The storey above the core decides

The storey that cannot see the building lean put a bending spine — a core, a braced tower — and a racking frame side by side on every floor of a twenty-storey building, and found that the storey-by-storey stability check is wrong twice over. The storeys near the top read a sway factor below the building’s, because they drift with the spine’s rotation and the check counts that drift as their own; the storeys near the ground read a factor far above it, because they barely drift, while the whole tower above them leans and their columns carry its gravity through that lean. The least storey factor was 19 per cent pessimistic, and the ground storey’s amplification was under-read by a factor of nine.

That building had its spine to the roof. Many do not. A core stops below the plant floors, a braced tower ends where the floor plate steps in, a podium of shear walls carries a frame above it. The question the earlier essay left was what happens at the interface, where a bending tower below meets a racking frame above and the check’s two errors would seem to meet.

The same building, with the spine stopped #

Keep the building exactly as before — twenty storeys of 3.6 m, a spine and a frame placed between bending and racking by $\alpha H = 1$, an equal lateral load at every floor, and the gravity load scaled so that the building’s elastic critical factor is 5 — and stop the spine at the top of storey 15. Above it, the five storeys are the frame alone.

Each storey’s own sway factor is the storey check’s quantity: the storey’s shear over its gravity load, times its height over its first-order drift. If the storey were a frame on a rigid base, that number would be its own critical factor, and the check reads the least of them as the building’s.

With the spine to the roof the factors run smoothly from above 40 at the ground to 4.04 near the top, and nothing in the column of numbers is the building’s 5. With the spine stopped they break into two pieces. Inside the spine they are large everywhere — above 100 at the ground, 10.5 at the spine’s top storey — because those storeys hardly drift. Above it they jump down, and the first storey of bare frame above the spine reads 4.999. That is the building’s critical factor, to the third decimal.

The interface is not where the check is least reliable. It is the one storey where the check is exact.

Why one storey is the whole building #

The buckling mode says why. With the spine to the roof it is the bending tower’s mode, each floor moving a little more than the one below, all twenty taking part. With the spine stopped, the mode is a step: everything below the spine’s top stays still, the first storey of frame above it racks, and every floor above that storey moves sideways by the same amount, carried along as a rigid block.

That is how a racking frame on a stiff base buckles — every column in the storey leaning on its neighbours as one, because the floor above ties their tops together. A storey of frame has a sideways stiffness and carries the gravity of every floor above it; its own critical factor is that stiffness times its height over that gravity. The first storey above the spine carries the most gravity of any frame storey — five floors’ worth here — with the same stiffness as the others, so it has the least factor and buckles first, and the floors above it ride on top. The spine below acts as a rigid base: at its top it is far stiffer against sideways movement than the storey of frame resting on it.

A storey that buckles on its own on a rigid base is exactly the storey the storey check was written for. Its sway factor is its critical factor, by definition, and its storey estimate of the second-order drift is the second-order drift.

Exact above, short below #

Above the spine the two lines are one line. The storey estimate $1/(1 - 1/\alpha)$ gives 25 per cent extra drift in the first frame storey, which is exactly what a full second-order analysis gives, and smaller exact amplifications in the storeys above.

Inside the spine the earlier essay’s error survives intact. The ground storey drifts very little under the lateral load, so its sway factor is large and its storey estimate of the second-order drift is 0.8 per cent. The true amplification there is 6.4 per cent, eight times as much. The ground storey’s columns carry the gravity of all twenty floors, and those floors are displaced sideways by the whole spine’s lean plus the frame storey’s racking; the moment that gravity makes at the ground is what the storey check never sees, because it looks only at the storey’s own drift.

So the two errors of the shared building do not meet at the interface. One of them disappears: the frame above the spine is checked exactly. The other stays where it always was, in the storeys that look stiff.

When the frame above is too stiff to buckle first #

The exactness rests on the frame above being the part of the building that buckles first. That is a comparison between two candidates — the first frame storey on its rigid base, and the spine with the frame shared below it — and it can go the other way.

Stiffen the frame above the spine and its first storey’s critical factor rises. Up to about twice the stiffness of the frame below, it is still the part that buckles first, and the check reads it exactly. Past about 2.3 times, the frame above is no longer the weakest part. The building’s critical mode moves down into the spine, the least storey becomes the spine’s top storey, and it reads 93 per cent of the building’s factor; stiffer still, 79 per cent — the same order of pessimism as the shared building’s 81.

That is a design condition and not a curiosity. The frame above a core is often the stiffer frame — plant floors with heavy bracing, a hat truss, an outrigger level — and as soon as it is, the storey check returns to reading the shared building’s mode, with the shared building’s errors.

How high the spine can go #

The same comparison can be made against the spine’s height, with the frame the same stiffness above and below.

Stop the spine anywhere up to storey 17 and the first frame storey above it decides the building, exactly. At 18 and above, the frame above has too little gravity on it to be the weakest part — two floors on its first storey rather than five — and the critical mode is the spine’s again. With the spine to the roof the least storey reads 0.81 of the building’s factor, the earlier essay’s 19 per cent.

So there is a sharp change, not a gradual one, between a building the storey check reads exactly and a building it reads with a fifth of pessimism, and the change happens when the top few storeys stop being able to buckle on their own. The check is exact when the building has an obvious weak storey, and uncertain when it does not.

The ground storey under the spine #

The under-reading inside the spine depends on how much spine there is. With the spine stopping at storey 5, the ground storey’s true amplification is 0.5 per cent and the estimate 0.1 — both small, because the stiff spine is short and the frame above takes almost all the lean. With the spine to storey 15, 6.4 per cent true against 0.8 estimated. With the spine to the roof, 19 per cent against 2.1.

The ratio between truth and estimate is between five and nine throughout. What the spine’s height changes is how much there is to under-read. A designer reading the storey check on a building whose core stops at storey 15 is told, correctly, that the frame above is at 25 per cent; is told, incorrectly, that the base of the core is at 0.8; and has no way, from the check alone, to know that the first is exact and the second is an eighth of the truth.

One amplifier, or one per storey #

There are two ways to use the storey factors, and the difference matters more here than in the shared building.

One amplifier for the whole building, taken from the least storey factor, is what EN 1993-1-1 does when it reads $\alpha_{cr}$ from storey drifts, the amplifier that a load which makes itself worse arrives at by summing its own series: find the smallest, and if it is below ten, amplify every lateral effect in the building by $1/(1 - 1/\alpha_{cr})$. With the spine stopped at storey 15 that is $1/(1 - 1/5) = 1.25$ everywhere. In the first frame storey it is exact. In the four frame storeys above, whose true amplifications are 19, 14, 9 and 4 per cent, it is safe. Inside the spine, where the truth runs from 6.4 per cent at the ground to 8.5 at the spine’s top, it is three or four times too large — and safe.

One amplifier per storey, each from its own factor, is the more refined reading and the one the figure of amplifications draws. Above the spine it is exact storey by storey. Inside the spine it is the reading that is eight times too small.

So the stopped spine sorts the two practices cleanly. The building-wide amplifier is exact where the building is weakest and conservative everywhere else; the storey-by-storey amplifier is exact where the building is a frame and unconservative where it is a tower. In the shared building, where the least storey read 19 per cent low, even the building-wide amplifier was 33 per cent against a true 19 to 26 — safe, and nowhere exact. The stopped spine is the case where the check’s simplest use gives the right answer at the storey that decides the building.

What the first frame storey is asked to be #

If the first storey above a stopped spine is where the building buckles, then that storey’s racking stiffness is the building’s stability, and its design is a stability design. Doubling the frame’s stiffness above the spine doubles the gravity the building can carry at the same critical factor — the whole of the improvement goes through that one storey, because nothing else is involved in the mode.

That stops at 2.3 times. Stiffen the frame above any further and the spine’s own mode, shared with the frame below, becomes the weaker one, and the building’s capacity stops rising with the stiffness above. The best frame above a stopped core is one about twice as stiff as the frame below it — stiff enough to lift its first storey’s critical load up to the spine’s own, and no stiffer, because past that point the weakness has moved somewhere the extra stiffness does not reach.

One storey, cut above and below #

The storey check’s free body is one storey, cut above and below, with the storey shear and the gravity of everything above it acting across the cuts — the same free body on which a gravity load is turned into an equivalent lean. Its sway factor is the ratio of the two moments on that free body: the shear times the storey height, which the frame resists, against the gravity times the storey’s own drift, which the frame must also resist.

That free body is complete when the storey’s ends do not rotate — when the floor below it is held, and the floors above it move as one. Above a stiff spine both conditions hold: the spine’s top is the rigid floor below, and the frame floors above the first storey move together in the mode. Inside the spine neither holds: each storey’s floors rotate with the spine, and the drift that matters to the gravity load is the accumulated lean of everything above, not the storey’s own share of it. The storey free body is the right free body for the frame above, and the wrong one for every storey that is part of the bending tower.

The arithmetic of the first frame storey #

Every number above the spine can be written down by hand.

The first frame storey carries the lateral load of the five floors above it, $5H$, and the gravity of the same five, $5P$. Its first-order drift is $5H/k$, with $k$ the storey’s racking stiffness. Its sway factor is

$$\alpha = \frac{5H}{5P} \cdot \frac{h}{5H/k} = \frac{k h}{5P}.$$

That is the critical factor of a storey of stiffness $k$ and height $h$ carrying $5P$ — the gravity at which $5P \cdot \delta / h$ equals $k\delta$. It has no $H$ in it and no spine, and it is the building’s critical factor because nothing else in the building buckles at a lower load. The storey above it carries $4P$ and has a factor of $kh/4P$, a quarter higher; the roof storey carries one floor and has five times the first frame storey’s.

A spine stiff enough to be a base #

The spine is far stiffer than the frame at its top. The exactness depends on the spine’s top being a rigid base for the frame. A slender core whose top rotates appreciably would turn the first frame storey’s floor into a hinge rather than a base, and its sway factor would no longer be its critical factor.

The frame’s racking stiffness is a single number per storey. A real frame’s storey stiffness depends on its beams and columns and on what is above and below it, which is why a frame is a girder stood on end rather than a spring; the model’s frame is a shear spring, which is the idealisation that makes a storey’s factor exact on a rigid base.

And every floor carries the same gravity and the same lateral load. A building that steps in — fewer floors above the spine but heavier plant on them — changes which storey carries the most gravity, and the weakest frame storey may not be the first one above the spine.

What the plots cannot tell a designer #

They cannot tell which part of a real building buckles first. The sharp change between an exact check and a pessimistic one happens where two parts of the building have equal critical factors — a building held in one part and not in another — and no storey-by-storey quantity can say which side of it a building is on. That needs the building’s own buckling analysis — the eigenvalue the check exists to avoid.

They do not show torsion. A core stopping partway up is usually a core stopping off-centre, and a building whose stiff part ends asymmetrically twists at the interface as well as racking.

And they do not show the transfer. Where a frame stands on a stopped core, the frame’s columns may not continue down through it, and a column that stops delivers its load sideways at exactly the level where this essay’s weakest storey begins.

What it comes to #

Stop the spine and the storey check becomes exact above it. The first storey of frame above the spine reads the building’s critical factor — 4.999 against 5.0 — and its second-order drift exactly.

Because the whole building buckles in that storey. The spine stays still, the first frame storey racks, and every floor above moves with it as one block.

Inside the spine the check is as wrong as ever. At the ground, 0.8 per cent estimated against 6.4 true.

And the exactness is lost when the frame above is stiff or short. Past 2.3 times the stiffness of the frame below, or with the spine past storey 17, the mode returns to the spine and the least storey reads a fifth low again.

Still open: the stiff hat on a shared building #

The exact case needs a weak storey above a stiff base. The commonest tall building has the opposite: a core to the roof with a stiff outrigger or hat truss at the top, which makes the perimeter columns work with the core and turns the top of the spine into something held. There the core’s top rotation is restrained, its bending mode changes from a cantilever’s to a propped one, and the storeys just below the hat — which in the shared building were the ones the check read most pessimistically — sit under a restraint that the storey free body cannot see. Whether a hat truss makes the storey check more pessimistic, less, or exact somewhere new is the question that follows this one up the building.