The arm that is reset by hand

The arm that slips on purpose gave a late-connected outrigger a joint that lets it move under the slow force and holds under the fast one. A friction joint that slips at 2 MN held 2 MN of creep force for ever and left the arm no margin against the wind on its worse side; a viscous damper with a time constant of ten days let in 2 per cent of the creep and held 99 per cent of a three-hour storm. The damper solved the problem, and it is a machine: a hydraulic device carrying several meganewtons, expected to work for fifty years in a building whose structural stiffness depends on it.

That essay ended on the middle path between connecting the arm once and letting it slip for ever. Connect it with shims, and every few years open the joint, change the shims to take out the differential since the last time, and close it again. Whether a handful of adjustments keeps the arm’s creep force as low as a damper does, how much of it creep takes back between them, and whether a building can be relied on to carry out maintenance its own stiffness depends on, was the question.

The same building, and a spanner every few years #

The building is the earlier essays’: 200 m of concrete, its arm making the columns work against the core’s bending, built a storey every week and fitted out over six months, its perimeter columns working at twice the core’s stress, with an outrigger three quarters of the way up whose two arms join the core to the columns with a rotational stiffness of 10⁸ kN·m a radian. Every storey of the columns and the core is cast on its own day, loaded as the floors above it go up, and creeps and shrinks by Eurocode 2’s rules for its own age and size. The columns shorten more than the core, and the arm, connected on day 385, resists the difference: it is a spring in series with concrete that itself creeps under the force the arm puts into it. Left alone, the arm ends its first fifty years carrying 3.48 MN of creep force. The design wind puts 5.80 MN into the same arm.

An adjustment is modelled as what a maintenance crew does. On the chosen day the joint is opened: the arm lets go of every kilonewton it holds, and the concrete that was carrying that force is unloaded by the same amount. The shims are changed to fill exactly the gap the joint now shows, and the joint is closed. From then on the arm collects the differential again from nothing. The concrete, unloaded, recovers part of the creep the arm’s force had caused — creep in reverse — and that recovery is counted too.

Adjusted at one, three and ten years, the arm holds 1.40 MN just before its first visit, 1.14 before its second, 1.21 before its third, and creeps back to 0.91 MN by the fiftieth year. Its worst force is 40 per cent of the 3.48 it reaches left alone.

That is the answer to the first half of the question, and it is not the damper’s answer. Shims take out the creep that has happened; they cannot stop the creep that is happening. Between visits the arm is a rigid connection again, collecting the differential exactly as it did before, and the force it reaches before each visit is the creep of that interval. A damper lets the creep through continuously and holds the arm near nothing at all times; a shimmed joint lets it through in steps and holds whatever the last interval brought.

What each visit removes #

The shims tell the same story in millimetres. The first visit, a year after connection, removes 6.3 mm; the second, two years later, 5.1; the third, seven years after that, 5.5. They are nearly equal, and that is not a coincidence of these dates. Creep in concrete grows roughly as the logarithm of time once the first months are past — the deflection that arrives three years late is the same curve in a beam — so equal ratios of time bring roughly equal increments of shortening. One year, three years, ten years are three equal steps on that clock, and each brings about the same differential.

Together the three visits remove 16.9 mm. The rest of the 23.4 mm differential the arm’s level sees after connection is the creep still to come after the tenth year, which the arm then carries for the remaining forty, and the elastic part that the arm and its concrete took up rather than the shims.

How many visits #

One visit at one year takes the arm’s final force from 3.48 to 2.58 MN — and the final force is also its largest, because everything after the visit accumulates without limit. A second visit, at twenty years, brings the final force down to 0.81, but the force just before it has reached 2.34. A third, spacing the three at one, about four and a half and twenty years, takes the worst to 1.46.

From four visits on, the worst force stops falling, at 1.40 MN, and the figure shows why. Every schedule here begins with a visit one year after connection, and the creep of that first year is 1.40 MN on its own. More visits later cannot reach back into it. The worst force a shimmed arm ever holds is set by its longest interval in creep’s own time, and the first year after connection is the longest interval of all: the concrete is youngest, the finishes have just arrived, and the shortening is fastest.

The first year is the longest #

The first year is the expensive one for three reasons that arrive together. The concrete carrying the outrigger level’s load is youngest, so it creeps fastest under what it already carries; the finishes arrive over the six months after topping out, loading every storey below the arm a quarter more; and the columns’ thicker sections, which creep later than the core’s thin walls, are only beginning their shortening. The differential the arm sees in its first year after connection is the largest it will ever see in a year.

Two ways of shortening that interval come out of the same model. A fourth visit three months after connection, before the one at a year, cuts the worst force from 1.40 to 1.20 MN. And connecting the arm later helps exactly as the arm connected late found it does: connected a year after topping out and adjusted two and nine years later, the arm never holds more than 1.14 MN; connected six months after topping out and adjusted at one, three and ten years, 1.06. Every one of these is a way of making the first interval shorter in creep’s own time, which is the only thing that lowers the worst force. A building whose outrigger is not needed for the wind until its occupants arrive can afford to leave the arm open through the fit-out and close it into a much slower differential.

Visits spaced like the creep #

The spacing matters more than the count. Three visits at one, three and ten years hold the arm to 1.40 MN. The same three spread evenly, at five, ten and fifteen years, let it reach 2.48 before the first visit, because five years of fresh creep is most of the creep there is. Three visits crowded into the first two years catch the early creep well but leave the arm to collect the next forty-eight years’ unchecked: 2.01 MN at the end. Three visits late, at ten, twenty and thirty years, leave the worst of all, 2.93 before the first.

The visits should be spaced like the creep, not like a calendar — close together early, wide apart later, in roughly equal ratios of time. A maintenance schedule written as “every five years” is the schedule that suits a load that grows linearly, and creep does not.

A visit that is missed #

The second half of the earlier essay’s question was whether a building can be relied on to do the maintenance. The figure prices the failure. Miss the first visit and the arm reaches 2.11 MN before the second; miss the second, 2.01 before the third; miss the third and the arm collects the whole of the creep after the third year and ends at 1.80. In each case a missed visit costs exactly the creep it was there to take out, added to the next interval’s. Nothing worse happens: the arm does not ratchet, and the next visit, when it comes, takes out the accumulated creep as easily as one interval’s.

That is a gentler failure than it might have been, and it is worth seeing why. The nut turned after the storm found that retightening a column base’s bolts after every storm made them ratchet, because the storm was a load the bolts yielded under and the retightening put back the preload that made them yield again. Creep is not like that. The arm does not yield; it holds the differential elastically, and opening the joint gives back exactly what the creep gave it. Re-shimming an outrigger is resetting a spring, not winding a ratchet.

The arm when the storm arrives #

The creep force matters because it is already in the arm when the wind arrives. On the arm’s worse side the wind adds to it, and the arm has to be designed for the sum. Connected and left, that sum is 9.28 MN, 60 per cent more than the wind alone. With the friction joint of the earlier essay slipping at 2 MN, 7.80. Adjusted at one, three and ten years, 7.21 at the worst moment just before a visit, and 6.72 for the last forty years. The shimmed arm needs a quarter more capacity than the wind alone asks for; the arm left alone needs three fifths more.

The damper needs about 2 per cent. That is what a machine buys, and it is a great deal. What the shims buy is most of the way there with nothing to fail but the schedule: no fluid to leak, no seals to age, no device whose stiffness under a storm has to be trusted. And the arm can be sized for the sum at its worst moment, with the visits’ dates written on the drawings, so that a missed visit is a known loss rather than an unknown one.

What the designer writes down #

A shimmed outrigger moves part of the structural design into a maintenance manual, and the figures say what the manual has to contain. The arm is sized for the wind plus the largest creep force the schedule allows — 7.21 MN on the worse side for visits at one, three and ten years — and the schedule itself is part of the design, as binding as a bolt size. The manual gives the dates, the expected thickness of shim at each, 6.3, 5.1 and 5.5 mm here, and the force at which a visit becomes urgent rather than routine. A shim thickness far from the expected one at any visit is a measurement that the building is creeping differently from its design, and is worth an engineer’s attention before the next storm rather than after it.

The manual also has to say what happens if the schedule lapses, and the figures make that a short paragraph: each missed visit adds its interval’s creep to the next, and the arm must have been sized for that or the next visit must come sooner. A design that sizes the arm for the worst missed visit — 2.11 MN with the first one missed — buys itself one lapse of grace for about 0.7 MN of extra capacity. That is the price of not trusting the owner, and for a building that will change hands several times in fifty years it may be worth paying.

The first interval, by hand #

The first visit’s 1.40 MN can be checked against the differential. In the first year after connection the arm’s level shortens against the core by 6.3 mm net of what the springs give back, the thickness of the first shim. The arm’s springs on one side have a stiffness $k = 2K/\ell^2 = 2 \times 10^8/30^2 = 2.2 \times 10^5$ kN per metre, so 6.3 mm would be 1.40 MN if the springs were elastic — and they nearly are over a year, because the concrete carrying the arm’s force has only begun to creep under it. The creep of that concrete is what lets the left-alone arm end at 3.48 MN after fifty years rather than at the 5.2 MN its 23.4 mm of differential would give an elastic spring.

A joint opened and closed exactly #

The joint is opened in a moment and closed exactly. A real adjustment unloads the arm with jacks, measures the gap, fits shims to the nearest millimetre or half-millimetre, and recloses with a preload; the force left after a visit is a fraction of what the shims could not quite match, not nothing. With shims in half-millimetre steps, that residual is up to a twelfth of the first visit’s force.

The building is unloaded while it is adjusted. An arm opened for adjustment carries nothing, and the core alone carries the building’s lateral load while it is open; a visit has to be made in calm weather and the joint reclosed before the next storm, or the arm must be adjusted one side at a time. The wind does not wait for the schedule.

Every storey creeps by the rule. The creep and shrinkage of real concrete differ from Eurocode 2’s predictions by a quarter either way, and the differential they drive is the one the columns shorter than the core first measured, and the floor that is level on one date found what that uncertainty does to a differential. A shimmed arm is one of the few places in a building where the differential is measured directly — the shim removed at each visit is a reading of it — so its maintenance record is also a record of how the building is actually creeping.

A bearing forty floors up #

They cannot show the joint itself: a steel bearing between the arm and the column’s belt, with jacking points, a shim pack and a way to see the gap, all of which have to be designed for access forty floors up and for a crew to work on fifty years after the building opened. They cannot show the owner who has to schedule the visits, nor what happens when the building changes hands and the drawings with the dates on them do not. And they cannot show the arm on the other side of the core, which shortens with the same columns and must be adjusted on the same days, so that the core is never held by one arm alone — nor a second outrigger further down the building, which the second arm found shares the core’s rotation with the first and collects its own differential on its own schedule.

Still open: the arm that reads its own force #

Every schedule here is fixed in advance, by dates. A joint with a load cell in it, or shims whose gap can be read, tells its owner how much creep force the arm is holding, and a visit could then be made when the force reaches a chosen limit rather than on a date. A schedule driven by measurement would adapt to a building that creeps faster or slower than predicted, and would put the visits exactly where the creep is. Whether a force limit of, say, a tenth of the wind’s could be held with a reasonable number of visits — and whether the first year after connection, when the creep is fastest and the building is busiest, can be served at all by a schedule that waits for the force to arrive — is the question a measured arm puts to the dates written on a drawing.