The breeze that uses up a chimney

The wind that brings its own frequency put a light steel chimney, 1.2 m across and swaying at 0.9 Hz, in a steady breeze of 6 m/s and found it swinging 181 mm across the wind: the vortices leaving it had matched its frequency and then locked on to it. Of the three things a designer can do about that — make it heavier or better damped, spoil the vortices with strakes, or accept the motion — the essay said the last is the commonest, and that accepting the motion means checking the fatigue at the welded details, with “the cycle count supplied by the wind statistics rather than by a machine”.

That count was named and not made. This essay makes it, for the same chimney, and asks the two questions a designer holding it would ask next: whether a stiffer chimney is a better one, and how much damping is enough.

A year of wind at the top of a chimney #

A fatigue check needs the number of cycles, and a vortex-shedding chimney oscillates only while the wind sits near its critical speed. So the number of cycles is the number of hours the wind spends there, times the chimney’s own frequency.

How long the wind spends at each speed is a property of the site. Hourly mean speeds at a fixed height are well described by a Weibull distribution, and for most places its shape parameter is close to two, which makes it the Rayleigh distribution:

$$p(U) = \frac{2U}{v_0^2}\,e^{-(U/v_0)^2}$$

with a single scale $v_0$. That is also the distribution the European wind code assumes when it counts vortex cycles, and the Annex’s note on $v_0$ is worth reading twice: it may be taken as a fifth of the characteristic mean wind speed at the height concerned — the fifty-year mean. A site whose fifty-year hourly mean at 32 m is 27.5 m/s gets $v_0 = 5.5$ m/s. The storm the chimney was designed for sets the scale of the breezes that will wear it out, which is a pleasing economy and also a reminder that the everyday wind is being estimated from the extreme one rather than measured.

The picture already contains most of the argument. The wind is commonest at 3.9 m/s and its mean is 4.9, and the chimney’s band starts above both. Even so the wind spends 2,537 hours a year inside it, which at 0.9 Hz is more than eight million cycles. The amplitude, though, is not uniform across the band: it is a narrow hump at the critical speed, and a cycle at the edge of the band at a quarter of the peak amplitude does not do a quarter of the damage, because fatigue damage grows as the cube of the stress range or faster.

Where along the wind the damage is done #

Turning the year into damage needs a stress. The chimney is a cantilever 32 m tall of 400 kg per metre, and its first mode is taken as the square of the height fraction. Swinging with a tip amplitude $y$, the inertia of that mass makes a base moment

$$M = \omega^2 m\, y\, \frac{H^2}{4}$$

and the wall that gives 0.9 Hz at that mass and height is a 7.6 mm tube, whose section modulus turns 181 mm at the tip into 69 N/mm² at the base — a stress range of 138 N/mm² every cycle at the peak of the band. The base is a welded detail, taken at category 71: a tube welded to a base flange or a ring stiffener, the kind of detail whose number is the stress range it survives two million times.

With a stress range at every speed and a number of cycles at every speed, the year’s damage can be summed speed by speed with Miner’s rule, the curve of the detail supplying how many cycles of each range it would survive.

The two curves barely overlap. The hours are spread across the whole range of breezes and peak at 3.9 m/s; the damage is a spike at the critical speed, with half of it done inside a band only 0.7 m/s wide. The lock-in band as drawn in the earlier essay is 2.7 m/s wide, and most of it does not matter. The fatigue of a vortex-shedding chimney is decided by how often the wind sits within a few per cent of one speed, and a wind record that bins its speeds a metre per second apart is barely fine enough to say how often that is.

That narrowness is what EN 1991-1-4’s formula packages. Annex E of EN 1991-1-4 writes the number of cycles as

$$N = 2\,T f_0\, \varepsilon_0 \left(\frac{U_{cr}}{v_0}\right)^2 e^{-(U_{cr}/v_0)^2}$$

with $T$ the life in seconds, $f_0$ the frequency and $\varepsilon_0$ a “bandwidth factor” between 0.1 and 0.3. Written out, that is the Rayleigh density at the critical speed times a band of width $\varepsilon_0 U_{cr}$ around it, times the frequency — every cycle counted at the full amplitude. The amplitude hump here does the same damage as a flat band of width 0.22 of the critical speed at full amplitude, so the Annex’s range of 0.1 to 0.3 brackets it, and the choice between its ends is a factor of three in the life.

The chimney that locks in most #

Now the first design question. A stiffer chimney has a higher frequency, and its critical speed $U_{cr} = f_0 D/\mathrm{St}$ rises in proportion. The intuition is that a higher critical speed is a rarer wind, so a stiffer chimney locks in less often.

The count says something different, because three things change together. The density $p(U_{cr})$ falls once the critical speed is past the commonest wind — that is the intuition. But the band the wind has to sit in widens in proportion to the critical speed, and the chimney does more cycles an hour in proportion to its frequency. Multiplied out, the cycles a year go as

$$U_{cr}^3\, e^{-(U_{cr}/v_0)^2}$$

which rises as the cube of the critical speed before the exponential takes over, and is largest where its derivative vanishes, at $U_{cr} = v_0\sqrt{3/2}$.

The chimney that locks in most often is tuned to a wind 22 per cent stronger than the climate’s scale: 6.7 m/s here, above the commonest wind of 3.9 and the mean of 4.9. The chimney of the earlier essay, at 6.0 m/s, sits just below that peak and collects 4.6 million damaging cycles a year. All three curves — the envelope and the Annex’s formula at both ends of its bandwidth range — peak in the same place, because the peak is a property of the Rayleigh distribution and the linear growth of the band and the frequency, not of the amplitude model.

So the question “is a stiffer chimney better?” has an answer that depends on which side of $1.22\,v_0$ the chimney is on. Below it, stiffening moves the chimney up the rising side of the curve and it collects more cycles; above it, stiffening moves it down the falling side.

Stiffening it by half buys five days #

To turn the count into a life the stress range has to be followed as well, and here the chimney is kind to the argument. Stiffening a steel tube of fixed diameter means thickening its wall. If the chimney’s mass per metre is held fixed — the lining and insulation, platforms and access fittings are much of it — then the frequency rises as the square root of the wall thickness while the section modulus rises with the thickness itself, and the base moment rises with the frequency squared. The two cancel exactly: the stress at the base for a given tip amplitude does not depend on how stiff the chimney is. And the tip amplitude at lock-in depends on the Scruton number, which holds mass and damping and not stiffness. At a fixed mass, every chimney in this family swings 181 mm and puts 138 N/mm² through its base at lock-in, and only the count differs.

In the climate of scale 5.5 m/s the chimney as drawn lasts 22 days. That is the first result, and it is blunt: the chimney the earlier essay described as entirely adequate for its fifty-year wind, which “accepts the motion”, cracks at its base in about three weeks of ordinary weather. Accepting the motion was never available to it at a Scruton number of 11.

The second result is the shape of the curve. Stiffening the chimney by half — a wall 2.25 times as thick, 17 mm instead of 7.6 — moves its critical speed from 6.0 to 9.0 m/s, and the life goes from 22 to 27 days, because the chimney has been moved from one side of the peak to the other and down only a little. The same chimney at half the frequency, with a quarter of the wall, lasts 72 days, three times as long, because its critical speed of 3 m/s is on the rising side and the band it needs is narrow and slow. The worst frequency for this climate is 1.04 Hz, and the chimney as drawn is within a few per cent of it.

The three climates move the trough. A sheltered inland site with a scale of 4.1 m/s is worst for a chimney at 0.77 Hz; an exposed coastal one at 7.2 m/s, for one at 1.34 Hz. The frequency to avoid is not a property of the chimney. It is the site’s wind scale divided by the Strouhal number and multiplied by the diameter, times 1.22. A chimney designed in one place and built in another can move from one side of its own worst case to the other without a line of its drawings changing.

None of the lives on the figure is acceptable, and that is also the point of drawing it. Stiffness moves the life by factors of two or three within a family of structures whose lives are measured in days. It is not the lever that fixes a lock-in fatigue problem; it is the lever that decides how bad the problem is before the real remedy is applied.

Twenty-two days, as a Miner sum #

It is worth seeing the 22 days as the S–N diagram sees it, because that is where the remedy comes from.

The year’s cycles form a nearly flat line out to the peak range and then drop vertically, because every speed in the band has a range below the peak and none has a range above it. The detail’s curve falls from upper left to lower right, steeply below the constant-amplitude limit, and ends at the cut-off, 28.7 N/mm², below which a cycle of constant amplitude does no damage at all in air. The chimney’s spectrum crosses the curve far above the cut-off. Its peak range, 138 N/mm², is survived 274,198 times and the year delivers the equivalent of 4.6 million: sixteen and a half lifetimes’ worth of damage every year. Ranges within a fifth of the peak do three quarters of the damage, which is the same spike as before read along the other axis.

Two things follow from the geometry of the plot. Any remedy that moves the vertical drop left — a smaller amplitude — moves the whole spectrum under the curve at once, because every range in the band is proportional to the peak. And if the drop moves left past the cut-off, the spectrum is entirely below it and the sum is zero. The cycles that do not count found a traffic spectrum losing nine crossings in ten to the cut-off, because the ranges in it were spread across two orders of magnitude and a cut-off sliced through them. A lock-in spectrum is not spread: it is one amplitude and its shoulders, and a cut-off does not slice it but takes all of it or none.

Damping that ends the count #

The remedy that moves the amplitude is mass and damping, as their product in the Scruton number, which is why a tuned mass damper or a hanging-chain damper is the usual retrofit. The amplitude at lock-in is inversely proportional to the Scruton number, so doubling the damping halves every range in the spectrum.

The curve has three parts, and each is the S–N curve’s shape seen through the amplitude. From 0.4 to about 1 per cent damping the peak range is above the constant-amplitude limit and the slope of three governs: doubling the damping from 0.4 to 0.8 per cent makes the life 8.6 times longer, a little more than the cube of two because the band’s shoulders are already falling onto the shallower part of the curve. From 1 per cent to about 1.9 the peak range is between the constant-amplitude limit and the cut-off, the slope of five governs, and the life climbs much faster: 1.1 years at 1.0 per cent, 9 at 1.5, 21 at 1.7, 35 at 1.8, 118 at 1.9. And at 1.92 per cent the largest range, 28.7 N/mm², reaches the cut-off and the life stops being a number.

In amplitude the difference is undramatic. The chimney still locks in at exactly the same breeze, as often as before; it swings 38 mm instead of 181. The mass that helps by being late explains how a small tuned mass supplies that much effective damping to a single mode, and it is the ordinary way to buy it, since a steel chimney’s own damping is fixed by its welds and its lining.

The design lesson is in the last tenth. The damping that gives a life of 35 years, 1.80 per cent, and the damping that gives an unlimited one, 1.92 per cent, differ by a tenth of the latter. A designer aiming for a fifty-year life with the slope-of-five arithmetic ends up within a few per cent of the damping that makes the question disappear, and would do better to aim for the cut-off directly and state it as the requirement: the largest lock-in range at the base shall be below the detail’s cut-off. That is a condition on the Scruton number alone once the geometry is known, and it needs no wind statistics at all — which removes from the calculation the least reliable number in it.

Why the count peaks where it does #

The peak at $1.22\,v_0$ looks like a coincidence of numbers and is not. The count is the product of three factors in the critical speed: the chimney’s frequency, which is proportional to it; the width of the band the wind must sit in, which is proportional to it; and the density of winds at it, which for a Rayleigh climate is itself proportional to it at low speeds and dies exponentially at high ones. Three powers of the critical speed against a Gaussian decay peak where $3 = 2(U/v_0)^2$.

For a Weibull climate of shape $k$ the same argument gives $U_{cr} = v_0\,((k+1)/k)^{1/k}$: 1.22 at the usual $k = 2$, 1.14 at a gustier inland $k = 2.5$, 1.41 at a $k = 1.5$ site dominated by a few strong winds. So the worst chimney is always somewhat above the scale, never at the commonest wind, and the offset is largest where the wind is least regular. The mode of the distribution, where the wind spends the most hours, is the wrong place to look, because the hours there are hours at a low speed, and a chimney that locks in at a low speed is a slow chimney that collects few cycles per hour in a narrow band.

This is the same arithmetic that makes the design wind a poor guide to fatigue in general. The fifty-year storm sets the strength; the Rayleigh scale sets the fatigue; and the wind standard joins them by a factor of five that is an assumption about the shape of the wind’s distribution at a site, not a measurement of it.

The base, by hand #

The whole of the headline number can be checked on the back of the drawing. The critical speed is $f_0 D/\mathrm{St} = 0.9 \times 1.2/0.18 = 6.0$ m/s. The Annex’s count with a bandwidth factor of 0.22 is

$$N = 2 \times 3.16\times10^7 \times 0.9 \times 0.22 \times 1.19 \times e^{-1.19} = 4.6 \times 10^6$$

a year. The range at the base is twice $\omega^2 m y H^2/4$ over the section modulus: $(5.65)^2 \times 400 \times 0.181 \times 32^2/4 = 593$ kN·m on a 7.6 mm tube of 1.2 m diameter, whose modulus is $\pi \times 0.6^2 \times 0.0076 = 0.0086\ \mathrm{m^3}$, giving 69 N/mm² and a range of 138. Category 71 survives $2\times10^6 \times (71/138)^3 = 274{,}000$ cycles of it. Four point six million over 274 thousand is 16.8 lifetimes a year, and a year over 16.8 is 22 days.

The damping that ends the count needs only the ratio of the peak range to the cut-off: $138/28.7 = 4.8$, so the Scruton number has to rise 4.8 times, from 11.2 to 53.5, and at a fixed mass the damping from 0.40 to 1.92 per cent. None of that uses the wind.

What the model assumes about the wind #

The wind is steady for an hour at a time. Lock-in needs the wind to hold inside the band for the forty-odd cycles it takes the amplitude to build up, and the count here gives every hour in the band the full envelope amplitude. Turbulent wind wanders in and out of the band within the hour, and the amplitude at lock-in in turbulent flow is lower than in smooth flow; both make the real count smaller. The Annex’s bandwidth factor is partly an allowance for this, which is why its range is wide.

The wind blows from every direction equally. A circular chimney does not care which way the wind comes from, and the Rayleigh distribution here is the omnidirectional one. A structure that is not axisymmetric — a lattice mast, a rectangular stack, a pair of chimneys that shelter each other — has a different critical speed and a different band in each direction, and needs the directional record.

The amplitude is EN 1991-1-4’s lock-in amplitude, a correlation-length and mode-shape model with empirical coefficients, applied at a Scruton number where it is least certain. At a Scruton number of 11 the amplitude could be somewhat lower in reality. The counting argument — where the count peaks, and that damping ends it rather than slowing it — does not depend on the amplitude model; the 22 days does. And the whole count presumes a bounded amplitude: a section that gallops instead, feeding itself from a steady wind, has no band to sit in and no amplitude to count at, and fails the first time the wind passes its threshold.

What the pictures cannot show #

That a cut-off is a fact about constant-amplitude cycles in air. The chimney is also buffeted along the wind by gusts in every storm, at stress ranges much larger than 28.7 N/mm² on a few occasions a year, and a crack started by those does not stop growing because the lock-in cycles are small; once a crack exists, cycles under the cut-off count again. The cut-off belongs to the water makes the same point for corrosion, which removes the cut-off entirely, and a steel chimney carrying flue gas at its base is not in dry air. So “unlimited” at 1.92 per cent is unlimited only for a base that is protected and not otherwise loaded; the honest requirement is the cut-off with margin, and a slope-of-five check of the remainder.

Nor can they show the weld. A category is a detail as drawn and inspected. A base weld made on site, in the wind, on a flange that did not quite sit flat, is a lower category, and moving from 71 to 50 shortens every life on these figures by a factor of nearly three and moves the damping that ends the count up by 40 per cent. The load that never came near failing anything is the general form of that sentence: the detail, not the steel, decides.

The same count on a cable, a lamp post and a stay #

The argument is not about chimneys. Anything slender and round in a wind locks in, and the count applies to each with its own critical speed. A lighting column 0.3 m across at 2.5 Hz locks in at about 4 m/s, close to the commonest wind of most sites, and collects cycles at nearly three times the chimney’s rate for the same hours; lamp posts crack at their base doors and base plates for exactly this reason, and the fix is the same damping-to-the-cut-off rule. The stays of a cable-stayed bridge have many modes and a critical speed for each, so the stay shaken along its own length is a spectrum of chimneys, each tuned to a different wind, and the one closest to $1.22\,v_0$ is the one to damp first.

And the argument inverts for a structure that is meant to be stiff for other reasons. A footbridge or a floor is stiffened to lift its frequency out of the range of footfall, which works because footfall has a fixed frequency; lifting a slender structure’s frequency out of a wind’s reach works only if it is lifted past the whole of the wind climate, and the peak at $1.22\,v_0$ says how far that is. The period nobody chose is the reminder that the frequency is usually a by-product of the design rather than a decision, and here it is a by-product with a price.

Still open: the chimney that is not alone #

Every number here is for one cylinder in free wind. Two chimneys side by side, or a chimney downwind of a building, see a wake instead of a wind: the vortices shed by the upstream body arrive at a frequency set by that body, and the downstream one can be excited at a speed its own critical speed does not predict, over a band much wider than its own. Ferrybridge was a failure of exactly that kind for cooling towers. Whether a group of identical chimneys collects more lock-in cycles than the same chimneys apart — because each widens the band the others respond in — and whether the worst spacing is the one that also makes the count largest in a Rayleigh climate, is the question that turns this one-cylinder count into a site plan.