The arm connected late

The arm that makes the columns work introduced the outrigger: a storey-deep arm joining a tall building’s core to its perimeter columns, so that when the wind bends the core the arm turns with it and pulls on the columns, which then carry part of the overturning moment as a tension–compression pair. The arm that ignores the shape of the wind found where to put it on a 200 m core and what it carries: under the same 6,000 kN of wind in three shapes, its best level hardly moved and its force changed by half.

Every one of those calculations had the arm connected from the start. That essay’s last section named the practice that complicates them. The core and the columns shorten by different amounts under the building’s weight and its creep, and an arm connected early would carry that difference as a force; so the arm is often left unconnected — its diagonals erected, its bolts left loose in slotted holes — until the building is nearly complete. Connected late, it starts with no force but misses none of the shortening that happens afterwards. The essay asked whether the force an outrigger collects from the core’s continuing creep is larger than the force it was designed to carry from the wind, and whether the date of the connection is the outrigger’s most important dimension.

The answer to the first is no, but not by as much as the practice assumes. The answer to the second is yes.

A building that shortens for fifty years #

The building is a concrete one, 55 storeys of 3.6 m, about 200 m tall, built at a storey a week, so that it tops out in week 55. Its perimeter columns work at 12 N/mm² at their base under the finished building and its core at 6 — columns usually carry about twice the stress of a core, which is sized for the wind. Both stresses fall with height as the load above them does. A quarter of the load is finishes, partitions and cladding, which arrive on day 450, some months after topping out.

Every storey is treated as its own concrete. It is cast on its own day; it is loaded a little more each week as the floors above it go up, and once more when the finishes arrive; and each load increment creeps from the day it arrives with EN 1992-1-1’s creep coefficient for the storey’s age then. Each storey also shrinks from the day it is struck. The columns are 600 mm across and the core’s walls 300 mm thick, so they have different notional sizes, which EN 1992-1-1 uses to set how fast each one creeps and dries. Summing the storeys below the outrigger’s level, 149 m up, gives each member’s shortening there at any date.

By topping out the columns have shortened 56 mm below the arm’s level and the core 49; the difference is 7.0 mm. Then the shortening goes on. After fifty years the columns have shortened 117 mm and the core 86, and the difference is 30.5 mm. More than three-quarters of the difference the arm would ever see arrives after the frame is finished.

The columns are shorter than the core found the shape of this difference up the building — largest near mid-height for the part caused by load, largest at the roof for the part caused by shrinkage. What it did not need was the timing, because a floor that is levelled as it is built only cares about what happens after it is placed. An outrigger connected late is the same kind of object: what matters is the part of the difference that happens after its connection, and that turns out to be most of it.

What the arm catches #

The arm is the earlier essays’ — $10^8$ kN·m per radian, its columns 30 m apart — so on each side it is a spring of 222 MN/m between the core and the column, in series with the concrete of the column and core that carry its force. From the day it is connected, every millimetre of differential below it is resisted.

Connected as soon as its own level is built, the arm ends after fifty years with 3.98 MN in it. Connected at topping out, 3.47 MN. After two years, 2.33; after five, 1.35. The design wind, a power-law profile of 6,000 kN in all, puts 5.8 MN into the same arm. So an arm connected at topping out carries a late force of 60 per cent of its wind force, and one connected at once 69.

That answers the essay’s first question. For this building the late force does not exceed the wind’s. But it is not small beside it, and two things about it make it count for more than its size suggests. It is permanent: the wind’s force comes and goes, and the late force is there on every day of the building’s life. And it acts the same way on both sides of the core — both columns held up by the arms as they try to shorten — where the wind’s force pulls up one column and pushes down the other. On the side where the wind already pulls the column up, the two add: 5.8 MN of wind and 3.5 of shortening, 60 per cent more than the arm was designed for if the late force was left out.

The date is the largest load #

The connection date moves the late force more than anything else in the problem. Each year of waiting removes the part of the differential that happens in that year, and the differential is front-loaded: creep and shrinkage both run fastest early. The late force halves by about three years after the first storey is cast, two years after topping out, and is under a tenth of the wind’s force if the connection waits twenty years — which no real building does, because the arm is wanted for the wind long before then.

That makes the date a design dimension in exactly the sense the earlier essay asked about. An outrigger’s stiffness, its level and its member sizes are drawn; its connection date is decided on site, often by the programme, and it changes the arm’s force by a factor of three between the plausible extremes. A specification that says “connect the outrigger after topping out” has left the arm’s largest permanent force to be chosen by whoever is managing the crane.

There is a wrinkle worth knowing. An arm left unconnected is not doing its job against the wind either, and the building has to stand up without it in the meantime — so a late connection is a decision to accept a softer building, with more drift and more core moment, for the months or years it waits. The trade is not free on either side.

Creep takes back a third #

The differential the arm sees after topping out is 23.5 mm, and a spring of 222 MN/m stretched by that much would carry 5.23 MN. The arm carries 3.47. The difference is the concrete it pushes against. The arm’s force goes down the column and into the core as a sustained load, and sustained load on concrete creeps; the concrete under the arm gives a little more each year, and the force relaxes as it does. The strain that was imposed found the general case: a force held by an imposed deformation leaks away as the material holding it creeps.

The same creep that drives the differential also relaxes the force it causes. It is the arrangement the jacks the arch forgets found in a fixed arch, from the other side: there a deliberate correction relaxed while the load’s own moment did not; here an unintended restraint relaxes, and that is what keeps the late force at two-thirds of its elastic value. The relaxation is largest for the differential that arrives soonest after connection, while the concrete under the arm is youngest and creeps fastest — another reason an early connection is not as much worse as the differential alone would suggest.

Slower, not only busier #

The usual account of differential shortening is about stress: columns carry more stress than the core, so they creep and shorten more. That is most of the force here, which falls steadily as the columns’ stress is lowered. But it does not fall to nothing. With the columns at exactly the core’s 6 N/mm², the arm still ends with 1.22 MN; with them at 4 N/mm², below the core’s stress, 0.47.

The remainder is timing. The columns are 600 mm across and the core’s walls 300 mm thick, and a thicker member dries and creeps more slowly: the creep that belongs to the member found that EN 1992-1-1’s coefficient grows with the notional size’s time constant, and the shrinkage does the same. The thin walls do most of their drying and much of their creep before the arm is connected; the thick columns are still doing theirs afterwards. So even equal stresses leave the columns shortening more after the connection than the core. Only when the columns are as thin as the walls and as stressed does the late differential vanish.

What the arm feels is not how much more the columns shorten, but how much more of their shortening is still to come. A composite or steel column, which does not creep, reverses the picture: then the core is the slow member and the arm is pushed the other way. Neither is wrong; each needs the timing, not just the stress.

Where the late force goes #

The late force is not a force the arm holds against nothing. Each arm holds its column up as the column tries to shorten past the core, so it takes some of the column’s gravity load off it and hands it to the core: the arm is a beam carrying load from the perimeter into the middle of the building. After fifty years, with the arm connected at topping out, that is 3.5 MN taken off each perimeter column below the arm and put into the core.

The column’s own load at its base is its stress times its area: 12 N/mm² on a column 1.2 m square is about 17 MN. A fifth of it has been moved to the core by an arm nobody designed as a transfer structure. The column below the arm carries less than it was sized for, which is harmless; the core carries more, which is not nothing, since the core’s walls are thinner and were sized for wind; and the foundations carry different loads from the ones they were designed for, the core’s piles more and the columns’ less. A building that was meant to settle evenly under its gravity load now has the late force moving load between its foundations for fifty years.

That is a familiar shape. The structure that was never complete found that every structure built in stages carries forces that depend on when each part became structural, and that the order of building is a load case. An outrigger connected late is that principle at its plainest. The arm becomes structural on one day, and every millimetre the building shortens after that day becomes a force in it and a load moved between the members it joins. The drawing shows the finished building and none of that history.

It also gives the reverse view of the same arithmetic. If the arm were connected on day one and stayed connected, it would have carried the whole differential as it accumulated — forcing the columns and the core to shorten alike below it, at the cost of a large force — and the building above would have been built level onto a core and columns that never shortened apart. Connecting late is a decision to accept the differential as a movement rather than as a force, for as long as the arm waits, and to take the rest as a force afterwards. The figures put a number on how the cost divides.

A stiffer arm catches more of it #

The two forces respond differently to the arm’s stiffness, and that is the last thing a designer needs to know. The wind’s force in the arm is the arm’s share of an overturning moment that is already there; a stiffer arm takes a larger share, but with diminishing returns, from 2.1 MN at a fifth of the drawn stiffness to 9.0 at five times it. The late force is the arm’s stiffness times a movement that is fixed by the building’s construction, less what creep relaxes; it rises nearly in proportion to the stiffness, from 0.7 MN to 17.4.

So the two cross. At the drawn stiffness the late force is 60 per cent of the wind’s; at about 2.3 × 10⁸ kN·m per radian they are equal; at five times the drawn stiffness the late force is twice the wind’s. An outrigger made stiffer to control drift is an outrigger made to collect more of the building’s own shortening, and past a point the shortening, not the wind, designs it. What the second arm is worth found diminishing returns in adding arms for drift; this is the same economy seen from the arm’s own forces.

What the calculation takes for granted #

The arm is one spring a side, connected all at once. Real outriggers are trusses whose diagonals may be connected one by one, and whose bolts may be tightened in stages; some are connected through jacks or friction devices that let them slip under the slow differential while holding against the wind. Those devices are the engineered answer to this essay, and the force they leave is whatever their slip setting allows.

The concrete under the arm is taken at one age. The force passes down a column and into a core whose storeys are of different ages; the relaxation is computed for concrete as old as the arm’s level, a simplification that slightly understates the relaxation in the older storeys below.

The core and the columns carry their own loads only. Once connected, the arm transfers gravity load from the columns to the core, which changes both members’ stresses and so their later creep. That transfer is the arm’s late force itself, and including it would lower the differential slightly — the arm relieves the very column that is loading it.

The air is the same from the first day. Every storey dries at 60 per cent relative humidity from the day it is struck. A building under construction is open to the weather for months, then enclosed and heated, and the drying — and so the shrinkage and the drying part of the creep — speeds up when the heating comes on. That moves some of the differential later, after topping out, which is the direction that makes a late connection’s force larger rather than smaller.

The wind’s force is the earlier essay’s, computed for a wind applied once to the finished building with the arm connected; the late force is a fifty-year figure. Comparing them is a comparison of a design force with a permanent one, which is the comparison a designer has to make.

Still open: the arm that slips on purpose #

The practical answer to a late force is not to wait longer but to let the arm move. A connection that slides under a slowly applied force and locks under a fast one — a friction joint set just above the late force’s rate, a hydraulic damper that flows under creep and stiffens under wind — would let the arm ignore the differential while still carrying the wind. Whether such a connection can tell a year’s creep from a gust’s thirty seconds well enough to do both, how much of the arm’s stiffness against the wind it costs to give it a way to slip, and whether a building that relies on it must then be inspected for whether it has slipped too far, is the question the late connection leaves for the joint that is designed to be late for ever.