The arm that ignores the shape of the wind

An outrigger makes a tall building’s perimeter columns carry part of its wind moment. A storey-deep arm connects the core to the columns; when the core bends under the wind it rotates at the arm’s level, the arm pushes one column down and pulls the other up, and the couple between them is a moment taken off the core. Where the arm goes is a compatibility calculation — the core’s rotation at that level, less what the arm’s own couple gives back, equals the rotation the arm and its columns allow — and the best level depends on whether the question is the drift at the top or the moment at the base. A second arm buys about half of what the first left.

Both of those essays loaded the core with the same wind at every height. That is the textbook load and it is not the wind. Wind speed rises with height above the ground, and the pressure goes as the square of the speed, so the profile a wind standard prescribes loads the top of a tall building harder than its base. An earthquake’s first mode loads a building as an inverted triangle. The earlier essays on outriggers had left the triangular profile as a question that would move the optima, and the expectation was that a load concentrated towards the top would pull the best arm level somewhere new.

Three winds with one total #

To compare the shapes fairly they must carry the same total force. The core is the 200 m core of the first outrigger essay, of bending stiffness $1.2 \times 10^{10}\ \text{kN·m}^2$, with an arm and its columns that resist the core’s rotation with 10⁸ kN·m per radian. The wind totals 6,000 kN in each of three shapes.

The power-law wind is a code shape: a speed growing as the height to the power 0.16, the exponent for open country, so a pressure growing as the height to the 0.32 — the same pressure that a building’s faces see as mostly suction once it has been turned into coefficients. It is closer to uniform than it looks, because most of its growth happens near the ground and the top half of the building sees nearly the same pressure throughout. The triangle is the extreme, and the shape the first mode of a cantilever puts on it in an earthquake. Between them they bracket any wind a designer would draw.

For the same total force, the triangle’s resultant is two thirds of the way up, a third higher than the uniform load’s, so its base moment is a third larger and its tip drift about half as large again.

Where the arm goes #

The single arm’s sweep, repeated for each shape, is the whole of the question.

The three curves are nearly the same curve. The triangle, with three quarters of its load in the building’s upper half against the uniform load’s half, moves the best level up by 1.2 per cent of the height — two and a half metres on a 200 m building, less than a storey — and the share of the drift the arm removes changes by less than a percentage point. The power-law wind is indistinguishable from the uniform one. The arm’s best level is a property of the core and the arm, not of the wind’s shape.

That runs against the expectation on two counts. The best level does move, but upward rather than downward, and it moves too little to change any design: an outrigger is placed at a plant floor or a refuge floor, which come in steps of a storey, and the step between the uniform optimum and the triangle’s is smaller than one of them.

Why the shape washes out #

The reason is in what the arm is matching. The arm works by stopping the core’s rotation at its level, and the drift it removes is the couple it carries times the drift a unit couple at that level removes — which is a property of the core alone, $x(H - x/2)/EI$ for a couple at height $x$. The couple is set by the core’s rotation at $x$ under the wind, and the rotation at a point is an integral of the moment below it.

Integrals forget detail. The rotation profile of a cantilever under any smooth load rising towards the top has the same shape — zero at the base, growing steeply, flattening near the top — and differs between loads mainly in its size, which scales the couple and the drift together and leaves the best level where it was. The triangle’s rotation profile is a little steeper near the top than the uniform load’s, which is why its best level is a little higher. The arm sees the wind only through an integral of an integral of it, and two integrations take most of the shape out of anything.

The deflected shape shows what the arm does with that. Below it, the arm’s couple acts against the wind’s moment over the whole lower part of the core, and the core bends much less there. Above it, the core is a cantilever 49 m long standing on a base that turns through the small angle the arm allows, and the top’s drift is that rotation times the height above it plus the short cantilever’s own bending.

What does move the arm #

The comparison worth making is with the thing that does move it.

The arm’s stiffness moves its best level across more than a third of the building: a soft arm belongs near the top, where the core rotates most and even a weak restraint has something to hold; a stiff one belongs lower, where it can take a larger share of the building’s moment and the core above it is long enough to matter. Across that whole range the two winds keep their best levels within two and a half per cent of the height of each other. The stiffness of the arm, which is set by the depth of the outrigger truss, the axial stiffness of the columns and the flexibility of the connections, matters between fifteen and thirty times as much to where the arm goes as the shape of the wind does.

That puts the design effort in the right place. A designer uncertain about the wind profile — and every wind profile is a fitted curve, with an exponent that depends on the terrain upwind — can stop worrying about where the arm goes. A designer uncertain about the arm’s stiffness — which depends on details that are often fixed late — cannot.

The base moment’s optimum, which moves more #

The first outrigger essay found that the arm’s best level depends on which question is asked: the level that removes the most tip drift is not the level that removes the most base moment. The same split survives a change of wind, and the base moment’s optimum is the more sensitive of the two.

For the core drawn, the arm that removes the most base moment sits at 0.485 of the height under the uniform wind, 0.497 under the power law and 0.519 under the triangle, removing 26.5, 27.6 and 29.1 per cent of the base moment. The shift, 3.4 per cent of the height, is nearly three times the drift optimum’s. The base moment is a statement about the bottom of the building, and an arm placed for it sits lower, where the load above it is a larger share of the whole and the share above changes more when the load is pushed upward.

Neither shift changes a design, for the reason already given: the gap between the two questions is ten times the gap between the winds. An arm placed for drift at three quarters of the height removes about a quarter of the base moment under any of the three winds, and one placed for base moment at half the height removes about two fifths of the drift. The choice of question is the decision; the wind’s shape is a refinement of it.

The terrain, which is the uncertain part of the profile #

The power law’s exponent is not fixed. It is a property of the ground upwind: about 0.1 over open sea, 0.16 over open country, 0.22 over suburbs and 0.3 over a city centre, where the buildings around a tall one slow the wind near the ground and leave the top exposed. For the same total force the exponent moves the resultant from 0.545 to 0.615 of the height across that range.

The arm does not notice. Its best level for drift runs from 0.748 of the height at an exponent of 0.1 to 0.753 at 0.3 — a metre on the building — while the couple it carries rises from 164 to 194 MN·m and the bare core’s drift from 0.56 to 0.66 m. The terrain changes everything about the outrigger’s design except where it goes, which is a convenient property for a decision made early, before the wind tunnel has measured the site.

The force in the arm, which does depend on the wind #

The wind’s shape has to go somewhere, and it goes into the arm.

The couple grows faster than the base moment. The triangle’s base moment is a third larger than the uniform load’s, and its arm carries nearly half as much again. The reason is where the extra load is. The arm restrains the rotation of everything above it, and under the triangle a larger share of the load is above the arm’s level — the part of the building the arm effectively holds up — so the arm takes a larger share of the extra moment than the core does. A designer who analysed the outrigger under a uniform wind and scaled its forces by the base moments would undersize the columns’ axial forces by about a tenth.

So the shape of the wind matters to the outrigger after all, but not to where it is. It matters to how strong it is, and to how much the columns it engages have to carry in tension and compression on top of their gravity load.

The drift, which depends on the wind and not on where the arm is #

The last comparison closes the argument.

The shape of the wind decides how far the top moves: the triangle drifts 47 per cent further than the uniform wind, with or without the arm, because its load is further up a cantilever whose drift grows as the cube of the lever. Where the arm is placed hardly matters: put at the uniform wind’s best level, the arm leaves the triangle’s drift 0.1 mm larger than at the triangle’s own best, out of 393 mm. A drift check under a code wind profile needs the profile; an outrigger layout does not.

The order of the decisions #

Read together, the figures sort an outrigger design into decisions that need the wind’s shape and decisions that do not, and that sorting is worth more than any of the numbers.

Where the arm goes does not need it. The level can be chosen with a uniform load, against the plant floors and refuge floors the building already has, and against the arm stiffness the outrigger truss can realistically be given — which is the variable that moves the answer. A wind tunnel test that arrives a year later will not move the arm.

How strong the arm and its columns are does need it, and needs it more than the base moment suggests. The couple grows half as fast again as the base moment when the load moves up the building, so the columns’ tension and compression should come from the profile itself, never from a uniform analysis scaled.

Whether the building meets its drift limit needs it most of all. The triangle’s top moves half as far again as the uniform load’s, with the arm or without it, and a drift check made under a uniform wind of the right total would pass a building that the standard’s profile fails by a margin no outrigger level could recover.

The pattern is a familiar one in this subject: the stiffest path takes the load, and the shape of the load decides how much. Where a member should go is a question about the structure; how much it must carry is a question about the load.

The arm, by hand #

For a uniform load and a rigid arm the best level has a closed form — the arm should sit where the couple removing the most drift is applied — and the stiff end of the stiffness figure approaches it, 0.56 of the height. For a finite arm the compatibility for one arm at height $x$ is

$$\theta_{\text{load}}(x) - \frac{x}{EI}M = \frac{M}{K}, \qquad M = \frac{\theta_{\text{load}}(x)}{x/EI + 1/K},$$

and the drift it removes is $M\,x(H - x/2)/EI$. The wind’s shape enters only through $\theta_{\text{load}}(x)$, the bare core’s rotation at the arm’s level. For the uniform load that is $w(H^3 - (H - x)^3)/6EI$; for the triangle it is a quartic in $x$. Near three quarters of the height both are close to their maximum and nearly flat, which is why the product that is being maximised has its peak in almost the same place for both — and why the peak’s height, the drift removed, scales with the size of $\theta_{\text{load}}$ while its position does not.

A cantilever core, rigid floors and a static wind #

The calculation rests on choices that limit it.

The core is a cantilever with a single stiffness. A real core changes section up the height, usually thinning towards the top, which moves the rotation profile and so the best level — by more, in practice, than the wind’s shape does.

The arm’s stiffness is one number. It is the arm truss and the columns’ axial stiffness and the joints between them in series, and the columns’ contribution depends on their length below the arm, which is longer for a lower arm: a lower arm is softer for the same truss. That coupling is real and is left out, as in the earlier outrigger essays.

The wind is static. A tall building’s wind response is partly resonant, and the resonant part loads it in the shape of its first mode — close to the triangle here — so the triangle is the more relevant of the extremes for the dynamic share. The essay’s conclusion survives that, since the triangle is one of the shapes compared.

Shear deformation of the core is ignored. A core with large openings deflects partly in shear, the other half of a tall building’s drift, and a core that deflects in shear rotates less, which moves the outrigger’s best level and reduces what it can do.

A second arm, and the floors that carry the wind in #

The figures cannot show the second arm. With two outriggers the best pair moves the same way as the single arm — from 0.35 and 0.70 of the height under the uniform load to 0.375 and 0.725 under the triangle for the softer core of the second outrigger essay — and the drift removed changes by a fraction of a percentage point. The argument is the same and is not drawn again.

They also cannot show the stresses in the floors. The arm’s couple enters the columns through the outrigger truss, but the wind enters the core through the floor slabs at every level, and under the triangle more of it enters near the top, where the floors also carry the arm’s own forces into the core. That is a diaphragm question, which the floor answers by being a beam lying down, and it is not the outrigger’s.

Still open: the arm that is connected late #

Every calculation here has the arm connected to the columns from the start. In practice the outrigger is often left unconnected until the building is nearly complete, because 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. Connected late, it starts with no force but has missed none of the shortening that happens afterwards — which, in a concrete core, continues for years. Whether the force an outrigger carries from the core’s continuing creep is larger than the force it carries from the wind it was designed for, and whether the date of the connection is the outrigger’s most important design decision, is a question about time that a static wind cannot ask.