Two motions with one name

Push a tall building sideways and it moves in two different ways at once.

A frame racks. Each storey is a rectangle that becomes a parallelogram: the columns bend in double curvature between floors, the beams bend to let them, and the sideways movement accumulates a bit at every level. How much each storey contributes depends on the shear passing through it, so the bottom storey — carrying the whole wind above it — racks most.

A cantilever bends. A shear wall or a core rotates about its base and its rotation accumulates upward, so the movement at any level is the integral of everything below. The sideways movement per storey grows with height, and the top storey moves most relative to the one below it.

Every real building does both, and the two profiles are the wrong way up for each other.

Which free body produced the number #

Take the building as a vertical cantilever with two independent flexibilities, and cut it at height $z$.

The bending flexibility relates the curvature to the moment of everything above the cut: $EI\,v'' = M(z)$. For a uniform wind $w$ over a height $H$ that integrates twice to

$$v_{bend}(z) = \frac{w z^2 (6H^2 - 4Hz + z^2)}{24EI}$$

The shear flexibility relates the slope directly to the shear: $GA\,v' = V(z)$, which integrates once to

$$v_{shear}(z) = \frac{w z (2H - z)}{2GA}$$

Two integrations against one. That is the whole difference, and everything else follows from it: a quantity obtained by integrating twice is small where the integrand has not accumulated, and a quantity obtained by integrating once is large where the integrand itself is large.

At the roof, $v_{bend} = wH^4/8EI = 222$ mm and $v_{shear} = wH^2/2GA = 144$ mm. At the base storey the first contributes essentially nothing and the second contributes all of it.

One number decides the split #

Non-dimensionalise and everything collapses to a single group:

$$\alpha H = H\sqrt{\frac{GA}{EI}}$$

The ratio of the two roof contributions is $(\alpha H)^2/4$, so below about one the racking term is four times the bending one and the profile is largest at the base; above about six the bending term is nine times the racking one and the profile grows upward. Between them the two mechanisms are comparable and the profile has an interior maximum.

The building here sits at $\alpha H = 2.48$, which is where most real tall buildings sit — not by design but because the two stiffnesses come from the same members and scale together. A core sized for strength has a bending stiffness of a certain order; the frame around it has a shear stiffness of a certain order; the ratio does not vary as widely as the buildings do.

Its worst storey is number 13 of 30. Racking’s worst is storey 1. Bending’s worst is storey 30. Their sum’s worst is neither of theirs, and it is at neither end of the building.

Why the roof number is the wrong one #

Roof drift is what gets quoted, limited and compared between buildings. It is height over 500, height over 400, whatever the client’s brief says.

It is not what damages anything.

What damages things is inter-storey drift — the relative movement of one floor with respect to the one below — because that is the deformation imposed on everything spanning between two floors. A cladding panel, a partition, a lift guide rail, a stair, a riser: each is fixed at two levels and each is racked through the drift of the storey it sits in.

The two are related by an average and not by a bound. The roof drift here is 366 mm over 120 m, which is 1 in 328, and if every storey drifted equally that would be every storey’s answer. The worst storey drifts 1 in 296, eleven per cent worse. On a building with a more extreme $\alpha H$ the discrepancy is larger: a pure cantilever’s top storey drifts about twice its average, and a pure frame’s bottom storey rather more than that.

A roof-drift limit is therefore a proxy that is right on average and wrong where it matters, and it is used because it is easy to state, easy to compare, and available from a single number in an output file.

The clearest way to see how little the roof figure carries is to draw a second building that reports the same one.

Stiffness applied at the wrong end #

The practical consequence of the two profiles is that stiffness is directional in its effect, and a measure aimed at the wrong quantity does nothing where it is needed.

Adding bending stiffness — a bigger core, wider-spaced mega-columns, an outrigger — flattens the upper part of the profile and barely touches the base. It reduces roof drift efficiently and reduces the bottom storey’s drift by almost nothing.

Adding shear stiffness — closer columns, deeper spandrels, more bracing bays — reduces the lower storeys and has little effect at the top.

So a building failing its drift check at storey 13 needs to know which component dominates there before anybody spends money. Reading only the roof figure, which is 61 per cent bending here, would point to the core; reading the storey-13 split points elsewhere.

Where the two components come from #

Both stiffnesses are assembled from members, and the assembly is worth stating because it explains why the ratio is what it is.

$EI$ for a core is the second moment of the walls about the building’s own centroid, which is enormous — it goes as the plan dimension squared times the wall area, and the plan dimension is tens of metres. For a framed tube it is the columns’ areas about the same centroid, reduced by shear lag, which is a real and often large reduction.

$GA$ for a frame is not a shear area at all; it is a fiction that reproduces the racking flexibility of a storey. A storey racks by the columns and beams bending, and the equivalent shear stiffness comes out as a harmonic combination of a column term and a beam term — so the softer of the two governs, and a frame with strong columns and shallow beams is a soft frame.

That harmonic form is the same one that appears in a built-up column’s shear flexibility, and for the same reason: two deformations in series, where the answer is dominated by whichever is larger.

The worst storey’s height is known from the massing #

The group $\alpha H$ decides everything about the profile, and it is worth writing it in a form a designer has before any member has been sized.

At the roof the two contributions are $wH^4/8EI$ and $wH^2/2GA$, so their ratio is

$$\frac{v_{bend}}{v_{shear}} = \frac{H^2 GA}{4EI} = \frac{(\alpha H)^2}{4}$$

which at $\alpha H = 2.48$ gives 1.54, and therefore the 61 per cent bending share quoted above.

Now ask what $\alpha H$ depends on. For a building whose lateral system scales with its plan — a core proportional to the footprint, a frame at a fixed column spacing — the bending rigidity goes as the plan dimension to the fourth power and the shear rigidity as its square. So $GA/EI$ goes as $B^{-2}$, and

$$\alpha H \;\propto\; \frac{H}{B}$$

$\alpha H$ is the building’s slenderness, near enough, times a constant that belongs to the structural system rather than to the size.

That is a genuinely useful thing to know at concept stage, because it makes the location of the critical storey a consequence of the massing rather than a discovery in the analysis:

A slender tower is bending-dominated and its worst storey is high up. A broad building of the same height is racking-dominated and its worst storey is at the bottom. The building drawn, at $\alpha H = 2.48$, has its worst storey at 13 of 30 — 43 per cent of the height — and that number was fixed by the plan and the elevation before anything structural was chosen.

Three buildings, one roof drift, and the worst storey at the bottom, at 43 per cent of the height and at 77 per cent of it. The quantity that moves it is the ratio of the two stiffnesses and nothing else — not the height, not the wind, and not the number the drift check is written against.

Two qualifications keep it honest. The proportionality assumes the lateral system grows with the plan, and a great many buildings have a core sized by lifts and stairs rather than by structure — a fixed core in a growing footprint drives $\alpha H$ up and the building toward cantilever behaviour whatever its slenderness suggests. And the constant of proportionality is not small: a braced frame, a moment frame and a shear-wall building of identical proportions sit at quite different $\alpha H$, which is the system’s contribution and is the one thing on this page a designer chooses.

There is a corollary worth carrying into the massing conversation itself, because it is one of the rare structural facts that is useful before a structure exists. Making a building narrower at constant height does not merely make it more flexible — it changes the kind of flexibility it has, moves the critical storey upward, and changes which stiffening measure will work. A tower that has been slimmed during design is not the same drift problem it was, scaled; it is a different problem with its worst storey somewhere else and a different answer to what should be stiffened.

What the relationship removes is the surprise. A drift check that fails at storey 13 of a moderately slender tower is not an anomaly to be investigated; it is where the check was always going to fail, and it could have been predicted from a sketch.

The drift that is not from wind #

Two other things move a tall building sideways, and both interact with the profile above.

Second-order effects. A building that has drifted carries its weight off the vertical, which produces an additional overturning moment and therefore additional drift. The amplification is $1/(1-\theta)$ with $\theta$ the stability index of the storey, and it is a storey quantity — so it amplifies most where the drift is largest and the axial load is greatest, which is the lower part of the building. A storey at $\theta = 0.15$ has its drift amplified by 18 per cent.

Plan torsion. A building whose centre of stiffness is not its centre of mass twists as well as translating, and the corner moves most. The drift at a corner can be half again the drift at the centre of the plan, and cladding is on the corners.

Both act on the inter-storey drift rather than on the roof figure, and both make the discrepancy between the two worse.

The limit is not one number either #

Once the two profiles are separated, the drift limit itself stops being a single figure and becomes a small family of them, each attached to something that is being protected.

Cladding is racked through the storey drift and its tolerance depends on how it is fixed. A panel on a sliding fixing accommodates a great deal; one on four rigid brackets accommodates the tolerance of the joints and no more. Limits of height over 400 to height over 300 per storey are typical, and the number is really a property of the facade specification.

Partitions and finishes are the sensitive case, and the relevant drift is the one after they were installed — which excludes everything the frame did during construction and includes only the wind and imposed movement afterwards.

Lift guide rails have a straightness requirement over their whole length that is a curvature limit rather than a drift limit, and it is the one criterion in the list that cares about the shape of the profile rather than its slope.

Occupant comfort is not a displacement at all.

Four criteria, three of which are about the storey rather than the building, and one of which is about neither. The habit of quoting a single roof-drift ratio survives because it is one number and because it correlates loosely with all four — and the correlation is the same average-not-a-bound relationship this whole essay is about.

Where the model stops #

The two mechanisms are treated as independent and simply added. In a real building they are coupled at every floor by the diaphragm, which is what makes a wall-and-frame pair stiffer than either — the wall holds the frame at the bottom and the frame holds the wall at the top. Adding the two profiles is a good approximation and it misses that interaction, which flattens the sum further.

$GA$ is a fiction fitted to a storey. It reproduces the racking flexibility of a regular frame and stops meaning anything where the frame is irregular — a transfer level, a setback, a missing column.

And the wind is uniform. A real wind profile grows with height, which shifts load upward, increases the bending share and moves the worst storey up.

Reading the split from an analysis that does not report it #

A frame analysis of a real building reports node displacements and nothing else. The decomposition into bending and racking is not an output, and it is worth saying how to recover it, because the recovery is what turns the observation above into a usable check.

Take two floors and look at the rotation of the storey between them. A purely racking storey has parallel floors: the columns bow but the floors stay horizontal, and the storey’s drift is entirely a shear distortion. A purely bending storey has floors that have rotated: the drift comes from the accumulated rotation below, and the storey is a rigid body tilted.

So the bending share of a storey’s drift is the floor rotation times the storey height, and the racking share is what is left. Both are available from the same output: subtract the tilt from the drift and the remainder is the shear. The check that follows — is the governing storey short of racking stiffness or of bending stiffness — costs two subtractions per storey and answers the question that decides where to spend money.

It also gives a warning that no drift figure gives. A storey whose racking share suddenly jumps is a storey where something in the frame has changed: a column has been transferred, a bracing bay has stopped, a beam has been made shallow for a services run. That discontinuity is a soft storey, and it is visible in the decomposition long before it is visible in the total.

What the pictures cannot show #

The whole calculation is static, and wind is not. The drift that damages cladding is the peak of a fluctuating response, and the drift that annoys occupants is an acceleration rather than a displacement — a quantity that does not appear anywhere on these figures and that frequently governs the design of a tall residential building outright.

Nor can the figures show the sequence. A building is built storey by storey and the columns shorten as the load accumulates, so the frame that resists wind is not the frame in the drawings; the differential shortening between core and columns has already put the floors out of level before any wind arrives.

A last omission: the figures show one direction. A building is asymmetric in plan more often than not, so the two stiffnesses differ between the two principal directions and $\alpha H$ has two values — a building can be a cantilever one way and a frame the other, with its worst storey at the top in one direction and near the base in the other.

The assumption the figure rests on #

The shear rigidity is 6.0 × 10⁵ kN and the bending rigidity 1.4 × 10⁹ kNm², both taken as constant up the height. Neither is: columns and walls reduce in size with height, so both stiffnesses fall — and they fall at different rates, which means $\alpha H$ is a function of height rather than a property of the building. A tall building’s real profile is the sum of two curves computed on a structure that changes, and the two-parameter model above is a first sketch of it rather than a description.

The same two mechanisms appear in plan rather than in elevation, which is the last thing the two-parameter model leaves out. A floor plate is a deep beam lying down, with its own bending and shear flexibilities, and the drift at a corner depends on which of the two dominates in exactly the same way — so a building can be bending-governed up its height and racking-governed across its plan, with the worst movement anywhere in the building at an intersection of the two.

The ladder from here #

Later rungs on this anchor: the coupling between the two mechanisms through the diaphragm, which is the wall-and-frame interaction proper. Second-order amplification storey by storey, and the stability index that decides when drift stops being a serviceability quantity. Drift under a realistic wind profile rather than a uniform one. The acceleration criterion, which is what actually governs tall residential buildings. And drift in seismic design, where the demand is a displacement rather than a force and the whole hierarchy of what to stiffen is turned upside down.