The storey that cannot see the building lean

Two essays on sway stability ended on the same sentence. Held, and not held set out the alignment chart for one column in one storey and conceded that “a multi-storey frame buckles as a system. The critical mode may involve several storeys at once, and the storey-by-storey calculation each column’s $G$ implies is a decomposition of convenience.” Counted, not checked, on the leaning columns a storey carries, said that “a storey-by-storey check can miss it — which is the storey’s version of the same trap this page is about, one level up.”

This essay makes the check and the whole-building analysis side by side, and finds that the storey check does miss something — but not the thing the warning expected, and not on the side either essay implied.

The check, and the building #

The storey-by-storey method is Horne’s, from 1975, and it is written into the steel code as the way to decide whether a frame needs second-order analysis. Push the frame sideways with a small horizontal load and measure each storey’s drift $\delta$. Then each storey has its own sway factor

$$\alpha_i = \frac{V_{H,i}}{V_{P,i}}\cdot\frac{h}{\delta_i}$$

the ratio of the storey shear the push produces to the gravity load the storey carries, times the storey height over its drift. The reasoning is that a storey’s gravity load $V_P$ acting through its drift $\delta$ is a sideways shear $V_P\delta/h$ — the P-Δ shear of the load that makes itself worse — and the storey becomes unstable when that equals what its stiffness can resist, $V_H$ for a drift $\delta$. The least $\alpha_i$ is taken as the frame’s critical load factor, and the same number, written as a stability coefficient $\theta = 1/\alpha$, is how the seismic codes decide storey by storey how much to amplify each storey’s forces.

The building here is the one every tall-building text starts from: a flexural spine — a braced core or a wall, bending as a cantilever — tied at every floor to a racking frame, each of whose storeys is a shear spring. One number places it between the two, $\alpha H = H\sqrt{S/EI}$, with $S$ the frame’s racking rigidity and $EI$ the spine’s bending rigidity: near nought a pure bending tower, large a pure racking frame. How a tall building stands still drew these two shapes and found that tying them together is where tall buildings live. The building has twenty storeys, and its gravity load is scaled so that its own elastic critical load factor — the eigenvalue of the whole assembly, which the storey method is trying to estimate — is 5.

Where each building drifts #

The drift profiles are the whole of the story, and they are opposite. A bending tower’s storeys drift more the higher they are, because each floor rotates with the whole of the spine below it: the ground storey, fixed at its base, hardly moves relative to the ground, and the top storey drifts fourteen times as much. A racking frame’s storeys drift most near the bottom, where the storey shear is largest, and least at the top. The dual building has its worst drift halfway up.

Every storey reads a different critical factor #

The storey factors inherit the drifts. The bending tower’s run from 3.88 at the top to more than 40 at the ground — for a building that has one critical factor, 5. The racking frame’s run from 4.5 near the bottom upward, and the dual building’s have their least at the storey that drifts most.

Taken as the storey method intends, the least storey factor is the building’s estimate, and it is low in every case: 3.88, 4.31, 4.52, against 5. The storey method’s critical factor is safe, and for a bending tower it is 22 per cent pessimistic. The limit can be had by hand. A pure bending cantilever with its gravity load spread evenly up its height buckles at $qH = 7.84\,EI/H^2$. Pushed by an even lateral load $w$ per unit height, its top rotates by $wH^3/6EI$, so its top storey drifts $wH^3h/6EI$ and carries a storey shear $wh$ against a gravity load $qh$: Horne’s factor there is $6EI/qH^3$. The ratio of the two is $6/7.84 = 0.77$, the value the sweep below reaches at its bending end. The storey method reads the top storey as the critical one because the top storey drifts most, and it drifts most because it is carried by the rotation of everything below it — rotation the top storey’s own stiffness has nothing to do with.

If that were the whole finding it would be reassuring. It is not, because the same factors are used a second way.

The ground storey’s amplifier #

A storey stability coefficient is also an amplifier. The seismic codes write $\theta = P\,d_r/(V h)$ for each storey — the reciprocal of the storey’s $\alpha$ — and multiply that storey’s forces by $1/(1-\theta)$ when $\theta$ passes a tenth. The logic is the same: the storey’s gravity acting through its own drift is an extra shear, and the storey’s forces grow by the factor it implies.

In the racking frame the storey estimate follows the truth closely, storey by storey, which is what the method was built for: a racking frame — a Vierendeel girder stood on end — really is a stack of storeys, each resisting its own shear with its own stiffness, and a storey’s gravity really does act through its own drift.

In the bending tower the two part company, and in both directions. Above storey 8 the storey estimate is too high — the top storey’s own factor says 1.347 where the building says 1.262. Below storey 8 it is too low, and at the ground it is very low: the ground storey reads its own amplifier as 1.019, a two per cent increase, and the building’s second-order solution gives it 1.199, a twenty per cent increase — ten times as much. That storey carries the largest forces in the building, and its forces are the ones the storey method has amplified least.

The reason is the mechanism the storey formula assumes. It charges each storey with its gravity load times its own drift. But a bending tower’s ground storey hardly drifts; what the gravity above it acts through is the lean of every floor above — the whole building’s sideways displacement, which the ground storey carries as overturning moment in its spine. Its own drift does not see that lean. The storey that is most loaded by the building’s lean is the one that cannot see it.

One lean, one amplifier #

What the storey method misses, the whole building makes simple. A building that bends leans under wind in very nearly the shape it would buckle in — the two curves for the bending tower are two per cent apart at worst, the same closeness of a static shape to a mode that a mode shape notices in dynamics — and a deflection that has the shape of the buckling mode is amplified by gravity as a whole, by the one factor $1/(1 - 1/\lambda)$. So in the bending tower every floor’s displacement, and so every storey’s drift, is amplified by nearly the same 1.25, from 1.199 at the ground to 1.262 at the top. The ground storey’s small drift is amplified exactly as much as the top storey’s large one, because both are parts of one lean.

The racking frame is the opposite case. Its buckling mode gathers into its lowest storeys, where the gravity above is heaviest, and its lean under wind is spread up the height; the two shapes are nearly half the roof’s displacement apart. There, a single global amplifier is the wrong tool and the storey method is the right one, since the frame’s storeys really do fail one at a time.

Wrong in both directions until the building racks #

Swept across every building between the two idealisations, the two errors move together. The critical factor the least storey gives is 0.77 of the truth for a bending tower, rising to 0.94 by $\alpha H = 100$; it is always safe. The ground storey’s estimated P-Δ increase is 0.09 of the truth for a bending tower, 0.37 for a building at $\alpha H = 10$, and only near a pure racking frame does it reach the truth. The storey method is safe as a classification and unsafe as an amplifier, and the second error is the larger one for every building that has a spine.

Most tall buildings have a spine. A braced core is a bending tower, because its storey drift comes mainly from its columns lengthening and shortening — the flexural component of two motions with one name — and a shear wall is a bending tower outright. The buildings that behave as racking frames are moment frames without cores, and those are the low ones.

Two ways to combine the storeys, and a bracket #

The storey factors are not wrong; the minimum is the wrong way to combine them. There is an exact identity that says what the right way is. The building’s critical factor is the smallest value of a Rayleigh quotient, stiffness energy over the work gravity does through the lean, taken over every possible shape of lean. Put the lean the building actually takes under the lateral load into that quotient, and after one summation by parts the stiffness energy becomes $\sum V_{H,i}\delta_i$ and the gravity work $\sum V_{P,i}\delta_i^2/h$. Since each storey’s own factor is $\alpha_i = (V_{H,i}/V_{P,i})(h/\delta_i)$, the quotient is

$$\lambda_R = \frac{\sum_i \alpha_i\,w_i}{\sum_i w_i}, \qquad w_i = \frac{V_{P,i}\,\delta_i^2}{h}$$

an average of the storey factors, each weighted by its gravity load times its drift squared. The storey method takes the least of the $\alpha_i$; the Rayleigh quotient takes their weighted average. The least of a set is below any average of it, and a Rayleigh quotient is never below the true critical factor, so the two bracket the truth: the minimum from below, the weighted mean from above.

How tight each side is depends on the building. For the bending tower the weighted mean is 5.11 against the true 5 — two per cent high — because its lean is so nearly its buckling mode that the Rayleigh quotient is nearly exact. For the dual building it is 5.09. For the racking frame at $\alpha H = 30$ it is 5.74, fifteen per cent high, and at $\alpha H = 100$ it is 6.10, because the racking frame’s lean and its mode are different shapes. The minimum does the opposite: 3.88 for the tower, 4.52 for the racking frame, 4.70 at $\alpha H = 100$ and converging on the truth as the frame racks.

So each combination is right for one kind of building. For a building that bends, the gravity-weighted average of its storey factors is its critical factor to two per cent; for a building that racks, the least storey factor is. The storey method as written uses the second everywhere, which is safe for the first kind of building and 22 per cent pessimistic — and then reuses the individual factors as storey amplifiers, which is where the danger was. The weighted mean has no such second use: it is one number for the building, and the amplifier that goes with it is one amplifier for every floor.

The weighted average costs nothing the storey check does not already have. It needs each storey’s drift under the lateral load and each storey’s gravity load — the two columns of the storey check’s own table — and one more column, their product with the drift, to weight by. And the drift profile itself says which end of the bracket to trust. If the ground storey drifts a small fraction of the storeys above it, the building bends, its lean is close to its mode, and the average is the number; if the ground storeys drift most, the building racks and the least storey is. A profile in between — the dual building’s, worst halfway up — is where both are within about fifteen per cent and a proper eigenvalue analysis earns its keep.

By hand, for the bending tower’s ground storey #

The ground storey’s own factor can be estimated in a line. A uniformly loaded cantilever’s slope near its base is $wH^2z/2EI$ at a height $z$, so the ground storey’s drift is about $wH^2h^2/4EI$, and its storey shear and gravity are $wH$ and $qH$. Its factor is

$$\alpha_1 = \frac{wH}{qH}\cdot\frac{h}{wH^2h^2/4EI} = \frac{4EI}{qH^2h} = \frac{4H}{h}\cdot\frac{EI}{qH^3}$$

and with $qH^3 = 7.84\,EI/\lambda$ that is $\alpha_1 = 4(H/h)\lambda/7.84$. For twenty storeys and $\lambda = 5$, $\alpha_1 = 4 \times 20 \times 5/7.84 = 51$: the ground storey of a twenty-storey tower reads a critical factor ten times the building’s, and its amplifier is $1/(1 - 1/51) = 1.02$. The truth at the ground, from the figures, is 1.20. The formula also says the error grows with the number of storeys: a forty-storey bending tower’s ground storey reads twenty times the building’s factor.

What a designer should take from it #

Three things, in the order they matter.

For a building with a spine, amplify the lean, not the drifts. A single amplifier $1/(1-1/\lambda)$, with $\lambda$ the building’s critical factor from an eigenvalue analysis or a conservative estimate of it, applied to every floor’s displacement and so to every member’s sway forces, is within five per cent of the second-order truth at every floor of the bending tower. The storey coefficient, applied storey by storey, is ten times too small at the storey that matters.

The least storey factor is a safe classification. If the question is only whether second-order effects need considering at all — the steel code’s “$\alpha_{cr} \geq 10$” — the least storey factor answers it safely for every building here, and for a bending tower pessimistically.

The ground storey’s P-Δ is overturning, not shear. In the spine, the extra forces gravity adds at the base are a moment — the building’s weight times its lean — carried as axial force in the core’s columns or as bending in the wall. A storey-shear formula cannot find them, however it is amplified, and they are the forces the foundations see. The arm that makes the columns work is the structural answer to the same moment: an outrigger that turns the core’s lean into column forces is also resisting its P-Δ.

Where the model stops #

The spine and the frame are continuous up the height and uniform. Real buildings step: the spine thins upward, the frame changes at a transfer level. A soft storey in a racking frame is exactly what the storey method finds best; a soft storey in a spine-dominated building is found neither by the storey method nor by the single amplifier, because the mode gathers into it and the lean does not.

The floors are rigid in plan and the building does not twist. A building with its spine off-centre sways and twists at once, and its critical mode may be torsional, which neither the storey method nor this model contains.

The load is an even push up the height. A triangular wind or seismic profile moves the drift profiles a little but not their character: the bending tower’s ground storey still drifts least and carries the most lean.

What the pictures cannot show #

That the storey method’s two readings are usually made by different people. The steel designer reads the least storey factor to decide whether second-order analysis is needed; the seismic engineer reads each storey’s coefficient to amplify its forces. The first reading is safe here and the second is not, and on a building with a braced core the second is the one that sizes the base of the braces.

Nor can they show the base that is not quite fixed. A spine on a flexible foundation rotates at its base, which adds a rigid lean to the whole building and makes the ground storey’s own drift even less representative of the lean above it.

Still open: the building that is two buildings #

The spine and the frame here share every floor, so they share one mode. A building whose spine stops partway — a podium with a tower on it, or a core that ends below the top storeys — is a racking frame above and a bending tower below, with an interface where the storey method’s two errors meet. Whether the storey coefficient at the interface is right, wrong or wrong in a new direction, and whether the single amplifier survives a mode that changes character halfway up, is the question that takes this argument from the idealised building to the ones with setbacks that are actually built.