The wall the twist was said to spare

The floor that twists in its own time gave a floor plate mass and shook it, and found that the standards’ ratio of worst edge to average edge meant one thing under a push and another under a response spectrum. Both of those readings were elastic. They took the walls’ stiffnesses as fixed and asked how far a floor would move if nothing ever yielded, which is a question no designer intends to have answered, since the forces such a floor is designed for are a quarter or a fifth of the ones the elastic analysis found.

So the question that essay left is the one that follows from the design. The walls will yield. A common and reasonable expectation is that the far walls, which the twist loads hardest, go first; that the centre of rigidity then slides towards the near side, because only the near walls are still stiff; that the eccentricity grows; and that the floor twists more as it goes on, each wall’s yielding making the next worse. If that is right then the ratio read at the design stage is an optimistic statement about the floor that will actually exist after a few seconds of shaking. It turns out to be right for one way of dividing the strength among the walls and exactly backwards for another, and the one for which it is backwards has a different problem, on the side of the floor nobody was watching.

Four walls and three ways to divide their strength #

The floor is the 30 × 18 m plate of the earlier essays, with its radius of gyration of 10.1 m. Four walls run along the direction of shaking, at 0, 10, 20 and 30 m across the plan, and each is elastic and then perfectly plastic, with no hardening and no loss of stiffness on the way back. Their stiffnesses fall in a straight line across the plan so that the centre of rigidity sits 4.2 m from the centre of mass, towards the stiff walls, and the stiffest wall has three times the stiffness of the softest. Cross walls that never yield bring the torsional radius to 10.6 m. That is the same elastic floor as before, Ω = 1.05, with a torsion-free period of 0.5 s.

A design shear $V$ has to be divided among the four walls so that the strengths sum to it, and equilibrium does not say how. Any division that sums to $V$ and balances the torque is a statically admissible state, which is the whole of what the bound from underneath asks of a design, and the theorem then promises that the collapse load is at least what was designed for. It does not promise anything about how far the walls will have to move in getting there.

Three divisions cover the ground. In the first, each wall gets the force an elastic static push at the centre of mass puts on it, torsion included: this is what a design office does when it reads the forces from its analysis and sizes each wall to the force it read. In the second, each wall gets its share of stiffness with the torque left out, which is what the analysis would give with no eccentricity. In the third, the four walls are equal, so that the strength has its centre exactly at the centre of mass while the stiffness does not.

The way to read a division is by what it does to each wall’s yield displacement, its strength over its stiffness, since that is how far the wall can go before it starts asking for ductility. Under the first rule those displacements are simply the ones the elastic push gives each wall, so they spread across the plan by a factor of 2.9: 0.59 on the stiff side and 1.72 on the far side. The stiff wall has been handed a low strength because the twist was calculated to help it, and it keeps its high stiffness, so it will reach that low strength after moving very little. Allowing for torsion has made the wall on the stiff side the wall with the least room on the plan.

What the push says about the order #

Pushed statically, the expected story is told by one of the three rules and not by the others. Where strength follows stiffness, the far wall reaches its strength first, at 0.58 of the design shear, because the twist gives it more movement than its neighbours and they all yield at the same displacement. The walls still elastic then have their centre 6.9 m from the centre of mass rather than 4.2. The next wall goes at 0.67 and the centre moves to 10.6 m; the third at 0.74, and it is at 15 m, on the last wall left. The tangent eccentricity has more than tripled. This is the runaway the expected story describes, and this push traces it exactly.

Where strength follows the elastic force nothing of the kind can happen. Each wall was sized for precisely the force this push brings it, so all four reach their strengths at once, at the design shear, and there is no order and no intermediate state with a moved centre. A push cannot distinguish this floor from a perfectly regular one except by how far the far edge has gone when the knee arrives. Pushed over until it will not stand made the corresponding point about a pattern of storey forces: a pushover reports on the load shape it was given, and if the strength was laid out from that same load shape then the pushover’s main finding is a tautology. The floor was not going to be pushed in that shape. It was going to be shaken, and the shape it takes in shaking is not the one the strengths were read from.

Shaken, the bill lands on the other side #

Twenty sampled records were used, each a filtered noise of the same character with a strong phase of about ten seconds, and each floor was given a strength set so that the torsion-free oscillator’s peak elastic force on that record is four times the design shear. A strength reduction of 4 is a typical choice for a moderately ductile wall system. Every result is a mean over the twenty, with displacements measured against the torsion-free oscillator’s own peak on the same record.

The figure is the finding. The two rules give near mirror images of each other. Where strength follows stiffness the far wall is asked for 8.2 and the stiff wall 4.2, the distribution the expected story predicts. Where the strength allows for torsion, the far wall’s demand does fall, to 4.3, which is what the allowance was for. And the stiff wall’s demand rises from 4.2 to 9.6, more than twice the far wall’s and higher than any demand under the rule the allowance was meant to improve. In nineteen of the twenty records the stiff wall is asked for more than the far wall. It is not a marginal effect of one record.

For scale, a symmetric floor on the same records, with no eccentricity at all, asks 7.3 of each of its walls at this reduction. A number well above 4 is not itself a surprise: the earthquake asks for a displacement explains why a stiff short-period oscillator is asked for more ductility than its strength reduction, and a torsion-free period of 0.5 s on a soil filter of 0.5 s is near that band. So the walls of the eccentric floor are not asked for wildly more than a symmetric floor’s. They are asked for it in a different place.

The arithmetic behind that place takes one line, and it is worth checking by hand. Ductility is a wall’s peak displacement over its yield displacement, and the design shear is a quarter of the elastic force, so a wall with a yield displacement of 0.59 on the earlier scale yields at 0.59 × 0.25 = 0.148 of the torsion-free peak. The floor’s stiff edge moves 1.43 of that peak. So the wall’s ductility is 1.43 / 0.148 = 9.7, against the 9.6 computed; the far wall’s is 1.87 / (1.72 × 0.25) = 4.3. The two edges’ movements differ by a factor of 1.3 and the two walls’ yield displacements by a factor of 2.9, so almost all of the difference in demand comes from the strength the design gave them and very little from the twist.

Whether the floor yields into more twist or less #

The question the previous essay ended on has a direct answer, and it depends on the rule. A floor whose strength allows for torsion twists less as it yields: 0.69 of a torsion-free floor’s movement while elastic, 0.45 at a reduction of 4 and 0.26 at 8. A floor whose strength follows stiffness does not. Its twist stays near 0.65 up to a reduction of 5, while the sway it is measured against grows, and then rises, to 0.84 at a reduction of 8. That floor does what the expected story says. The other does the reverse.

The reason the twist falls is the same thing that makes the stiff wall pay. Twisting a floor about its centre means one side moving forward while the other moves less, and the walls resist that only as far as their force can change with their movement. A wall at its strength has stopped resisting: its force stays at the plastic value whichever way the rest of the floor goes. Once all four walls are at or near their strengths, the torque they apply is a fixed number, the strength’s own first moment about the centre of mass, and when the strengths have been laid out so that this moment balances, the plastic floor has almost no torque to twist it with. What is left to restrain or drive the twist is the cross walls and the inertia. A floor whose strengths are centred on the mass translates in its plastic excursions, and every excursion moves both edges alike. A floor whose strengths are off the mass does not, because its plastic force is an eccentric force.

This is the strength eccentricity, and it is a different quantity from the stiffness eccentricity on which the whole of the earlier reading rested. Elastic analysis cannot see it, because in an elastic floor there is no such thing as a wall’s strength. Under the second rule the strength’s centre sits the full 4.2 m off the mass, with the stiffness’s. The third rule puts it on the mass exactly, and the first comes within 0.16 m, which is why their two curves run together in the figure.

A ratio that improves for the wrong reason #

A measure of twist that divides by the twist found that the ratio saturates as the twist grows. Here it falls, with the same weakness showing from the other side. As a floor yields, its translation grows quickly and its twist does not, so the ratio’s denominator is taken over by sway and it heads for one. A floor at a reduction of 4 whose strengths allow for torsion has a ratio of 1.18, below the 1.2 limit, while elastic it read 1.40, and the same floor designed for a reduction of 8 reads 1.05: nearly perfectly regular.

The reading is true as far as it goes. This floor really does move nearly uniformly when it yields heavily. But a reader who took 1.05 for evidence that nothing about the plan matters would miss everything in the previous figures, because the ratio measures edges and the demand is not set by the edges. It is set by the edges divided by the yield displacements, and the yield displacements are where the plan’s asymmetry now lives. At a reduction of 8 the walls are asked for 35, 21.7, 16 and 12.8: a spread of 2.7 between the two sides of a floor whose ratio says it is regular.

The edges show what the ratio is averaging. Below a reduction of about 2 the two rules are indistinguishable, because neither floor has yielded much. Beyond it they separate, and both have their near edges catching up, since a yielding floor sways further and the twist has less to do with how far anything goes. But the far edge of the floor whose strengths follow stiffness is moving 1.3 times as far as the other floor’s at a reduction of 8: 3.69 against 2.76. A designer who cares about the far edge’s cladding, or its columns’ drift, has a real reason to prefer the first rule, and it costs the wall on the other side.

The rule neither end of the argument uses #

Since one rule overloads one side and the other rule the other, there has to be a division in between that overloads neither, and it is easy to find. Going a share of the way from the stiffness rule to the elastic rule moves both walls’ demands smoothly, and they cross about halfway: at a share of 0.5 the worst wall on the plan is asked for 6.1. That is 26% less than the stiffness rule’s worst, 36% less than the elastic rule’s, and also less than the 7.3 a symmetric floor asks of each of its walls on the same records. The allowance for torsion, taken at full value, has been overdone by about a factor of two as far as the walls’ ductility goes.

This is what the moment that was moved on purpose did along a beam, moved here to a plan. Equilibrium says the strength may be placed anywhere that sums to the shear and balances the torque. The elastic distribution is one admissible choice, attractive because a program prints it, and it has no particular claim on the ductility demand. Weaker in one place, and better on every average found the same in elevation: a building’s strength need not follow its elastic forces, and following them does not make the demand even.

The finding about a half should not be carried away as a number. It belongs to this floor, these walls and this set of records, and the mix that balances the two sides will move with period and torsional radius. At a stiffer torsional radius of 13 m the two rules ask 6.9 and 7.0 at their worst and the difference is much smaller; at 16 m, 7.6 and 6.7, and the choice of rule has nearly ceased to matter. It matters for floors near Ω = 1 and below it. What travels is the shape: a demand that falls on one side as it rises on the other, and an elastic allowance that lands past the crossing.

Whether the ratio at the design stage predicts anything #

The last question was whether a ratio of 1.2 at the design stage says anything about the floor once its stiffest wall has gone. Across twenty-five plans, it does not say much. Floors reading below the limit while elastic and floors reading well above it have worst edges at a reduction of 4 in the same band of about 1.8 to 2.2, when the strengths allow for torsion. The highest points in the figure are at the left: plans that are torsionally flexible, with radii of 6 and 8 m, whose elastic ratio is low for the reason the previous essay gave, since combining modes forgets signs and the combined edges look alike. Those floors read most regular and move furthest once they yield, up to 3.54 times a torsion-free oscillator when their strengths ignore torsion.

The rule used to divide the strength has more effect than the ratio does. At the low end of the ratio the two series sit far apart; at the high end they sit together. So a limit on the elastic ratio, used as the test of which plans are allowed the higher strength reductions, is sorting floors by something only loosely related to how they will behave. The torsional radius is the thing to look at, and it is a number the same analysis prints.

How much of this is the records #

A mean of twenty records is not a precise number. On the stiff wall at a reduction of 4 the individual records give ductilities from 3.5 to 17.3, with a median of 9.9, and running sixty records in place of twenty lowers the mean from 9.6 to 8.7. The far wall is steadier, 2.6 to 7.9 with a median of 3.9, and the twist steadier again, 0.33 to 0.63. The comparison between the rules is firmer than any of the numbers on its own, since each record is applied to both, and the stiff wall’s excess shows up in nineteen of the twenty. But quoting a ductility of 9.6 to two figures would claim more than this sample has earned, and a floor designer who needs one should be reading a distribution rather than this mean.

Period changes the size and not the pattern. At a torsion-free period of 1 s the walls under the elastic rule ask 9.5 and 4.3, nearly as here. At 2 s, 13.1 and 6.0, against 5.3 and 13.6 under the stiffness rule. At 0.2 s, where everything yields much further for the same reduction, the mirror image holds at 36 and 16 against 13 and 31.

The assumptions under the picture #

A wall’s stiffness was held fixed while its strength was changed. This is the assumption most likely to mislead. A reinforced wall’s yield displacement is set mainly by its geometry, its yield strain and its length, not by how much steel it has, so adding strength adds stiffness nearly in proportion. Paulay’s argument, made over twenty years ago, is that this makes a wall’s stiffness a result of the design rather than an input to it, and on that view the second rule in the figures, where every wall yields at the same displacement, is not a rule anyone chooses so much as what a set of walls of one length tends towards. Each of the three rules is therefore a statement about walls that could have their strength set freely, which steel braced bays nearly can and walls of equal length nearly cannot.

One storey and rigid floor. A real building’s twist runs through its height, and the floor is a beam lying down is a reminder that the plate itself has a stiffness. A multi-storey floor has a plastic mechanism to form as well as a yield sequence, and nothing here says how the demand would stack over storeys.

Perfectly plastic, in one direction. The walls neither harden nor lose stiffness in the way a cracked wall does, and the ground moves along one axis while the cross walls stay elastic throughout. They are what keep the floor’s torsion stiff when every wall along the push has yielded, and they are shaken in the other direction in any real event.

Filtered noise is not an earthquake. The records have no pulses, no directivity, and no rotational component. They share one spectral shape, and the point of them is to compare strength rules on identical input. The statement that one wall’s demand exceeds another’s is much firmer than the number given for either.

Still open: the cross walls that are being shaken too #

Every cross wall here stays elastic, because the ground moves along one axis only. When the elastic torsional stiffness of a plan comes mostly from the walls across the shaking, a floor that loses all its walls along the push is held against twist by those alone — and a real earthquake loads them at the same time in their own direction. If they yield as well, the floor has no torsional stiffness left for that part of the record, whatever the strength eccentricity says. Whether two-way shaking gives back the runaway twist that a one-way record took away, and whether a plan should therefore be read with how a tall building stands still in mind as having two sets of walls that need to stay elastic in turn, is what the other axis would ask of this one.