The hat that hides the lean

A storey-by-storey sway check asks each storey of a building one question: how much does its own gravity load, acting through its own drift, add to its own sway? The answer is the storey’s sway factor, $\alpha = (V_H/V_P)(h/\delta)$ — the storey shear over the storey’s gravity, times the storey height over the drift that shear causes — and the amplification it predicts is $1/(1 - 1/\alpha)$. Held, and not held showed why a restraint changes everything about a column’s critical load. The storey that cannot see the building lean put that check against a building of a bending core and a racking frame sharing every floor, and found it wrong in both directions at once: too pessimistic where the frame governs, too optimistic in the lower storeys where the core bends and the storeys lean with everything above them.

The storey above the core decides took the core away from the upper floors and found the check exact at the interface, because a racking frame on a stiff base is the one structure for which a storey’s own lean is the building’s. Its last section named the opposite case, and it is the commoner one. A tall building’s core usually runs to the roof, and at the roof — or at a plant floor part-way up — a hat truss or an outrigger ties it to the perimeter columns. That restraint makes the columns work with the core: as the core tries to rotate, the arms push one line of columns down and pull the other up, and the columns’ axial stiffness resists the rotation.

The question is what the storey check makes of a building whose core is held at the top.

The building, held at the top #

Take twenty storeys, a core and a racking frame sharing every floor, with the two balanced so that the building’s $\alpha H$ — its height times the square root of the frame’s racking stiffness over the core’s bending stiffness — is 1. The core does most of the lateral work, as it does in most tall buildings. Scale the gravity load so that the building’s own elastic critical factor is 5, which is the range where second-order effects are not negligible but are not dominant. Load it with the same lateral force at every floor.

Then add a hat truss at the roof whose rotational stiffness on the core is ten times the core’s own $EI/H$ — a moderate arm on perimeter columns of ordinary size.

Without the hat, the top storeys read the least sway factor: 4.04 at storey 17, 4.06 at the roof, all within a few per cent of each other and all below the building’s critical factor of 5. The check is pessimistic there, as the shared-building essay found, because the frame’s storeys near the top are where the frame does most of its racking and the storey sees it.

With the hat, the top storeys read very different numbers. The roof storey’s sway factor is 15.47, the one below 11.1, the one below that 8.7. The hat has held the top of the core, the core no longer rotates freely there, and the roof storey’s first-order drift under the same lateral load has fallen to about a seventh of what it was. The least sway factor in the building has moved down to storey 9, where it reads 4.14 — still below 5, still on the safe side, and now at mid-height.

With the same restraint placed at storey 10 instead of at the roof, the picture inverts again: the least factor is back at the top, 3.85, and the outrigger’s storeys read a local bulge where the core is held.

The building still leans #

A sway factor of 15 at the roof says that the roof storey, considered alone, would amplify its drift by $1/(1 - 1/15.47) = 1.069$: 6.9 per cent. That is what a storey with that much stiffness under that much gravity would do. The question is what the roof storey actually does.

The true second-order increase in the roof storey’s drift is 24 per cent. Down the whole height, the true increase stays between 20 and 26 per cent, close to the building’s own $1/(1 - 1/5) = 1.25$, because one buckling mode dominates the response and every storey’s drift is amplified roughly as that mode is. The storey estimate wanders from 3 per cent at the ground to 32 at mid-height and back to 7 at the roof.

Without the hat the roof storey estimated 33 per cent against a true 26 — too much, which is the safe direction. Under the hat it estimates 6.9 against 24: 28 per cent of the truth. The hat has moved the top storeys from the side of the check that wastes material to the side that misses a load. At storey 9, where the least factor now sits, the estimate is 32 per cent against 26, still pessimistic.

Drift the top storeys did not make #

The storey estimate’s assumption is that each storey’s second-order drift is caused by its own lean: its own gravity load, acting through its own drift, resisted by its own stiffness. Drawing what that assumption predicts against what happens storey by storey shows where it fails.

Without the hat the roof storey’s own lean would add 0.325 of the largest first-order storey drift, and it actually gains 0.263: its own lean overstates what happens, because the core below it carries some of the roof storey’s P-Δ shear down to storeys with spare stiffness. Under the hat the roof storey’s own lean would add 0.019, and it gains 0.065 — three and a half times as much.

The difference is drift the roof storey did not generate. It belongs to the storeys below, whose gravity loads lean through their drifts and whose P-Δ shears the core carries up the building. A core is a continuous beam; a moment put into it at storey 8 bends it at storey 18 as well. With the hat holding the core’s top rotation, the top storeys are where the core’s bending under the lower storeys’ P-Δ shows as racking of the frame, and the top storeys’ own stiffness — which the hat has made large — is the wrong denominator for it.

This is the general reason a storey check fails: it is exact only when every storey’s second-order drift comes from its own lean, which is to say when the building’s first-order deflected shape and its buckling mode distribute the lean in the same proportions. A racking frame on a stiff base does that. A core and frame together almost never do, and a hat on the core makes the top storeys the place where they differ most.

The stiffer the hat, the worse the reading #

The hat’s stiffness is a design choice, and it moves the error continuously.

With a very weak hat, a tenth of the core’s $EI/H$, the roof storey’s estimate is about 1.2 times its true second-order drift — the shared building’s safe-side error. The estimate falls through the truth at a hat stiffness of about 0.7 times the core’s $EI/H$, is 0.28 of it at 10, and 0.07 at 1,000, where the hat is effectively rigid and the roof storey reads a sway factor of 60.

Meanwhile the building’s critical load rises: to 1.81 times its unhatted value at a stiffness of 10, to 1.98 at 1,000. The hat is doing what it was designed to do. The building is much stiffer and much stronger against sway, and the least storey factor stays between 0.82 and 0.84 of the building’s own critical factor throughout — so a designer who takes the least factor in the building as the building’s, which is how EN 1993-1-1’s storey formula is normally used, remains on the safe side.

The danger is in reading each storey’s own factor as that storey’s amplifier. Any procedure that does — a per-storey amplifier for sway moments, such as the B2 factor of the American steel specification’s approximate second-order analysis, or a decision that storeys reading α above 10 may be analysed to first order — reads the top storeys under a stiff hat as needing almost nothing. They need a quarter more.

Where the outrigger sits #

A hat is an outrigger at the roof, and an outrigger can be placed anywhere. The question of where is usually answered for drift; the storey check gives it a second answer.

Holding the arm’s stiffness at ten times the core’s $EI/H$ and moving it down the building, the building’s critical load is greatest — 3.06 times its value with no arm — with the outrigger at storey 11, a little above mid-height. That is not where an outrigger is placed for least drift, which depends on the load’s shape and moves with the arm’s stiffness; stability and drift are different questions with different optima, as the arm that makes the columns work found for drift and moment.

The roof storey’s reading moves through the truth as the outrigger rises. With the arm low in the building the roof storey is free, the core above the arm behaves like the cantilever of the shared building, and the estimate is above the truth — up to 1.35 times it. At storey 12 the roof storey’s estimate is exact, 1.00 of the truth. Above that it falls short, to 0.28 with the arm at the roof. So the storey check is exact somewhere new: not at the arm, but at the roof, when the arm sits at about six tenths of the height. That is a coincidence of the building’s proportions rather than a theorem, and it would move with $\alpha H$ and with the arm’s stiffness, but it marks the boundary between the two kinds of error. A building with two arms, which the second arm found worth about half of what the first arm left of the drift, would put the upper arm’s storeys on the optimistic side and leave the storeys between the arms to whichever side their position decides.

The core that does the work #

The effect depends on how much of the lateral work the core does.

Where the core does nearly all of it, at $\alpha H = 0.1$, the hat multiplies the critical load by 2.23 and moves the roof storey’s estimate from 1.35 times the truth to 0.29 of it. At $\alpha H = 1$, from 1.24 to 0.28, with a gain of 1.81. Where the frame does the work, at $\alpha H = 10$, a hat on a slender core changes the critical load by nothing, and the roof storey already reads below the truth without one — about half — and somewhat further below with one. A hat on a frame-dominated building does little for its stability and does not change what the check gets wrong about it; a hat on a core-dominated building transforms both.

That is the uncomfortable conjunction. The buildings where a hat is most worth building — tall, slender, core-dominated — are the ones where it most distorts what the storey check reports, and in the direction that misses load.

What the missing drift costs the columns #

A storey’s sway moments scale with its drift, so an error in the amplification is an error of the same size in the moments the columns are designed for. In the roof storey under the hat the estimate amplifies the first-order sway moments by 1.069 and the building amplifies them by 1.243. A column designed to the storey estimate is designed for 14 per cent less sway moment than it carries — and the same holds, by 12 and 9 per cent, in the two storeys below.

Those are not ordinary columns. Under a hat truss the top storeys’ perimeter columns are the ones the arms push and pull: they carry the hat’s axial forces as well as their own gravity and the sway moments of their storey, so they are among the most heavily loaded members in the building in combined compression and bending. The storey check under-reads their moment exactly where their axial load is highest. In a building without a hat, the top storeys’ columns are the lightly loaded ones and a pessimistic storey reading wastes a little steel there; with a hat, an optimistic reading takes margin from the columns that can least spare it.

Lower down the hat changes less. The least storey factor has moved from the top of the building to storey 9 and reads 4.14 rather than 4.04 — marginally closer to the building’s 5 — so a designer who takes the least factor and applies it everywhere stays on the safe side by about the margin the unhatted building had. The error the hat introduces is local to the storeys it holds, and it is invisible to the one reading of the storey check that is safe.

Why the check cannot see the hat #

A storey free body has three things in it: the shear the storey carries, the gravity load above it, and its drift. It has nothing about where the drift came from. Under a hat, the top storeys’ first-order drift is small because the hat holds the core, and their second-order drift is large because the lean of everything below is carried up the core into them. The storey check divides one by the other and calls the ratio the storey’s stiffness.

The only cure is to stop reading storeys as independent. The building’s own critical factor, found from the whole structure — by an eigenvalue analysis, or from the least storey factor taken as the building’s and applied everywhere — is on the safe side here, because it treats the building as the single buckling mode it is. A second-order analysis of the whole frame is exact by construction. What fails is the convenience of the storey-by-storey reading, in the one kind of building where its convenience looks most justified: storeys that read stiff are given credit for stiffness that belongs to an arm at the top of the building, not to them.

What the model holds #

The building here is a core and a racking frame as continua: no individual columns, no axial shortening of the core, and a hat that is a rotational spring on the core rather than a truss on real columns. The perimeter columns’ own axial flexibility is folded into the hat’s stiffness; their shortening under gravity, which in a real building loads the hat and sends some gravity into the core, is left out, and so is any sway of the hat truss itself.

The lateral load is uniform up the height, and the gravity load is the same on every floor. Under a wind that grows with height, the top storeys’ first-order drift is larger and the storey factors there smaller, which would narrow but not close the gap the hat opens. And the critical factor is held at 5 in every comparison: the hatted building carries more gravity at that factor than the unhatted one, because the hat has raised its critical load. A designer who keeps the gravity load fixed instead would see the hatted building’s critical factor rise to nine, its amplifications fall, and the same proportion of them missed at the top.

The two errors of the storey check #

The storey that cannot see the building lean and the storey above the core decides found the storey check pessimistic where a frame racks freely, optimistic in the core’s lower storeys, and exact above the top of a core that stops. A hat on a core that runs to the roof adds a fourth case: optimistic in the top storeys, by a factor of three or four, because the hat has made their own stiffness large and left their lean unchanged.

In all four the cause is the same. The storey check is right only when a storey’s lean is its own. A building with any continuous spine — a core, a braced tower, a stiff wall — shares its lean among its storeys, and the arms, hats and belts that make such a building efficient are exactly the elements that decide which storeys receive the lean the others generate.

Still open: the hat that is itself a storey #

The hat here is a spring at a floor. A real hat truss is a storey deep — often two — with diagonals that rack and chords that stretch, and it has its own sway stiffness as a storey of the building. Whether the hat storey reads a sway factor near infinity because its diagonals are so stiff, whether the storey immediately below it, which carries the hat’s reactions into ordinary columns, becomes the weakest in the building, and whether a belt truss that ties the perimeter columns together without connecting to the core changes the storey readings at all, is what a hat drawn as a truss rather than a spring would ask of this one.