The nut turned after the storm

The base that lets go and closes again put a preloaded column base through a storm: four cycles of a moment that reversed between ±170 kN·m, enough to lift the plate off its bed each way and to yield the bolts on the tension side. The bolts stretched, kept 83 per cent of their preload, and then stopped stretching. The second cycle traced the first cycle’s return and the third and fourth lay on top of it. The base settled into a looser joint and stayed there.

A base that has lost a sixth of its preload is a base somebody will want to repair, and the repair is obvious and cheap: put a spanner on the nuts and turn them until the bolts carry 150 kN again. The base is then, by every measurement an inspector can make, the base it was before the storm. This essay asks what the next storm does to it, and the one after that.

The same base, and a spanner #

The base is the one the earlier essays built: a 500 × 400 × 20 mm plate on a grout bed that pushes back but cannot pull, held down by two M24 grade 8.8 bolts in each of two rows 60 mm in from its ends, each preloaded to 150 kN. Each bolt row is a spring of the bolts in series with the plate’s bending at the row, elastic until the bolts reach their yield of 226 kN, then stretching permanently with a slight hardening. The bed is elastic and cannot pull.

A storm is four cycles of a moment from zero to +170 kN·m, back through zero to −170 and back to zero. Between storms, the nuts are either left alone or retightened. Retightening is modelled as what a spanner does: each nut is turned until its row carries the chosen preload again with no moment on the base. The turn takes up the bolt’s permanent stretch, so the joint is as tight as it was new — but the stretch is still in the steel. The next storm starts from the same joint and adds its own stretch to the bolt’s.

The hero figure is the result. Bolts left alone after the first storm stretch 0.12 mm in it and nothing in the nine after. Bolts retightened to their original 150 kN after every storm stretch 0.12 mm in every storm: the first storm’s stretch, nine more times, 1.25 mm after ten. The two strategies are identical in the first storm and part company in the second, and they never meet again.

Why the first storm stretches the bolt and the second does not #

The difference between the two lines is the difference between shakedown and ratcheting, and it is visible in a single storm’s loop.

The base left alone has been loosened by its first storm. Its bolts are 0.12 mm longer than they were, so at zero moment they clamp less — 124 kN rather than 150 — and the base lifts off its bed sooner, at 46.0 kN·m rather than 57. But a looser base reaches the storm’s peak moment with its bolts carrying less force, because less of their capacity was spent on preload, and the storm that stretched them once no longer reaches their yield. Every later storm goes round an elastic loop. The first storm made the base as loose as it needed to be to survive the storm. That is what the structure that settles down called shakedown: a residual state — here, the stretch in the bolts — in which every later cycle is elastic.

The retightened base has been given back exactly what shakedown took away. It is as stiff as it was new, lifting off its bed at 57 kN·m, and it reaches the storm’s peak with its bolts carrying exactly what they carried in the first storm. So they yield exactly as they did in the first storm, and stretch by the same amount. The storm has not changed and the base has been put back to where it was before the storm; the storm therefore does the same thing again.

The earlier essay on shakedown found that ratcheting needs two loads of different kinds — one that cycles and one that stays, so that each cycle’s yielding is pushed the same way by the load that stays. Here the second load is the spanner. The storm cycles; the retightening is a load applied between storms that always pushes the bolts back to the preload they cannot keep. Neither alone ratchets — the same pairing the beam the wind walks without pushing found in a portal frame, where a gravity load that stays and a wind that reverses walk a beam that either would leave alone. The storm alone shakes down after one storm. The retightening alone is just a nut turned once. Together, every storm ends with the bolts a little longer, and every retightening starts the next storm from a state in which they will be longer still.

The preload a storm can live with #

The first storm’s loss is not a defect to repair. It is the storm telling the base how much preload it can keep.

For a storm to ±170 kN·m, a bolt tightened to anything up to 128 kN comes out of the storm with all of its preload: the dotted line, on which nothing is lost. A bolt tightened to more than 128 kN comes out with less than 128, and the harder it was tightened the less it keeps — 124 kN from 150, 118 from 175, 112 from 200. A bolt tightened past the storm’s limit spends its extra preload on yielding sooner, stretches further, and loses more than the extra.

So for each storm there is a largest preload the bolts can keep through it, and it is a little above what the first storm leaves in bolts tightened to the usual value. For a storm to ±165 kN·m it is 146 kN, very nearly the 150 the bolts were given; for one to ±175 kN·m, 101 kN. The steepness of that change is the steepness the earlier essay found near the base’s capacity, which is 183 kN·m: a storm to 170 kN·m, 93 per cent of the capacity, left bolts tightened to 150 kN with 83 per cent of it, and one to 180 kN·m, 98 per cent, left them with 30.

Tightening harder does not keep more preload through a storm. It keeps less. The bolts’ preload and their reserve against the storm come out of the same yield strength, and what is spent on one is not available for the other. The bolt that was already stretched found the same arithmetic for a single bolt in a joint: a preload uses up part of the bolt’s capacity before any load arrives.

What every storm costs #

Retightened to the original preload, the bolts stretch in every storm. How much they stretch depends on how far above the storm’s limit the retightening goes.

The stretch each storm adds is zero up to the storm’s limit and then grows almost linearly with the preload above it. To ±170 kN·m, retightening to 150 kN — the original preload, 22 kN above the limit — costs 0.12 mm a storm; to 200 kN, 72 kN above, costs 0.43 mm. Each kilonewton of preload above the limit costs between five and six thousandths of a millimetre in every storm. The rate is set by the bolt row’s stiffness: everything a bolt carries above what the storm leaves has to be shed by the bolt stretching, and a bolt tightened past the limit ends the storm a little below it, so it sheds a little more than its excess.

The storm’s size moves the limit, not the rate. To ±165 kN·m the limit is near 150 kN and retightening to the original preload costs almost nothing, 0.02 mm a storm; to ±175 kN·m the limit is near 105 kN and the same retightening costs 0.28 mm. A base that sees storms near its capacity cannot be retightened to its original preload without stretching every time.

How many storms a bolt has in it #

A stretch that accumulates is a stretch that will one day be too much.

The measure of what a bolt can take is its elongation at fracture, and for a grade 8.8 bolt the standard guarantees 12 per cent over a gauge of five diameters: 14.4 mm for an M24. Retightened to 150 kN after every storm to ±170 kN·m, a row reaches that after 116 storms; after every storm to ±175 kN·m, 52. Retightened to 200 kN, 33 storms of ±170 kN·m. A base on a building that sees a design storm every year, retightened faithfully after each, uses up its bolts’ guaranteed elongation in a working life.

That number is generous to the bolt in one way and harsh in another. The stretch is spread here over the bolt’s whole free length, 256 mm, and in a real holding-down bolt it concentrates in the threads, where the section is smallest; a thread that has taken 14 mm of stretch has necked long before. On the other side, a real storm reaches its peak moment once or twice, not four times, and the next storm is rarely as large as the one before. The count is a scale, not a prediction: tens to hundreds of storms, rather than none or unlimited.

What is not in doubt is the difference between the two strategies. Bolts left alone after the first storm, or retightened only to what it left, have the count at the end of a missing line: no number of storms of the same size ever stretches them again.

A rigid range bought with stretch #

Retightening is not foolish. It buys something real, and the last figure prices it.

The rigid range is what a preload is for. The base that is rigid until the bed lets go found that a preloaded base is several times stiffer than an unpreloaded one up to the moment at which its plate lifts off the bed, and that moment is proportional to the preload. Left alone after the first storm, the base’s rigid range is 46.0 kN·m. Retightened to 150 kN after each storm, it is 56.6 kN·m again, a quarter more, and the base is as stiff under the everyday wind as it was the day it was built — at the price of 0.12 mm of stretch in every storm. Retightened to 200 kN, 75.5 kN·m at 0.43 mm a storm.

Whether that trade is worth making depends on what the rigid range is for. If the column’s design assumed a fixed base for its sway stiffness — the fixed base the bolts decide — then the rigid range is load-bearing, and losing a quarter of it changes the frame’s drift; the retightening is then maintaining the structure’s assumed behaviour, and the stretch is its cost. If the base was designed as nominally pinned, as the pinned base that is not pinned found most bases are, then the lost rigid range costs the frame nothing that was counted, and the retightening buys stiffness nobody relies on with ductility the bolt cannot get back.

Between the two ends of the line sits the choice the figure makes plain. Retightened to 124 kN — what the first storm left — the base has 47.0 kN·m of rigid range and adds no stretch in any later storm of the same size; it is the shaken-down base with its nuts snugged, and it costs nothing. Every kilonewton above that buys about 0.37 kN·m of rigid range and costs five to six thousandths of a millimetre of stretch in every storm after. The line is straight because both the rigid range and the stretch are proportional to the preload above the storm’s limit, so there is no point on it that is a better bargain than any other; there is only a decision about how much stiffness the frame needs and how many storms the bolts must survive.

None of this is fatigue. The load that never came near failing anything is about cracks growing under stress ranges well inside the elastic range, millions of cycles at a time; a ratchet is plastic stretch accumulating a storm at a time, tens or hundreds of events, and a bolt can run out of either without the other noticing.

What an inspection should record #

The figures suggest a different instruction from “retighten to the specified preload”. A base found after a storm with less than its specified preload has been told, by the storm, what preload it can keep through storms of that size. Retightening to what was found — or to a little below it — restores the joint to the shaken-down state the storm chose, costs no stretch in a storm of the same size, and keeps whatever rigid range that preload gives. Retightening to the original specification restores the rigid range the storm took away and arms the next storm to take it again.

An inspector cannot see the stretch. The torque that goes into the thread found that a torque reading measures friction more than preload, and a bolt that has been stretched and retightened reads the same torque as a new one. What can be recorded is the preload found before retightening, which a load-indicating washer or an ultrasonic gauge will give, and the number of times each bolt has been retightened. A bolt retightened many times to a preload it keeps losing is a bolt whose ratchet is running.

The stretch, by hand #

The stretch each storm adds can be estimated from the preload a storm takes. A row of two M24 bolts, each 353 mm² of thread over a free length of 256 mm, in series with the plate’s bending at the row, has a stiffness of 444 kN a millimetre, so every kilonewton a bolt loses is 2/444 = 0.0045 mm of its row’s stretch. Retightened to 150 kN, a bolt comes out of each storm with 124, having lost 26 kN, and 26 × 0.0045 = 0.12 mm, the figure’s value. Retightened to 200 kN it comes out with 112, having lost 88, and 88 × 0.0045 = 0.40 mm, against the figure’s 0.43: the extra is the slight hardening, which lets the yielded row carry a little more at the storm’s peak and so asks it to stretch a little further for the same loss.

The count follows: 14.4/0.12 = 116 storms at 150 kN, and 14.4/0.43 = 33 at 200 kN.

Where the spring model stops #

The bed is elastic and does not change. Grout under a plate that has lifted and closed many times crushes locally at the plate’s edges and loses some of its stiffness, which the model holds constant. A crushing bed would take some preload with it each storm, retightened or not.

The bolts harden only slightly and forget it. A real bolt stretched plastically hardens, and a bolt retightened after hardening yields a little later in the next storm; the model’s slight hardening resets with each retightening, so its ratchet is a little faster than a hardening bolt’s would be. Hardening slows a ratchet; it does not stop one, because the retightening puts back the preload the hardening would have protected.

Every storm is the same storm. Real storms vary, and a ratchet is driven by the large ones; a base retightened after small storms, which did not stretch it, loses nothing by being retightened.

The nut, the anchorage and the column above #

They cannot show the nut, whose threads take the same plastic stretch as the bolt’s and which can strip before the bolt fractures. They cannot show the anchorage’s bond in the concrete below, which is cycled with the bolt every storm. And they cannot show the column above, which sways a little more in every storm the base is left loose and a little less in every storm the base is kept tight — the frame-level half of the trade, which is where the decision to retighten is really made.

Still open: the bolt that is retightened with a washer that gives #

A base that must keep its rigid range without ratcheting needs a preload that a storm can take away and give back. A disc-spring washer under each nut — a stack of conical springs much softer than the bolt — makes the row a soft spring in series with a stiff one: a storm that stretches the bolt slackens the springs by the same amount, and a soft spring loses little force for a given stretch. Whether a row with a spring washer can be preloaded above the storm’s limit without yielding, because the washer, not the bolt, absorbs the movement — and whether the softer row then lifts the base off its bed sooner, giving back the rigid range the preload was for — is the question a joint made of springs in series leaves for a base that sees storms.