What the second arm is worth

The arm that makes the columns work puts one outrigger on a core and finds where it belongs: a couple applied at about 57 per cent of the height removes the most drift, and a couple applied much lower removes the most base moment. Two questions, two best heights, and neither of them is the middle.

Real buildings do not have one outrigger. They have two, sometimes three, at plant levels chosen years before anybody computed anything — and the arithmetic of how many turns out to be more interesting than the arithmetic of where.

Each arm is worth about half of what the one before it left, which is the shape of every well-behaved optimisation and is the reason nobody builds four.

Which free body produced the number #

The free body is the core, cut free of everything, with a couple applied to it at each outrigger level.

That is the whole model and it is worth stating what has been left out to get it: the outrigger and its columns are a rotational spring, the perimeter columns’ axial stiffness is inside that spring, and the floor slabs connecting core to perimeter carry shear and no moment. What crosses the cut at an outrigger level is a couple and nothing else — which is what an outrigger is for and what a floor slab cannot supply.

The compatibility is one equation per outrigger: the core’s rotation at level $i$, less the rotations the couples at every level $j$ have removed, equals the rotation the outrigger and its columns permit under the couple they carry. With two outriggers that is two simultaneous equations, and with four it is four — and the coupling between them is why the optimum levels move.

The rotations interact. A couple at 71 per cent of the height removes rotation everywhere below it as well as at its own level, which changes what the lower arm is being asked for. That is why the pair’s optimum is 35 and 71 rather than the single optimum repeated, and it is why the answer is a simultaneous solve rather than a superposition of independent decisions.

The pair, and how flat it is #

The pair’s optimum is a point on a surface, and the shape of the surface near it is what a designer actually needs.

That flatness is the practical result. An outrigger level is chosen by the architecture and confirmed by the mechanics, not the other way round, because a storey-deep truss has to go through a floor that can afford to lose its perimeter — a plant floor, a refuge floor, a mechanical level — and those are placed for reasons of duct runs and lift zoning.

The mechanics’ contribution is to say which ranges are acceptable, and the answer is: nearly all of the middle two thirds. What it rules out is an outrigger near the top, which removes almost nothing because there is very little core rotation left above it to hold, and one at the very bottom, which does the same for the opposite reason.

Why the base moment behaves differently #

The dashed curve in the first figure is the other question, and it is much less rewarding.

One arm removes 51 per cent of the core’s base moment. Two remove 65, three 73 and four 74 — and the fourth is worth one point, against the 1.5 points of drift it bought.

The reason is where each quantity is generated. Drift is an integral of curvature over the whole height, so removing rotation anywhere helps and removing it high up helps most, because everything above rides on it. Base moment is a local quantity, and only the couples themselves reduce it, one for one. Adding arms high up removes drift and does very little to the base.

Which produces the design rule this anchor is really about: an outrigger system is a drift device. A core sized by its base moment — a short building, a heavy wind, a slender plan — gets much less from outriggers than a core sized by its drift, and above about thirty storeys the drift is what governs, which is why outriggers appear at that height and not below it. It is the same crossover that decides where a deflection comes from in any cantilever: bending dominates and the fourth power of the height takes over.

What the arm has to be #

None of the numbers above survive if the outrigger is not stiff, and the sensitivity is worth knowing before choosing a truss depth.

The rotational stiffness of an outrigger is three flexibilities in series: the truss’s own bending and shear, the columns’ axial extension over the height above and below, and the connection between the truss and the core. On a real building the columns are usually the softest of the three — a column stretching over sixty storeys is a long spring — and that is why an outrigger is more effective in the middle of a building than near either end: it has columns on both sides of it to work against.

It is also why the belt truss exists. An outrigger reaches only the columns it is directly connected to, which on a rectangular plan is four or eight; a belt truss round the perimeter at the same level engages the rest of them by spanning horizontally between them, so the couple is delivered to twenty columns rather than eight.

That is not a small refinement: the axial stiffness of the couple’s columns is proportional to their total area, and a belt truss can double or triple it — moving the arrangement along the flat part of the curve above, where more stiffness buys little. A belt truss is worth most exactly where the outrigger is worth least, which is the honest way to state the trade.

The same arithmetic on a taller core #

The sequence in the first figure belongs to a 40-storey building, and the reason to run it again on a taller one is that the answer does not scale the way the drift does.

That similarity is the useful part. The optimum levels are fractions of the height and the reductions are percentages, and both are governed by one dimensionless group — the core’s rotational flexibility against the outrigger’s. A designer moving from a 40-storey building to a 70-storey one is at the same point on the same curve provided the core and the outrigger have grown together, which they usually have because both were sized by the same drift limit.

Where it stops being true is at the extremes of that ratio. A very stiff outrigger on a flexible core is on the plateau of the fourth figure and every additional arm is nearly free of diminishing returns; a flexible outrigger on a stiff core is on the steep part, where the first arm is worth much less than 85 per cent and the second is worth almost as much as the first. The shape of the sequence is a property of the ratio rather than of outriggers, and reading which regime a building is in comes before deciding how many arms to draw.

Why the answer is the same on both buildings #

That coincidence deserves an explanation rather than a shrug, because it is what makes any of this transferable.

The core’s rotation under wind goes as $wH^3/EI$. The couple needed to remove a unit of it goes as $EI/H$. The outrigger’s own flexibility is $1/K$. Divide one by the other and the whole problem depends on a single group — $KH/EI$ — with the load falling out entirely, because both the demand and the capacity are proportional to it.

So two buildings with the same $KH/EI$ have the same optimum levels and the same reductions, whatever their heights, loads or core sizes. The 40-storey and 70-storey towers in the figures were given properties that happen to land at nearly the same value of it, which is not a fluke: both cores were sized by the same drift limit, and a drift limit is a statement about $wH^4/EI$ that fixes the group once the outrigger is proportioned to the core.

The general shape of that is worth carrying. A well-posed optimisation usually collapses onto fewer variables than it started with, and finding the group is what turns a family of answers into one curve — the same move that reduces a stay’s damping to a single length and a suspension deck’s behaviour to one μ.

What to carry away #

Three sentences, and the middle one is the one worth arguing with a project manager about.

An outrigger system is a drift device: it removes five sixths of a core’s top drift with one arm and half the base moment, and the two objectives have different optimum heights and very different diminishing returns. It works on the core’s bending and on nothing else, so a building whose drift is mostly frame shear is one outriggers cannot help.

The second arm is worth about half of what the first left, the third half of that, and past three there is nothing worth a storey. Two is the answer on nearly every building, and it is an answer that arrives from the arithmetic rather than from precedent.

And the levels are flat over the middle two thirds of the height, so they are chosen by the plant floors. The mechanics’ job is to say which floors will do, which is a much easier conversation than the one about where the optimum is.

The forces in the arm, which are not from the wind #

One number has been missing from every figure here, and it is the one an outrigger is actually detailed for.

The couple the compatibility returns is the wind couple: on the pair drawn it is a moment of some tens of thousands of kilonewton-metres at each level, delivered as a tension–compression pair in the perimeter columns. That is what the truss members are sized for, and it is a straightforward number.

What arrives afterwards is not. The core is concrete and shortens under its own load, by creep, over years; the perimeter columns are usually more heavily reinforced, less heavily stressed, or steel, and shorten differently. An outrigger connecting the two is the member that objects to the difference, and the force it develops depends on the differential movement rather than on the wind.

On a tall concrete building that force can exceed the wind force, and it grows for the life of the structure. The standard remedy is a construction sequence: the outrigger’s diagonals are erected but left unconnected — with slotted holes, or an un-welded joint — until enough of the shortening has happened, and then made continuous. That is a piece of programme with a structural consequence, decided between an engineer and a contractor, and it is the reason an outrigger drawing carries a note that no analysis produced.

Where the model stops #

The couple is applied instantly and the building is built over three years. The columns are shorter than the core by the time the frame is topped out, and an outrigger connected before that differential has occurred is the member that objects to it. The forces from differential shortening in an outrigger can exceed the wind forces it was designed for, and the standard remedy — leaving the connection unmade until most of the shortening has happened — is a construction sequence decision that no analysis on this page contains. It is the clearest case in the collection of a member whose governing action is neither a load nor a resistance but a mismatch of two lengths.

The core is a prismatic cantilever. A real core changes section up the height, opens for lift lobbies, and is not fixed at the base but sits on a raft with a rotational stiffness of its own.

The wind is uniform. A triangular pressure profile moves both optimum levels downward, by a few per cent of the height, and the flatness of the curve absorbs the difference.

The core is treated as bending only. A real core has a shear flexibility as well, from its own openings and its coupling beams, and an outrigger does nothing about it — so the reductions here are of the bending part of a drift that has three parts, and the fraction of the total is smaller than every percentage on this page. How a tall building stands still is where that division is made.

And the outriggers are in one plane. A building with outriggers in two directions has arms that intersect at the core’s corners, and the two systems share columns — which couples them, and which is a three-dimensional problem the plane compatibility above cannot pose.

What the pictures cannot show #

The cost. Every arm on the first figure is a storey of the building’s perimeter given over to a truss, at the height where floor area is most valuable, plus the columns sized for a couple they would not otherwise carry, plus a connection to a core wall that has to deliver a very large moment into concrete.

Against that is a core that can be smaller, and on a slender tower that saving is real: the difference between a 92 per cent reduction and an 82 per cent one is the difference between a core that satisfies a drift limit and one that does not. The right number of outriggers is where the marginal storey stops being cheaper than the marginal core, and neither of those prices is a mechanical quantity.

They also cannot show the alternative systems. An outrigger is one of four ways to make a tall building stiff — the others being a tube, a diagrid, and simply making the core bigger — and the comparison between them is a comparison of plans and elevations rather than of curves.

The assumption the figure rests on #

That the outrigger’s stiffness is independent of the others.

It is not, because they share columns. Two outriggers at 35 and 71 per cent of the height are connected by the same perimeter columns, and the axial force one of them puts into a column changes the extension the other one sees. In the model here each arm has its own spring $K$, and in the building there is one set of columns carrying the sum of both couples.

The effect is a coupling term the compatibility above omits, and its sign is unhelpful: the upper arm’s couple stretches the columns over the height above the lower arm, which is exactly the length the lower arm relies on for its own stiffness. Two outriggers are softer than two independent springs, and the more of them there are the worse the approximation gets — which is the second reason, after the diminishing returns, that the sequence in the first figure flattens faster in a real building than it does here.

The ladder from here #

Later rungs on this anchor: the columns’ shared flexibility taken into the compatibility, which is the coupling this page ends on. The belt truss as a structure, with its own spanning action and the number of columns it can reach. Optimum placement under a triangular wind profile rather than a uniform one. The outrigger as a damper location, since a level with a large relative rotation is a good place to put something dissipative. The construction sequence with creep, shrinkage and a delayed connection, which is where an outrigger’s real forces come from. And the comparison this anchor eventually has to make: an outrigger against a tube, against a diagrid, against a bigger core — four ways of buying the same stiffness, priced in different currencies.