ties cannot lift its first frequency unless the line is carried to the deck. Both answers treated the deck as the fixed thing the stay hangs from. This one lets the deck move.

A swing in tension that pushes nothing across #

The size of the swing is a short calculation. A deck moving ±10 mm vertically at the anchorage moves the anchorage 4.2 mm along a stay inclined at 25°, since only the part of the movement along the stay stretches it. The stay resists that as a spring of EA/L, softened a little by its sag: Ernst’s equivalent modulus takes 1.1 per cent off the 1,400 MN of axial stiffness for a stay this short and this tight, and the spring is 11.5 MN a metre. So 4.2 mm of stretch is 49 kN on a tension of 3,500 kN, a swing of 1.39 per cent. It is a small number, and it is the only input the rest of the argument has.

What it does follows from the relation the tension measurement rests on. The square of every frequency of a taut string is proportional to its tension, so a 1.39 per cent swing in tension is a 1.39 per cent swing in the stay’s stiffness against sideways motion, repeated at the deck’s frequency. Nothing pushes the stay across. The stiffness that holds it straight is being turned up and down.

That can feed a motion the stay already has, and the way it does is worth following. A stay bent into its first mode is longer than a straight one. If the tension rises while the stay is out at the end of a swing, the deck is stretching a bent stay, and the work goes into the swing. If the tension falls while the stay is passing back through straight, it takes nothing out, because a straight stay has no extra length for the tension to act through. The deck adds energy by tightening the stay at the ends of its swing and slackening it in the middle, and a stay reaches the end of a swing twice in every cycle, once on each side. So the deck that pumps a stay is a deck moving at twice the stay’s frequency — and that pair of frequencies is not one an ordinary check compares, because an ordinary check looks for a deck and a stay whose frequencies are equal.

The transverse motion of one mode, with its tension swinging, is Mathieu’s equation with damping added: the acceleration, plus $2\zeta\omega$ times the velocity, plus $\omega^2(1 + \varepsilon\cos\Omega t)$ times the displacement, is zero. Here ω is the stay’s own circular frequency, Ω the deck’s, ζ the damping ratio and ε the swing, 1.39 per cent. Whether a small disturbance grows under that equation is decided by following two independent motions through one whole cycle of the deck and asking whether the pair of them has grown by the time the deck is back where it began. That is Floquet’s test, and the chart above is its answer across a range of deck frequencies and swings.

The swing has to beat four times the damping #

The shaded regions are where a disturbance grows, and they have two features that decide everything else.

They are narrow. Each one stands on the line where the deck’s frequency is exactly twice the stay’s, and even at a swing of 6 per cent the lightest region spans only 1.970 to 2.030 times the stay’s frequency. A deck a few per cent away from the right frequency does nothing, however hard it moves.

And each stands on a tip. With no damping the dashed V reaches down to no swing at all: an undamped stay grows under any movement of the deck, however small, provided the frequency is exactly right. With damping the tip lifts, and it lifts to a swing of exactly four times the damping ratio — 0.4 per cent for a stay damped at 0.1 per cent, 2 per cent at 0.5, 4 per cent at 1.

The factor of four comes from averaging the equation over a cycle. Write the swinging stay as one part in step with a cosine and one in step with a sine, let the two parts change slowly, and average what the swinging tension does to them over each cycle. Their sum grows at ω(ε/4 − ζ) and their difference decays at ω(ε/4 + ζ). A quarter of the swing drives the one; the damping holds both back. Growth needs a quarter of the swing to exceed the damping ratio, and nothing about the stay’s length, tension or mass is in that condition. Floquet’s test knows none of the averaging, and puts the tip in the same place to four significant figures.

A stay cable has almost no damping of its own, and 0.1 per cent is the value used for this one. The threshold swing is then 0.4 per cent, and the deck movement that produces it is ±2.9 mm. Nobody standing on a deck moving three millimetres would notice it.

Growth measured in minutes, with no limit in sight #

The histories show the averaged result doing exactly what it said. Below the threshold, at ±2 mm, the start splits into a part that decays fast and a part that decays slowly, and once the fast part has gone the slow part halves every six minutes. Above it the amplitude grows as an exponential, which is a straight line on these axes: tenfold every 491 s at ±5 mm, every 147 s at ±10 mm, every 61 s at ±20 mm. The dashed curves are the averaged solution drawn from its two rates alone, and they lie on the integrated ones — including the offset that puts every growing curve a factor of $\sqrt{2}$ below a single exponential, because half of the start went into the part that decays.

Two things about these curves separate them from the resonance a machine drives in a floor. A stay pushed sideways at its own frequency builds up along a curve that flattens at a limit, and the limit is the static deflection magnified by one over twice the damping ratio. Here there is no static deflection to magnify and no limit: the damping decides whether the motion grows, not how large it gets, and above the threshold the linear equation grows for ever. And the swing in tension does nothing to a stay that is perfectly still. It only amplifies a disturbance that is already there, which in practice is never missing — the wind across a stay and the traffic on its deck supply one continuously.

The other thing the curves say is how slow this is. Tenfold in 147 s at ±10 mm is tenfold in 148 cycles of the stay. From a millimetre to a few hundred millimetres takes several minutes, and for all of those minutes the deck has to hold its frequency at twice the stay’s. That condition turns out to be the harder one to meet, and a later section returns to it.

The damper that was too small for everything else #

A viscous damper near the anchorage was capped by its position: fixed at 2 per cent of the stay’s length, it can supply about 1 per cent of critical and no more, whatever size it is made. Against rain and wind that ceiling is the difficulty, because those mechanisms can demand more.

Against a moving deck the demand is a quarter of the swing, and a quarter of the swing is small.

Read along the straight line. The stay on its own grows under ±2.9 mm of deck. With the damper at 2.4 m from its anchorage, at its best size, it is damped at 1.02 per cent, the threshold swing is 4.1 per cent, and the deck has to move ±29 mm before anything grows. The damper whose ceiling was the whole difficulty against the wind multiplies the deck movement this stay can take by ten, because the threshold is proportional to the damping and the damping is exactly what the damper buys.

The curve above the straight line is the other arrangement, with the deck at the stay’s own frequency rather than twice it. The tension then rises once in each cycle, tightening the stay at one end of its swing and slackening it at the other, so the energy it adds at one end it removes at the other, and to first order the two cancel. What survives is second order in the swing — an eighth of its square against the damping ratio — and the threshold is a swing of $\sqrt{8\zeta}$: 8.9 per cent at the stay’s own damping, which takes ±64 mm of deck. The damper raises it only as a square root, to about ±205 mm. The dangerous ratio is two to one, and the one to one case needs a deck moving twenty-two times as far.

The damper’s curves carry the same point across the modes, and that matters because a deck does not only move at one frequency. The tension is shared by the whole stay, so a deck movement swings it by the same fraction for every mode, and every mode needs the same quarter of that swing in damping. The damper’s ceiling is also the same for every mode, set only by its distance along the stay. So one damper, sized between the best sizes of the first few modes, gives each of them about four fifths of 1 per cent, and that covers a deck moving about ±23 mm at twice any of their frequencies. Protection against a moving deck is one number for the whole stay, and it is a number the damper that failed elsewhere already supplies.

A band a few tens of kilonewtons wide #

The narrowness of the regions in the first chart has a consequence that decides when any of this happens at all.

At ±10 mm of deck the region is 0.33 per cent of the stay’s frequency either side of the exact ratio, which for this stay is a deck between 2.006 and 2.019 Hz. The stay’s frequency goes as the square root of its tension, so the same band in tension is twice as wide in proportion: 0.67 per cent, 23 kN either way. A stay 23 kN tighter or slacker than the one the deck is tuned to does not grow at all; it decays at its own damping, as if the deck were still.

Twenty-three kilonewtons is not much tension for a stay to gain or lose. A difference of 10 °C between the stay and the deck changes it by 166 kN, seven times the band, because a difference in thermal strain between the stay and the deck acts through the same axial stiffness that the deck’s movement does. Traffic changes the tension as well, by amounts that depend on the bridge and are not computed here. So the model does not describe a stay that is either inside the region or outside it. It describes a stay that drifts through the region as its temperature and its load change, and grows only during the minutes when the deck is moving at one steady frequency and the tension happens to sit within 23 kN of the value that matches it.

That turns the growth time from a curiosity into the controlling number. From 1 mm, the stay at ±10 mm takes 394 s to reach nine tenths of the amplitude it eventually settles at. An episode needs the deck frequency and the stay’s tension both to hold still for longer than that, and when either wanders the stay decays again. The model predicts bursts rather than a steady state.

The same band is a statement about what can be known in advance. The stay’s frequency is read to find its tension, and to decide whether a given stay sits inside this band, both its frequency and the deck’s would have to be known to a third of a per cent, in service, at the temperature of the day. That precision is available from a long record on a finished bridge. It is not available from a design model. Tuning a stay away from twice a deck frequency is therefore not a design strategy; supplying a quarter of the swing in damping is.

The stay stops itself by stretching #

The linear equation says the growth never stops. It stops because a swinging stay is a longer stay.

A stay bent into its first mode with amplitude q is longer than its chord by q²π²/4L, and stretching it that much through its axial spring adds tension. The added tension goes as the square of the amplitude, so it is nothing at a millimetre and a great deal at a few hundred. It raises the stay’s frequency — the stiffness that comes from a cable’s shape, appearing here in time rather than under load — and a stay whose frequency rises moves away from half the deck’s.

The stay swings its way to the edge of the band. At 362 mm the swinging adds up to 31 kN at the ends of each swing, and three quarters of that, 23 kN, in the average that sets the stay’s frequency. Twenty-three kilonewtons is the half-width of the band. The amplitude settles exactly where the stay’s own stretching has carried its frequency to the edge of the region it grew in, which is the balance averaging expresses in closed form; the integration of the full equation, which consults no averaging, arrives at the same amplitude to within a millimetre.

It does not arrive there directly. The detuning that stops the growth is set by the amplitude, and the amplitude takes time to respond to it, so the stay carries on growing for a while after it has passed the balance: it overshoots to 446 mm, falls back below the plateau, and rings down onto it over the following minutes, each crest a little nearer. The ringing is a second, much slower oscillation — of the amplitude of the swing rather than of the stay — with a period of about two and a half minutes, and the first crest is 23 per cent above the value the stay ends at.

The plateau is hundreds of millimetres, and it depends only weakly on how hard the deck moves. Doubling the deck movement to ±20 mm raises it from 362 to 521 mm, a factor of 1.44 rather than 2, because well above the threshold the plateau grows only as the square root of the deck’s movement. The damping has almost no say in how large the plateau is once the stay is well above the threshold; its whole effect is on whether the threshold is crossed. With the damper at 1.02 per cent and the deck at ±10 mm there is no plateau, because there is no growth.

A 362 mm swing at 1 Hz for as long as the conditions hold is a great many cycles of bending at the anchorage, where the strand enters the socket and the cable’s own bending stiffness concentrates the curvature. It is an amplitude that decides a fatigue life rather than a strength, and it is why the threshold is worth more than the plateau.

Which stays in a fan are exposed #

A deck has several modes, each with its own period, and a fan has many stays. Which of them meet at two to one is a question with a simple shape.

The stays in a fan are designed to broadly the same stress in the same strand, so the wave speed, the square root of tension over mass per metre, is nearly the same in all of them — 242 m/s here — and each stay’s frequencies fall as one over its length. A deck mode at a given frequency then picks out a length for each of the stay’s modes: the length whose first frequency is half the deck’s, the length twice that whose second frequency is, and so on. For deck modes at 0.9, 1.4 and 2.0 Hz the exposed first modes are at 268, 173 and 121 m, and a 242 m stay meets the 2.0 Hz mode in its second.

The 120 m stay is 0.8 m from one of those lengths, and the band that matters is a third of a per cent in frequency, which in a fan at one wave speed is a third of a per cent in length — 0.4 m either side. Its frequency at the design tension is a little above half the deck’s, so it is outside the band; 21 kN less tension brings it in, and it stays in until 67 kN less. A stay 1.3 °C warmer than its deck loses the first 21 kN. That is the practical reading of the whole chart. A list of stay lengths against deck frequencies finds the stays that are near a two to one ratio; it cannot find the ones that are inside the band, because the band is narrower than what any of those frequencies are known to. The list says which stays to watch. The damping says whether it matters.

What the one-mode stay leaves out #

Only one mode moves at a time. Each is taken in its own exact shape, the half sine an estimate of a period by its shape would choose, and the stretching is computed for that shape alone. A stay swinging in its first mode changes the tension every other mode feels, so a large motion in one mode is itself a swing in tension for the others, at twice its frequency — which is exactly the ratio at which the second mode of a taut string sits. The modes exchange energy through that coupling, and nothing here follows it.

The deck moves as a pure sine at one frequency. Real deck motion under wind or traffic is narrow-band and its frequency wanders. A deck driven by vortices shed at a rate the wind speed sets comes closest to holding one, and only while the wind holds its speed. Every growth rate above assumes the frequency holds for minutes.

Only the movement along the stay is counted. The deck’s ±10 mm also has a component of 9.1 mm across the stay, which drives the stay directly at the deck’s frequency. At twice the stay’s own frequency that is far from any of its resonances, and it adds a small forced motion the stability chart does not include.

The sag enters only through Ernst’s modulus. On a long, shallow stay the sag ties the in-plane symmetric modes to the tension at first order, which makes the one to one case a first-order effect after all. For a 120 m stay at this tension that tie is weak; for a stay several hundred metres long it is not.

The damping is viscous and constant. A strand bundle’s own damping depends on amplitude, and the wind supplies damping of its own that can add to the structural damping or, in rain, subtract from it. The threshold moves with every one of those.

And what no chart here shows is where in the stay the damage goes. A taut string has no bending stiffness and no socket. The amplitude a figure prints is a mid-span amplitude; the stress it implies is at the anchorage, and it needs a model of the anchorage to find.

The assumption the charts rest on is the first of these, and it is worth naming because every number depends on it: the deck holds a single frequency long enough for the stay to answer it.

Still open: whether a wandering deck ever holds a stay inside the band long enough #

Every growth drawn here needs the deck to stay within a third of a per cent of twice the stay’s frequency for several minutes, and a deck driven by turbulence or by traffic does not move at one frequency. Its motion is spread over a band, its dominant frequency drifts, and its amplitude comes and goes. Under a tension that swings randomly rather than as a sine, a Floquet multiplier no longer decides stability; what decides it is whether the mean square of the stay’s motion grows, and that depends on how much of the deck’s motion falls inside the narrow band rather than on how large its peak is. Whether real deck motion concentrates enough of itself within a third of a per cent of the right frequency, for long enough, is a question about the statistics of the deck — and it decides whether a quarter of the swing in damping is the requirement, or only a margin on one.