The force read off a frequency

A cable-stayed bridge has fifty or a hundred stays and every one of them has to be at the right tension, because the deck’s profile and the tower’s verticality depend on the whole set being right together. There is no way to measure that tension directly. The stay is anchored inside the deck at one end and inside the tower at the other, both anchorages are grouted and inaccessible, and nothing along its length carries a gauge.

What the stay does have is a natural frequency, and a natural frequency is a force with the units taken off it.

The relation, and its inverse #

A taut string with tension $T$ and mass $\mu$ per metre has

$$f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}.$$

Everything about the measurement is in reading that backwards:

$$T = 4\mu L^2 f_1^2.$$

$\mu$ is known from the strand schedule, $L$ from the drawing, and $f_1$ from an accelerometer taped anywhere on the cable for thirty seconds while the wind or the traffic excites it. The stay drawn returns 3.50 MN from 1.006 Hz, and the measurement is repeatable to about a per cent.

The relation is worth reading for what it does not contain. There is no modulus in it, no area, no material property of any kind — only a mass, a length and a time. So the measurement is indifferent to what the cable is made of, to how many strands it has, to whether they are galvanised, and to how much of its area has corroded away. It measures the force, not the stress, and it measures it through a quantity that has nothing structural in it at all.

Nothing else in structural engineering measures a force this well without touching it. A strain gauge measures a strain and needs a modulus and an area; a load cell has to be in the load path from the beginning; a jack measures a force it is applying rather than one that is there. A frequency measurement needs a clock.

The two corrections, and which one matters where #

The string is an idealisation in two directions, and the two spoil it in opposite regimes.

Bending stiffness. A stay is not a string; it is a bundle of strands inside a sheath, and it has some flexural rigidity. The parameter is $\xi = L\sqrt{T/EI}$, and every mode is raised by roughly $1 + 2/\xi$. For the stay drawn $\xi = 410$ and the correction is half a per cent on the frequency, a per cent on the tension. It matters most for short stiff cables — a short stay near the tower, a hanger on an arch — where $\xi$ can fall to fifty and the correction reaches ten per cent.

Sag. A cable hangs, and the sag stiffens the low symmetric modes because deflecting them requires the cable to stretch. Irvine’s parameter is

$$\lambda^2 = \left(\frac{wL}{H}\right)^2\frac{LEA}{H L_e},$$

which for the stay here is 0.13 against a crossover at $4\pi^2 = 39.5$. The stay is deep in the taut-string regime and the sag correction is below a thousandth. For a suspension bridge’s main cable, or a long shallow stay at low tension during erection, $\lambda^2$ is large, the first symmetric mode rises past the first antisymmetric one, and the two swap order — so the frequency an accelerometer picks up may not be the mode the formula assumes.

Why the spacing matters more than the value #

The frequencies are useful for measurement and they are a problem for everything else, and the reason is the spacing rather than any particular value.

A beam’s modes go as $n^2$: 1, 4, 9, 16 times the first. So a beam has a first mode and then a large gap, and a designer who puts the first mode above the pacing band is finished — nothing else is near.

A cable’s go as $n$: 1, 2, 3, 4 times the first, for ever. There is no gap anywhere. Whatever frequency arrives, there is a mode within half a spacing of it, and for the stay drawn that is 0.5 Hz.

So a cable cannot be detuned. Every mitigation for a lively cable is a dissipation measure rather than a frequency measure: a damper at the anchorage, a cross-tie between adjacent stays, a helical fillet on the sheath to spoil the rain-wind mechanism. Damping is the only thing that stops it, and on a cable that is not a figure of speech but the consequence of an arithmetic progression.

Which free body produced the number #

An element of the cable, of length $ds$, displaced sideways by $y$. The tension enters one end and leaves the other at a slightly different angle, so its resultant is transverse and equal to $T\,\partial^2y/\partial x^2$ per unit length — the same deviation force a curved tendon puts on the concrete round it, arising here from the curvature of the mode shape rather than from a designed profile.

Setting that against the element’s inertia gives the wave equation

$$T\frac{\partial^2 y}{\partial x^2} = \mu\frac{\partial^2 y}{\partial t^2},$$

whose solutions are sinusoids travelling at $c = \sqrt{T/\mu}$. The modes are the standing waves that fit between the anchorages, and the arithmetic progression is nothing more than the fact that a length holds one, two, three half-wavelengths.

The restoring force is geometric, not material. The cable has no bending stiffness worth mentioning and no elastic restoring force at all in the transverse direction; what pulls it back is the tension acting through the curvature it has acquired. That is the same geometric stiffness that a compression member loses, with the sign reversed — a tension gains stiffness where a compression sheds it, and the whole of cable dynamics is that one sign.

The mode a measurement actually catches #

An accelerometer taped to a cable and left for thirty seconds records whatever the wind happened to excite, which is usually several modes at once. Reading a tension off it means picking one and knowing which.

The safe way is to take the spacing rather than any single peak. Because the modes are an arithmetic progression, the difference between consecutive peaks is $f_1$ whether or not the first one was captured — so a spectrum showing peaks at 3.02, 4.03, 5.03 and 6.04 Hz gives $f_1 = 1.006$ without the fundamental ever having appeared.

That is the practical reason the progression matters, and it is a rare case of a structural property being convenient. A beam’s spectrum cannot be read this way at all: peaks at 4, 9 and 16 times a fundamental have differences of 5 and 7 times it, and recovering $f_1$ needs the mode numbers to be identified first.

The parametric case #

There is one excitation mechanism a cable has that no other member does, and it follows from the geometry rather than from any force.

When the deck a stay is anchored to moves, the stay’s length changes slightly — the anchorage points get closer together and further apart — so its tension oscillates. A member whose stiffness oscillates is a parametric system, and a parametric system is driven most strongly at twice its natural frequency rather than at its natural frequency.

So a deck oscillating at 2 Hz excites a stay whose natural frequency is 1 Hz, through a mechanism that has nothing to do with any transverse force on the cable. The response can grow without an obvious cause, and the frequency ratio that produces it — 2:1 — is one nobody looks for when a structure is checked mode by mode.

What the measurement is for #

Reading one stay’s tension is not the point. Reading fifty of them and comparing is.

A cable-stayed deck’s profile is decided by the whole set of stay forces acting together, and the set is adjusted during construction and again at completion — shims at the anchorages, a turn on a threaded socket, sometimes a re-stressing operation years later. What the engineer needs is not an absolute force but a comparison between the measured set and the design set, stay by stay.

That changes which errors matter. A systematic error in the mass per metre or in the end-fixity model shifts every reading by the same fraction and cancels out of the comparison; a random error from a bad accelerometer placement does not. So the measurement’s repeatability is worth more than its accuracy, and the practice is to measure every stay the same way with the same equipment on the same afternoon.

There is a second use, and it is the one that outlives the construction. A stay measured every few years gives a trend, and a trend is a corrosion and relaxation monitor: a stay losing force faster than the others is a stay with something wrong inside a duct nobody can open. The model of a structure is worth what its measurements say it is worth, and here the measurement is available for the price of a technician with a laptop.

Where the model stops #

The ends are pinned. A stay is grouted into a steel guide pipe for a metre or two at each end, so the effective length is shorter than the free length and the boundary is somewhere between pinned and fixed. This is the single largest uncertainty in the measurement, it moves the answer by a per cent or two, and it is why the practice is to measure several modes and fit rather than to use $f_1$ alone.

The mass per metre is the strand’s. A stay in service carries its sheath, its grout, its damper, occasional ice and — in the rain-wind mechanism that plagues them — a rivulet of water running down it. Each of those is a per cent or so on $\mu$ and therefore on the tension.

The cable is straight. For the measurement it is close enough; for the sag calculation the geometry matters, and an inclined cable’s transverse weight component is $\mu g\cos\theta$, so a nearly-vertical stay and a nearly-horizontal one have different sags at the same tension.

Damping is not in any of this. All the frequencies are undamped ones, which is right to a fraction of a per cent at the one per cent damping a stay has — and the reason the measurement works is that the damping is so low, because a heavily damped member has no clean peak in its spectrum to read.

The stay is treated as a single member. In service it is fitted with a damper near the deck anchorage, which shortens the effective length for the higher modes and not for the first, so the progression stops being exactly arithmetic — and the departure from it is itself a measurement of whether the damper is connected.

And no drawing here shows the cable moving. Every figure is a spectrum or a calibration curve, and the object they are about is a 120 m member tracing an amplitude of a few hundred millimetres at its middle in a mode nobody chose, driven by a wind nobody measured.

The same relation, everywhere else #

The string equation is one of the two or three most reused results in physics, and it is worth noticing how much of this collection it already underlies.

A stressed ribbon is a deck that is a cable, so its own modes are this arithmetic progression and its liveliness has the same cause. A hanger on an arch, a guy on a mast, a tie-down anchor on a tank: all of them are strings, all of them can be measured this way, and all of them have no gaps in their spectra.

A prestressing tendon inside a duct is a string with friction along it, and the frequency method is the standard way of checking whether one has been grouted properly — a debonded length rings at a frequency corresponding to its own free length rather than to the member’s.

And the reverse case is instructive. A member that returns a frequency very much lower than its geometry suggests has lost tension, and a member that returns one very much higher has gained restraint it was not supposed to have. Both are diagnoses available from a measurement that touches nothing.

The ladder from here #

Later rungs on this anchor: the multi-mode fit, which extracts tension, effective length and bending stiffness together from four or five measured frequencies and is what commissioning actually uses. Rain-wind induced vibration, which is a cable-specific instability caused by a water rivulet changing the section’s aerodynamics, and the helical fillets that defeat it. Cross-ties between stays, which couple several cables into one system with a different and sparser spectrum — the one intervention that does change the frequencies. Dampers at the anchorage, and the awkward result that a damper at 2 per cent of the length can add at most about 1 per cent of critical damping however good it is. Parametric excitation resolved properly, with the deck motion as a time-varying tension and a Mathieu equation underneath. And the same measurement applied elsewhere: a tie-down anchor, a prestressing tendon in a duct, a guy on a mast, and the strings of a piano — all of them forces read off a frequency, by the same relation, for the same reason.