How a tall building stands still

A tall building’s vertical structure is a solved problem: the columns get bigger toward the bottom and the arithmetic is a running total. Its lateral structure is not, and past about ten storeys it is the thing that decides the whole form of the building.

Two arrangements are available and they resist in completely different ways.

The two shapes are the interesting part before any of the numbers. A wall is a vertical cantilever and its slope accumulates upward, so its worst storey drift — the relative movement between one floor and the next, which is what cracks partitions and jams lifts, and which stiffness rather than strength decides — is at the top. A frame racks storey by storey in proportion to the shear it carries, so its worst drift is at the bottom. They are the wrong way up for each other, and that is exactly why the combination works.

What each one is doing #

A shear wall or a core is a cantilever with an enormous second moment of area, and its tip deflection under a uniform wind is the familiar $qH^4/8EI$. The fourth power of the height is the reason walls run out: doubling the height of a building multiplies the wall’s deflection by sixteen while multiplying the wind load by four and the wall’s own stiffness by nothing at all.

A moment frame has no such expression, because it does not bend as a whole. Each storey racks by $V_i/k_i$ where $V_i$ is the shear above that storey, and the total sway is the sum of those racking movements. That makes a frame’s deflection go as the square of the height rather than the fourth power, which sounds better and is offset by $k$ being small — a frame is a much less efficient use of material for lateral stiffness, and it is used because it leaves the floor plan open.

Each storey of the framed building is a portal under a horizontal load, solved on its own: the columns bend in double curvature, the beams bend to restrain them, and the storey racks by an amount proportional to the shear passing through it. The racking of each storey is added to the ones below, which is why a frame’s sway accumulates from the bottom and its deflected shape has its greatest slope where the shear is largest.

The interaction reverses, and that is the whole trick #

Tie the two systems together with floors that cannot stretch, and neither can take the shape it wanted.

That sign change is the reason the pair beats springs in parallel. In the lower storeys the frame wants to rack a long way and the wall will not let it, so the wall takes the shear. In the upper storeys the wall wants to lean out and the frame will not let it, so the frame takes more than the applied shear and hands the excess back to the wall as a restraint.

Each system is stiff exactly where the other is weakest, and the interaction converts the wall’s surplus stiffness at the bottom and the frame’s at the top into a structure with neither system’s worst drift. The 24% bonus over the parallel-spring estimate is that conversion measured.

It also explains a detail that puzzles people looking at a frame analysis of a real building: the beams connecting a core to the surrounding frame carry substantial forces near the top of the building, in a direction that looks backwards. They are the interaction force, and near the roof it points the other way.

One number decides which building this is #

Everything on this page — the shape, the reversal, the size of the bonus — is governed by a single dimensionless group, and it is worth extracting because it turns a family of drawings into one curve.

The wall contributes a flexural rigidity $EI$; the frame contributes a shear rigidity $GA = k h$, since a storey drift $V/k$ over a height $h$ is a shear strain $V/kh$. Those two combine into a length

$$\ell_s = \sqrt{\frac{EI}{GA}}$$

and the building’s behaviour depends on the height only through the ratio $\alpha H = H/\ell_s$.

Both quantities can be read straight off the two single-system drifts printed above. The wall alone reaches 146.5 mm, and $\delta = qH^4/8EI$ gives $EI = 8.19\times10^8$ kNm²; the frame alone reaches 140.0 mm, and $\delta = qH^2/2GA$ gives $GA = 7.0\times10^5$ kN. So

$$\ell_s = \sqrt{\frac{8.19\times10^8}{7.0\times10^5}} = 34.2\ \text{m}, \qquad \alpha H = \frac{70}{34.2} = 2.05$$

The two curvatures in that drawing are the whole of the shear length’s meaning. It is the height at which the two systems change places. Below it the wall’s flexural stiffness is the cheaper way to resist; above it the frame’s shear stiffness is. A building shorter than its own shear length behaves as a cantilever with a frame attached, and a building much taller than it behaves as a frame with a wall attached, and the interesting ones are within a factor of a few.

The bands are the design content:

At 2.05 this building is squarely in the middle band, which is why it gets a 24 per cent bonus over the parallel-spring estimate: the bonus is the interaction, and it goes to zero at both ends of the table.

And it says exactly what happens on scaling. The forty-storey version has the same $\ell_s$ and twice the height, so $\alpha H = 4.09$ — still interacting, but moving toward frame-dominated, which is why its reversal point slid from 80 per cent of the height to 70. Doubling a building’s height and keeping its behaviour would require four times the core stiffness at unchanged frame stiffness, since $\alpha H$ carries $\sqrt{EI}$ underneath. Nobody has ever quadrupled a core. That, rather than the fourth power on its own, is why the systems change at height.

The outrigger, and where it belongs #

The next move, once a wall and a frame together are not enough, is to connect the wall to the perimeter columns with a deep truss or a storey-deep wall at one level — an outrigger. Its effect is not to add lateral stiffness directly but to apply a restraining moment to the wall, using the perimeter columns as a couple.

The shape of that curve is the design rule, and it is not obvious. The wall’s rotation is largest at the top, which suggests putting the outrigger there; the length over which the restraining moment reduces curvature is largest at the bottom, which suggests the opposite. The product of the two peaks somewhere near the middle, and every study of the problem since the 1960s has found the optimum between half and six-tenths of the height for a single outrigger.

Two consequences follow that are pure engineering economics. Putting an outrigger at 55% of the height means giving up a storey of lettable perimeter in the middle of the building, which is worse commercially than giving one up at the top — so real buildings often place them at plant levels, accepting a few per cent of stiffness to put the structure where the lifts and the air handling already are. And two outriggers at roughly a third and two thirds do better than one at the optimum, with diminishing returns after that.

Two thirds of a frame’s flexibility is in its beams #

The frame was given one number, $k =$ storey stiffness, and it is worth opening because what is inside it decides where money spent on a moment frame goes.

Rack one storey of a regular frame and both the columns and the beams bend. They are in series — the column’s rotation at a joint is only resisted to the extent the beam resists it — so the storey stiffness is

$$k = \frac{12E}{h^2}\left[\left(\sum \frac{I_c}{h}\right)^{-1} + \left(\sum \frac{I_b}{L}\right)^{-1}\right]^{-1}$$

which is the standard subassembly result and reads exactly like two springs in series, because it is.

Put ordinary members into one bay: a 3.5 m storey with two 305 × 305 × 137 columns ($I_c = 3.28\times10^{-4}$ m⁴) and one 533 × 210 × 92 beam over a 6 m span ($I_b = 5.52\times10^{-4}$ m⁴):

$$\sum \frac{I_c}{h} = 1.87\times10^{-4}, \qquad \sum \frac{I_b}{L} = 0.92\times10^{-4}\ \text{m}^3$$

The beam is the softer half by a factor of two, and it contributes 67 per cent of the storey’s sway flexibility — despite having the larger second moment, because its span is 6 m against the column’s 3.5. The pair gives $k = 12{,}700$ kN/m, and the $7.0\times10^5$ kN of shear rigidity read off the drift above is about sixteen bays of this kind, which is what a perimeter frame on a plan of this size has.

So the instinct to answer a sway problem by enlarging the columns is mostly wasted. Doubling $I_c$ here takes $k$ from 12,700 to 15,200 — 20 per cent — while doubling $I_b$ takes it to 19,100, or 50 per cent, and doing both gives the full factor of two. It is $I/L$ rather than $I$ that is being summed on each side, and the beams have the long $L$.

That is also why a frame’s stiffness collapses when the bays get wider. Going from 6 m to 9 m bays with the same beam divides $\sum I_b/L$ by 1.5 and takes $k$ down to 9,500 — a 25 per cent loss of lateral stiffness from a decision made entirely about the floor plan, before any lateral calculation was run.

Every wall is also a foundation problem #

The lateral load a building resists has to leave it somewhere, and for a wall or core system it leaves through a footprint much smaller than the building.

A 20-storey building 70 m tall with a core 8 m across, under a wind resultant of 800 kN at mid-height, delivers 28,000 kNm of overturning moment into a foundation eight metres wide. Divide: the resultant lands 1.4 m from the core’s centre if the core carries 20,000 kN of gravity load, against a middle third of ±1.33 m — so the foundation is on the point of lifting on the windward side, from a wind the superstructure barely noticed.

The consequence is that a concentrated lateral system concentrates its foundation problem too, and the choice between a core and a distributed system is partly a choice about whether the overturning goes into a few piles in tension or into many columns in compression. A framed tube spreads it over the whole perimeter and the uplift per column is small; a single core takes all of it and needs tension piles, which are a different and more expensive thing than compression piles.

That is also why the systems that work at height are the ones with the largest plan dimension: they reduce the foundation forces in exactly the proportion they reduce the drift, and both improvements come from the same lever arm.

The drift limit is what all of this is for #

None of these systems is close to failing. The wind moment on a 20-storey building is well within what any sensible core can carry, and the calculation above is not a strength calculation at all.

The limit that governs is drift: the top deflection against the height, and the storey drift against the storey height. The building above reaches 58.0 mm over 70 m, which is H/1208 — comfortably inside the H/500 that is the usual expectation. Take the frame away and it is H/478, which is not.

Two quite different things are being protected. The overall drift matters for occupant comfort and for the tolerance of anything spanning between the building and its neighbours. The storey drift matters for whatever is attached: a partition, a cladding panel or a lift guide rail is asked to absorb the difference between one floor’s movement and the next’s, and it usually fails at a fraction of a per cent of the storey height. That is why the shape of the deflection matters as much as its size, and why a system with a large top drift and even storey drifts can be preferable to one with a small top drift concentrated in three storeys.

The same reasoning applies to the interacting pair with the numbers moved and the shape preserved: the worst storey is not at either end, and the total drift does not say where it is. A stiffening measure that improves the roof figure by taking the bending component down can leave the worst storey exactly where it was.

And the lateral system decides more than the static drift. The storey stiffnesses that resist wind are the ones that fix the building’s natural periods, so the same choice governs the response to an earthquake and to gusting — a matter this collection treats as a set of periods rather than one. A stiffening measure changes every one of those answers at once, and not all of them for the better.

Scale changes which system is possible #

Everything above was a 20-storey building. The arithmetic does not survive being scaled without changing the answer.

The fourth power is doing that. A wall’s contribution falls away as the building grows, so the systems used at height are the ones whose stiffness comes from plan dimensions rather than from a core: the framed tube, the braced tube, the outrigger system, the bundled tube. Each is an attempt to use the full width of the building as the depth of a cantilever, which is the same statement about depth that governs a beam, applied to a building standing on end.

That is the sense in which a tall building is a different structure rather than a bigger one. The lateral system is chosen by the height, and the height is limited by the lateral system, and the vertical load-carrying arrangement has almost nothing to do with either. What the two arrangements share is that the forces in both follow stiffness rather than load, so every part of the answer depends on every other part.

Second-order effects arrive last and decide the argument #

A leaning building carries its own weight eccentrically, and the moment that produces is a load nobody applied.

For most buildings the amplification is 5–15% and the design deals with it by amplifying the first-order answer. For a very tall or very flexible one it becomes the governing consideration, and the honest reading is that the lateral system’s stiffness has stopped being a serviceability matter and become a stability matter: a building whose sway amplification approaches two is a building approaching its own buckling load, and no amount of member strength addresses that.

The choice is made before any of this is calculated #

One more thing is worth saying plainly, because everything above reads as an optimisation and none of it is one in practice.

The lateral system is chosen at the first sketch, from a very short list, and on grounds that are mostly not structural: a core is where the lifts and the stairs are, so a building with a central core has one because the lifts had to go somewhere. A perimeter frame keeps the plan open and puts columns where the facade wants them. An outrigger costs a plant floor. A braced bay blocks a window.

What the arithmetic then does is tell the designer whether the arrangement that the building wants is stiff enough, and by how much it misses. The calculation is a check on a decision made by a plan, and that is a different activity from choosing the stiffest system available — which would be a braced tube on every building over twenty storeys, and which nobody builds because nobody wants a facade of diagonals over their windows.

The one place the structure genuinely leads is at the top of the range. Past about sixty storeys the systems that are stiff enough are so few that the structural arrangement dictates the plan, the facade and the internal layout — which is why very tall buildings look like each other and moderately tall ones do not.

What the picture cannot show #

The wall is a prismatic cantilever with no openings in it. A real core is perforated by doors at every level, so it behaves as a pair of coupled walls whose stiffness depends on the beams over the openings — and those coupling beams carry enormous shears and are the most heavily worked members in the building.

The floors are rigid in their own plane and the model has one lateral freedom per level. Where a floor is long, thin, or has large openings, the two systems do not move together, the diaphragm bends, and the sharing computed above is optimistic.

Nothing here is three-dimensional. A building with its core off centre twists in plan under a wind that pushes it sideways, and the torsional response is often the case that governs the corner columns. That requires a plan model rather than an elevation, and it is where the missing three equations bite hardest.

Where the ladder goes #

The first rung is the coupled wall — two walls and the beams between them — which is the commonest real core and whose behaviour interpolates between two independent cantilevers and one big one, depending entirely on how stiff the coupling beams are.

The second is the tube, where the perimeter of the building becomes the structure and the interesting quantity is shear lag: the corner columns take more than their share and the middle of each face takes less, so the tube behaves as though it were narrower than it is — the same effect a bolted angle suffers, at a hundred times the scale.

The third is the one this essay has treated as static and is not. Wind is a fluctuating load and a tall building’s response to it is dynamic; what a designer is really controlling with all of this stiffness is an acceleration at the top floor, and the limit that governs is a matter of what makes people uneasy rather than anything about the structure.