The floor that twists in its own time

A measure of twist that divides by the twist read the seismic standards’ test for a twisting floor — the worst edge’s movement over the average of the two edges’ — against the movement it is meant to flag. Under a static push the ratio saturates. The twist that moves the far edge moves the average too, so the ratio divides the twist by itself, and at a torsional radius a third of the plan’s length it never passes 1.76 however eccentric the plan, while the corner’s own movement keeps growing. That essay’s floor, 30 by 18 m with its centre of rigidity 4.2 m off the centre of mass and a torsional radius of 10.6 m, read 1.49.

Everything there was a push. That essay ended on a floor whose torsional radius is close to its mass’s own radius of gyration, so that it has a twisting mode at nearly the frequency of its swaying one. Shaken, such a floor does not take up the push’s deflected shape: it sways and twists in two modes that share the motion. The standards’ dynamic procedures read the same ratio from a response-spectrum analysis, and the question left was whether that ratio keeps the static one’s saturation or loses it, and whether a floor can read regular one way and irregular the other.

The floor given mass #

The floor is the earlier essay’s, given mass spread uniformly over its 30 × 18 m: a radius of gyration of 10.1 m about its centre of mass. Its walls have a translational stiffness and a torsional stiffness about the centre of rigidity, and the ratio of the two radii, $\Omega = r/\rho$, is the ratio of the floor’s uncoupled twisting frequency to its swaying one. On its own, without the eccentricity, the floor would sway with a period of 0.5 s. The ground shakes along the push with a design spectrum flat to 0.6 s and falling beyond it.

With the centre of rigidity off the centre of mass, sway and twist cannot happen separately. The floor has two modes, each a sway and a twist together, and each is excited by the ground in proportion to how much it sways. A response-spectrum analysis finds each mode’s peak, works out what each edge does in each mode, and combines the two modes’ edge movements into one peak per edge — here by the CQC rule at 5% damping, which allows for the two modes’ peaks arriving partly together. The ratio is then the larger combined edge over the average of the two.

Two modes, each a sway and a twist #

The earlier essay’s floor has Ω = 1.05: its torsional radius is almost exactly its radius of gyration, so on its own it would twist at nearly the frequency it sways. Two oscillators tuned to nearly the same frequency and coupled together share their motion between two modes rather than keeping one each — the effect the mass that helps by being late put to work on purpose — and here the coupling is the eccentricity. Coupled by its 4.2 m, the floor’s two modes are 0.60 and 0.40 s. In the first, the floor turns about a point 1.2 m inside its near edge, which barely moves while the far edge moves 1.95 times as far as a torsion-free floor would. In the second, it turns about a point three quarters of the way across, 22 m from the near edge, and the near edge takes the motion.

Combined, the far edge moves 1.95 times a torsion-free floor’s and the near edge 0.67, against 1.72 and 0.59 for the static push. The ratio is 1.49 both ways — a coincidence of this floor. The far edge moves 13% further dynamically than the push says, and the ratio does not show it.

The verdicts part on either side of Ω = 1 #

Across Ω the two readings do opposite things. The static ratio falls steadily as the floor gets stiffer in torsion: at 4.2 m of eccentricity, from 2.46 at Ω = 0.5 to 1.15 at Ω = 2, because a stiffer floor twists less under the same torque. The dynamic ratio rises from 1.17 at Ω = 0.5 to a peak of 1.55 near Ω = 1.2 and then falls with the static one. At 2.1 m of eccentricity the same: the static ratio falls from 2.05, the dynamic peaks at 1.42 near Ω = 1.15.

The two curves cross close to Ω = 1. Above the crossing the dynamic ratio is the larger: a floor slightly stiffer in torsion than it sways has its two modes close together, each with a large twist, and the far edge collects both. A floor 1.5 m eccentric at Ω = 1.14 reads 1.17 statically, regular, and 1.37 dynamically, irregular and near the extreme limit. Below the crossing the static ratio is far the larger, and that is the half of the figure the earlier essay could not have predicted.

The corner itself #

The ratio was always meant to stand for the corner’s movement, and the corner’s movement tells a simpler story. At Ω = 0.5 the static push moves the far edge 4.16 times as far as a torsion-free floor; the response spectrum, 2.19. A floor that flexible in torsion has its twisting mode at a long period, where the spectrum asks for less, and the ground excites that mode only through the little sway it carries. The push, which knows nothing about periods, applies the whole storey force as a torque. From Ω = 1 upwards the dynamic corner moves as far as the static one or further, by up to a tenth.

So both readings of a torsionally flexible floor are inflated statically, the ratio and the corner alike, and both are smaller dynamically. What is odd is how much smaller the dynamic ratio is: the corner still moves 2.19 times a torsion-free floor’s, more than any floor stiffer in torsion, and the ratio reads 1.17.

What the push cannot see #

The push does not know the twisting mode has a period. It applies the storey force at the centre of mass as a static load, so the torque about the centre of rigidity is the force times the eccentricity whatever the floor’s frequencies, and a floor flexible in torsion turns through a large angle under it. A shaking floor’s twist is set by how hard the ground excites its twisting mode, and that depends on two things the push leaves out: how much of the floor’s mass moves with that mode along the direction of shaking, and where the mode’s period falls on the spectrum.

Both work against a torsionally flexible floor’s twist. Its twisting mode carries little sway, so the ground, which pushes only along the push, excites it weakly; and its period is long, so the spectrum, falling beyond 0.6 s, asks less of it. The spectrum is not a load said it plainly: each point on a spectrum is the peak of one structure’s own response, and a structure that answers slowly is asked for less force. The push treats the twisting floor as though it answered instantly.

Combining the modes forgets which way each edge moved #

The reason is in how the modes are combined. A floor at Ω = 0.6 has a first mode at 0.93 s that is mostly twist: its far edge moves 2.01 times a torsion-free floor’s one way and its near edge 0.87 the other way. Its second mode, 0.45 s, mostly sways. The combination takes each point’s modal movements and returns one peak, a square root of a sum of squares and cross terms, and a square root has no sign. The near edge’s combined peak is 1.41 and the far edge’s 2.02, and nothing records that in the dominant mode they moved in opposite directions.

The ratio’s average is the average of those two magnitudes, 1.71. The centre’s own combined peak is 0.83. A floor twisting about its middle has two edges moving far in opposite directions and a centre barely moving, and the ratio reads its average from the edges’ sizes rather than their signed sum: it divides the far edge by something near the far edge itself. The static ratio divides by the twist once; the dynamic one, once the twist dominates, divides by it twice.

What the ratio divides by #

Across Ω the denominators show it directly. At Ω = 2, where the floor barely twists, the dynamic average, the centre’s peak and the static average are all 1.04: the floor moves almost uniformly and every way of averaging it agrees. As Ω falls the centre’s own peak stays close to one — the centre of a twisting floor is not where the twist is — while the average of the edges’ magnitudes climbs to 1.87 at Ω = 0.5, above the static push’s 1.69. The static average was already inflated by the twist; the dynamic one is inflated by the twist with its sign removed.

The standards could have asked for the centre’s movement, which a response-spectrum analysis gives as readily as the edges’, and the ratio of worst edge to centre would have been 2.55 at Ω = 0.5 rather than 1.17. Or they could have asked the analysis for the twist itself, the floor’s rotation combined over the modes, which has no such problem. The ratio of edges is a static idea carried into an analysis whose combined results cannot be added and averaged the way static ones can.

What a designer can read instead #

The analysis that produces the misleading ratio produces everything needed to avoid it. The floor’s rotation is a single quantity in each mode, and combined over the modes it is a peak twist with no sign problem, because a rotation has one value for the whole floor. The worst edge’s peak over the torsion-free floor’s peak — the amplification of the corner, which the earlier essay preferred to the ratio — is read from the same combined edges and is honest, because it divides by a number the twist has not touched. And the centre’s own combined peak is there to divide by instead of the edges’ average.

None of these is the ratio the standards tabulate their limits against, and the limits of 1.2 and 1.4 were set with the static ratio in mind. Reading the dynamic ratio against them answers a different question from the one the limits were written for, and on this floor it answers it wrongly a third of the time. A designer who wants to know whether a floor twists should read the twist, from the modes, where the spectrum has already put it.

A structure has more than one period made the case that modes are independent oscillators, each with its own shape, and that a building’s response is their sum. The ratio of edges is a quantity defined on the sum. Once the sum has been replaced by a combination that keeps magnitudes only, any quantity that relied on signs — an average of two points moving opposite ways, a difference of drifts — has to be computed in each mode before the combination, not after it.

Where the readings disagree #

Over 256 floors, with eccentricities from 1% to 21% of the plan’s length and Ω from 0.5 to 2, the two readings agree on 178 and disagree on 78. They disagree in two bands. Torsionally flexible floors with small eccentricities are flagged by the push and passed by the spectrum — 41 of them. Floors a little stiffer in torsion than in sway, in a band rising with eccentricity, are passed by the push and flagged by the spectrum — 37.

The second band is the one a designer would not expect. A floor whose walls are arranged so that its torsional radius slightly exceeds its radius of gyration — the arrangement a designer reaches for to make a plan resist twist, walls pushed out towards the perimeter — is the floor most likely to read regular under the push and irregular under the spectrum, because its two modes are close and each carries twist. The first band is the one a designer would act on wrongly. A floor that twists most has its twist least visible in the ratio its own dynamic analysis reports.

The modes, by hand #

With no eccentricity the floor’s two modes are uncoupled: a sway at 0.5 s and a twist at $0.5/\Omega$ s. Eccentricity couples them through the stiffness matrix’s off-diagonal term, $-Ke/\rho$, which for 4.2 m on a radius of gyration of 10.1 m is 0.42 of the sway stiffness. The two coupled frequencies are the roots of a quadratic whose sum is fixed by the uncoupled ones and whose product falls by the square of that term; at Ω = 1.05 they move apart from 0.50 and 0.48 s to 0.60 and 0.40.

The CQC rule weights the cross term between two modes by a correlation that depends on their frequency ratio and their damping. At 5% damping and a period ratio of 0.67 it is 0.055, so the two modes’ peaks are nearly independent and the combination is nearly a square root of a sum of squares: here the two rules agree to within two and a half per cent everywhere on the sweep. The eccentricity that couples the modes also separates them, and the floors in which CQC and SRSS would differ most — modes within a few per cent of each other — are floors with almost no eccentricity, whose ratio is near one either way. That is where the twist the combination rule invents found the square-root rule inventing a torque five times too large, and where the gap between two buildings found two neighbours whose periods agree swaying in step.

What the one storey leaves out #

One storey, one floor. A building has a mode of sway and twist for every storey, and its torsional modes are global, as the earlier essays’ list of later questions said. The single floor isolates the coupling of one sway with one twist, which is the mechanism; a tall building’s ratio is read storey by storey from modes that are each a mixture over the height.

No accidental eccentricity. The standards add an accidental eccentricity of 5% of the plan, which the earlier essay included and this one leaves out so that the comparison is between two readings of the same floor. With it, the static ratio rises as the earlier essay found, and a dynamic analysis has to be run twice, with the mass shifted each way, before its ratio can be read.

One spectrum shape. The spectrum is flat to 0.6 s, and the floor sways at 0.5 s. A floor further into the falling branch has its twisting mode asked for even less when it is flexible: at a swaying period of 1 s the dynamic ratio at Ω = 0.6 falls from 1.18 to 1.08, and the disagreement widens. The factor of two belongs to one mode found the same in another guise: a dynamic answer belongs to the mode that carries it, and a factor read off one mode does not transfer to another.

Elastic walls. A wall that yields changes the centre of rigidity during the shaking, and a floor whose stiff side yields first becomes more eccentric as it is pushed. That is a time-history question, beyond a spectrum.

Still open: the floor whose walls yield in turn #

The two readings here are both elastic, and both take the walls’ stiffnesses as fixed. A floor that twists in its own mode under a real earthquake loads its far walls hardest, and if those walls yield first the centre of rigidity moves towards the near side, the eccentricity grows, and the floor’s twisting mode lengthens while it is being excited. Whether a floor designed with walls whose strengths follow their elastic shares yields into a more torsional floor or a less torsional one — and whether a spectrum ratio of 1.2 at the design stage says anything about the twist the floor will have once its stiffest wall has cracked — is the question the corner that moves most puts to every plan whose strength and stiffness are placed by the same walls.