The arm that slips on purpose

The arm connected late followed a 200 m concrete building whose outrigger was left unconnected until the frame topped out, so that the core and the perimeter columns could shorten by their different amounts without it. Most of the difference had not happened by then. Three-quarters of the fifty-year differential arrived afterwards, the arm collected 3.5 MN of it, and that force — 60 per cent of what the design wind puts into the same arm — sat in the arm permanently and in the same direction on both sides of the core. The connection date was the largest load the arm had.

Waiting longer helps, slowly. The practical answer is to let the arm move: a connection between the arm and the column that gives way to the slow force of creep and holds against the quick force of the wind. Two kinds of joint offer to do that. A friction joint — a slotted bolted connection clamped to a set slip load — slides whenever the force passes the slip load and holds below it. A viscous damper — a piston in a cylinder of oil — flows at a rate proportional to the force on it, so a slow push meets almost no resistance and a fast one meets a great deal. This essay asks whether either can tell a year’s creep from a storm well enough to do both jobs.

The same building, with a joint in each arm #

The building is the earlier essays’: 55 storeys of 3.6 m, 200 m tall, built a storey a week, its perimeter columns working at 12 N/mm² and its core at 6 at the base, with an outrigger at 149 m whose rotational stiffness with its columns is $1.0 \times 10^8$ kN·m a radian. Each storey of the columns and the core is cast on its own day and loaded floor by floor as the building rises, and then by the finishes, which here are fitted out evenly over six months rather than placed in a single day. Each storey creeps and shrinks by EN 1992-1-1 from its own casting, and the differential shortening at the arm’s level is the columns’ shortening below it less the core’s.

The arm is connected when the frame tops out, on day 385, and from then on it resists whatever differential is still to come — through the joint. The arm and the concrete that carries its force are a spring in series, the concrete creeping under the arm’s force, as before. The joint is a third element in the same series. A friction joint is rigid until the arm’s force reaches its slip load and then slides by whatever the springs do not take. A damper flows at a rate equal to the force divided by its constant, which is written as the arm’s stiffness times a time constant τ: a damper with τ of ten days, held at a fixed displacement, lets its force decay to a third in ten days.

The hero figure is both joints against the rigid arm. Rigid, the arm collects 3.48 MN over fifty years, the earlier essay’s result. Through a friction joint set to slip at 2 MN, it collects exactly as the rigid arm does until it reaches 2 MN, after 3.7 years, and then holds 2 MN for the rest of its life. Through the damper, it collects at most 0.08 MN, while the building is being fitted out and the differential is moving fastest, and lets almost all of it go again within weeks.

A friction joint holds its slip load for ever #

The friction joint’s line is flat because a friction joint relieves a force only by sitting at its slip load. Below it, the joint is rigid and the arm collects creep force exactly as before. At it, the joint slides by whatever the differential adds, and the arm carries the slip load and no more. Creep never stops pushing the same way, so the joint never leaves its slip load: it has been brought to it and kept there by a force that does not reverse.

That is the friction joint’s flaw, and it shows the moment the wind arrives.

The wind pushes the building sideways and puts tension in the arm on one side of the core and compression in the arm on the other. The creep force is the same in both arms, because the columns on both sides shorten more than the core. So on one side the wind adds to the creep force and on the other it takes away. A friction joint at its slip load has no margin in the direction it is already sliding: on the side where the wind adds, the joint slides at once, and that arm does nothing. On the other side it can take its slip load plus the creep force before it slides the other way, 4 MN, still short of the wind’s 5.8.

Before creep has brought the joint to its slip load, the margin is larger on the first side, but it starts at 2 MN on both — already less than the design wind. A friction joint set low enough to let creep go is set too low to hold the wind.

No slip load does both #

The next figure sweeps the slip load to look for a setting that works.

The dashed line is the creep force the arm keeps: it rises with the slip load until the slip load reaches the 3.48 MN a rigid arm would collect, and then it is the rigid arm’s force, because creep no longer reaches the slip load. The solid line is the wind the worse side can take: the slip load less the creep force. It is zero for every slip load below 3.48 MN — wherever the joint actually relieves creep, the worse side has nothing — and only above that does it start to rise. It reaches the design wind’s 5.8 MN at a slip load of 9.28 MN, the wind and the creep force together. A joint set there never slides under creep, and the arm carries the whole 3.48 MN of creep force exactly as if it were rigid.

So a friction joint is either an arm that slides in a storm or an arm that does not relieve creep. There is no slip load that does both, and the reason is not a property of this building. A friction joint distinguishes forces by their size, and the creep force and the wind force are of the same size. What separates them is their speed.

A damper that is slow enough for the storm #

A viscous damper distinguishes forces by their rate, and the first question is whether it is slow enough to hold a storm.

A gust lasts seconds and a damper of any useful time constant is effectively rigid against it. The difficulty is the storm’s sustained wind, which holds roughly steady for hours. Under a steady push a damper’s force decays exponentially with its time constant: the arm takes the wind’s full 5.8 MN at first and lets it go as the damper flows. A damper of a tenth of a day keeps 29 per cent of the wind after three hours and almost nothing after twelve; one of a day keeps 88 and 61; one of ten days keeps 99 and 95.

Whatever the arm lets go, the core takes, and the building sways further. The arm that makes the columns work found what an outrigger’s stiffness is worth in drift and in the core’s base moment. A damper that has let half the arm’s force go has given back about half of that during the storm. A time constant of a few days or more holds the arm’s share through any storm a design wind describes.

And fast enough for the fit-out #

The other bound is the creep. A damper’s force under a slow push is its constant times the rate, so the longer the time constant, the more of the creep force it lets in.

The two curves cross the useful region from opposite sides. The storm line passes 95 per cent at a time constant of 2.4 days; the creep line passes a tenth of the rigid force at 45 days. Between them the damper does both jobs, and at ten days it does both comfortably: 2 per cent of the creep force, 99 per cent of a three-hour storm.

The window is a factor of 19 in time constant, and that is far narrower than the ratio of a year to three hours, about three thousand, would suggest. The reason is when the differential is fastest. A year’s creep is not spread evenly over the building’s life: in the months after the arm is connected, while the finishes go on and the floors above are still young, the differential moves at nearly two centimetres a year, ten times its rate a few years later. That is the rate a damper has to keep up with, and at that rate a damper with a time constant of months collects real force. It collects it during the fit-out and lets it go within weeks once the rate falls, which is why the damper’s line in the hero figure has a hump in its second year and nothing after.

What the joint has to move #

A damper that lets the creep force go does so by moving, and it moves by nearly the whole differential.

The damper’s piston has moved 23.3 mm after fifty years, all but a tenth of a millimetre of the differential since connection, and it is still moving a little each year. The stroke it is built for has to include that, in one direction, plus the storm’s movement either way about wherever the creep has left it. A damper detailed for the wind alone, with a stroke of a few millimetres centred on its mid-position, runs out of stroke in creep within a few years and then becomes a rigid strut that collects every millimetre after.

That gives the joint designed to be late for ever an inspection to go with it. The damper’s position is a measurement of the building’s differential shortening since the arm was connected, and a reading that departs from the predicted line in this figure says either that the building is shortening differently from its prediction — which the column that fails years later would want to know — or that the damper has stopped flowing, which an outrigger designer would want to know more. A friction joint offers the same reading, and a less useful one: it records only what passed its slip load.

What a damper costs the arm #

A damper in series with an arm is a spring in series with a spring, as the damper is only as good as its brace found for dampers in a frame: under a quick force it is as stiff as the oil and the cylinder let it be, and that stiffness is in series with the arm’s. Nothing here charges the damper any flexibility under a gust. A real damper’s stiffness against a fast load, its oil’s compressibility and its mountings, would soften the arm against the wind by whatever share of the series flexibility it adds — a design detail for the damper’s maker, and one the arm’s stiffness requirement sets.

The damper also changes what a second arm is worth. What the second arm is worth found that a second outrigger lower down adds less stiffness than the first, and the arm that ignores the shape of the wind that an arm’s value hardly depends on how the wind is spread up the building. Both results assumed arms that hold whatever they are given. An arm with a damper holds a gust and a storm but not a decade, so for every load that lasts longer than its time constant the building behaves as if the arm were not there — which is exactly the late force the damper was put in to refuse, and also any long-term lean the building takes on from settlement or from an uneven load. A building whose outriggers are expected to resist a permanent lateral load cannot use dampers in them; one whose outriggers exist only for wind can.

There is also a gain the figures do not count. A damper in an outrigger dissipates energy whenever the building sways, and an outrigger is where the core’s rotation and the columns’ axial movement differ most, so a damper there adds damping to the building’s sway. That is a reason outrigger dampers are fitted on buildings that have no differential shortening to speak of, and here it comes free with the creep relief.

The damper’s force from its rate #

The damper’s creep force is its constant times the rate. The arm’s spring, $k = 2K/\ell^2$ with $K = 10^8$ kN·m a radian and a lever of 30 m, is 222,000 kN a metre; a time constant of ten days makes the damper’s constant 2.2 million kN·day a metre. During the fit-out the differential at the arm moves at about 18 mm a year, 0.05 mm a day, and the damper’s force is 2.2 × 10⁶ × 0.05 × 10⁻³ ≈ 110 kN, against the calculation’s peak of 84 kN; the fit-out’s load arrives in steps, and the damper lets each go before the next arrives. A rigid arm at that point carries about 1.26 MN.

The storm’s retention is the exponential of minus the storm’s length over the time constant: three hours is an eighth of a day, and exp(−0.125/10) = 0.988.

The friction joint’s needed slip load is the arithmetic of the two sides: the worse side’s margin is the slip load less the creep force, and for it to reach 5.80 MN with a joint that is never reached by creep, the slip load must be 5.80 + 3.48 = 9.28 MN.

Where the series model stops #

The joint is in series with the arm alone. A real outrigger’s arm is a truss two storeys deep, and a joint at its column end is in series with the truss, its connection to the column and the column itself; the model lumps all of that into one spring and puts the joint at the end of it.

The damper is linear. Real viscous dampers are often built with a force that grows as a power of the velocity below one, so that they resist slow movement more, relative to fast, than a linear one does — which would narrow the window this essay found, since the creep rate is the slow movement.

The wind is one steady storm. A design wind is a mean that holds for an hour or more and a fluctuation about it; the damper is rigid against the fluctuation, and the storm figure is about the mean.

The oil, the seals and the other columns #

They cannot show the damper’s temperature, which changes its oil’s viscosity and so its time constant through the year by a factor of two or more, nor the seals that have to hold that oil in for the life of the building. They cannot show the building’s other columns, which are not outrigger columns and shorten without any joint at all. And they cannot show the cost of the maintenance a damper needs and a bolted connection does not — which is, in the end, why most late-connected outriggers are bolted.

Still open: the arm that is reconnected #

A middle path between connecting the arm once and letting it slip for ever is to connect it with shims and adjust them. Every few years the joint is unloaded, the shims are changed to take out the differential since the last adjustment, and the joint is closed again. Whether a handful of adjustments — at one, three and ten years — keeps the arm’s creep force as low as a damper does, how much of it creep takes back in the years between adjustments, and whether a building can be relied on to carry out maintenance that its own structural stiffness depends on, is the question the damper leaves for a joint that is adjusted by hand. The force that is capped on purpose is where a deliberate limit on a force first appeared, and the force that is whatever it needs to be is the friction that made the slip load a limit rather than a value.