The floor that cannot keep up with the column

The column that shortens on every cycle bent a steel column’s hinge back and forth while it carried its share of a building’s weight, and found that the hinge does not settle. Each half-cycle, the flange that yields in compression shortens; the other flange, carrying its part of the column’s load, never reaches yield in tension to pull it back. An I-section at three tenths of its squash load, cycled to three times its first-yield curvature each way, came back from every cycle eight yield strains shorter. The weight and the curvature make the ratchet between them, and neither alone does.

Every column there kept its load whatever happened to its length. That essay ended on the fact that a column in a frame cannot. As it shortens, the floors it carries bend down towards it, and a beam whose middle support has dropped lifts some of its load off that support and onto the ends. A column that shortens faster than its neighbours should therefore lose load, ratchet more slowly, and give its neighbours load until they ratchet faster. Whether that evens the columns out, or whether the load it gives away pushes its neighbours into ratcheting as well, is a question no single column can ask.

A stocky hinge has a threshold #

The columns here are stockier than the essay’s sections: an I-section of 15,000 mm² and 300 mm depth, the proportions of a heavy universal column, with a squash load of 4,125 kN in steel of 275 N/mm². Stockiness turns out to matter before anything else does.

The slender section of the earlier essay ratchets at almost any compression. This one does not ratchet at all below 0.15 of its squash load when cycled to three first-yield curvatures each way, and above that its rate rises almost in a straight line, 13 yield strains a cycle for every unit of compression ratio. A thicker flange and web give the section enough material near its centroid that a small compression can be carried by fibres that never yield, so the yielding at the flanges closes into a stable cycle; past the threshold the compression needs the yielding fibres as well, and they walk. The threshold falls as the curvature grows: 0.22 at two first-yield curvatures, 0.11 at four.

The frame puts its three columns either side of that line. Six floors of two 8 m bays under 25 kN/m load the interior column with 1,500 kN, 0.36 of its squash load, and each exterior column with 450 kN, 0.11. The interior column ratchets; the exterior ones do not. Their average, 0.19, is above the threshold.

Each column has a hinge at each end of its ground storey, and a hinge a section deep turns a yield strain into $2 \times 300 \times 275/205{,}000 = 0.80$ mm of shortening. The interior column on its own would shorten 2.3 mm a cycle.

The floor as a hand-over #

Each floor is a beam continuous over the three columns. When the interior column drops Δ below the other two, the floor behaves as a 16 m beam on its two ends with a spring in the middle, and takes $6EI\Delta/L^3$ off the middle and puts half of it on each end. For a beam of $294 \times 10^6\ \mathrm{mm^4}$ over 8 m spans that is 0.72 kN for every millimetre the middle drops, on each of six floors: 4.3 kN a millimetre, all of it lifted off the interior column and handed out, every floor above moving with the ground-storey column beneath it.

That is the mechanism the settlement that matters is the difference found under a building whose ground gives way unevenly. A superstructure evens out a difference in its supports’ movements in the only way it can — by carrying the difference itself as a force. Here the support that gives way is a column shortening in the middle of an earthquake.

An earthquake is too short for the hand-over #

Over twenty cycles, a long earthquake, the interior column shortens 41.6 mm in the frame against 46.2 mm on its own. The floors have lifted 180 kN off it, an eighth of its load, which has brought its compression down from 0.36 to 0.32 of its squash load and its rate down by about a fifth by the last cycle. The exterior columns have taken 90 kN each and are now at 0.13, still under the threshold: they have not shortened at all.

In an earthquake, the frame barely slows the column that is shortening. It shortens nine tenths as far as it would alone, and the floors are bent by almost the whole difference. The hand-over is real, but it is slow: at 2.3 mm a cycle the floors lift 10 kN a cycle off a column carrying 1,500, and the 700 kN they would need to lift to bring every column to the same compression is seventy cycles away even at that first rate, which falls as the column slows.

Why the hand-over is slow #

A floor hands out an elastic difference at once. If the interior column were simply shorter than the others — built short, or squeezed more because it carries more — the floors would bend the moment it was loaded and the loads would settle where the stiffnesses put them, all in one step. A column’s own elastic shortening is fast and small: 1,500 kN on a 4 m column of 15,000 mm² shortens it 1.9 mm, the exterior columns shorten 0.6 mm under their 450 kN, and the floors lift 6 kN off the interior column for the difference.

A ratchet is neither. It is a shortening the column imposes on itself a few millimetres at a time, and the floors can only answer each step after it has happened. The rate at which they hand load out is the product of two numbers that have nothing to do with each other: how fast the column shortens, set by its section, its load and the earthquake’s curvature, and how much load a millimetre of shortening moves, set by the floors’ stiffness. The hand-over finishes when the floors have moved the column’s excess, and that takes as many cycles as the excess divided by the load moved per cycle.

It has the shape of the columns that are shorter than the core, where the columns and the core of a tall building shorten by different amounts as storeys are built on top of them and the floors between them tilt by the difference — with an earthquake’s cycles in the place of storeys. Both are deformations that grow while the structure is in use, and in both the floors can only follow the difference and carry what following it costs.

The difference stops, eventually #

The hand-over does finish. As the interior column loses load its rate falls, and as the exterior columns gain load they cross the threshold and begin to ratchet themselves. The difference between them grows more and more slowly and stops at 161 mm, the drop at which the floors have lifted 700 kN off the interior column and given 350 kN to each exterior one. At that point all three carry the average, 0.19 of the squash load, ratchet at the same rate, and shorten together: the floors are carried down evenly and stop being bent any further.

The columns alone have no such limit. Each ratchets at its own rate for ever, and the difference after 400 cycles is 925 mm. So the frame does bound the difference, and it bounds it at a number fixed by the loads and the floor’s stiffness and nothing else — the excess the interior column has to give away, divided by the floors’ stiffness against its dropping. The answer to whether the columns equalise is yes, at a differential of 161 mm.

The floors pay in bending #

What the floors hand out, they carry. A floor beam continuous over a column hogs over it, 200 kN·m here under 25 kN/m, and sags 113 kN·m in its spans. As the column drops, the hog falls, passes through nothing after 37 cycles, and becomes a sag: once the columns have equalised, the beam sags 266 kN·m over the column it was designed to hog over, and its largest span moment has risen to 355 kN·m, close to its plastic moment of 405. The floor has become a 16 m beam spanning between the exterior columns with a weak spring in the middle, and it is being asked to carry most of a load it was designed to share with a support.

That is the cost of the frame bounding the difference. A floor stiff enough to hand out the interior column’s load quickly is one that takes the load as bending quickly, and the floors here reach their plastic moment only because the beam happens to be just strong enough. A lighter floor, or a heavier load, yields its beams over the interior column before the columns have equalised. The beams then hinge, the hand-over stops growing, and the interior column goes on ratcheting at whatever load the yielded floors have left it.

Stiffer floors hand it out sooner #

The pace of the hand-over is the floors’ stiffness against the column’s rate. For six floors of the 457 mm beam, half of what the interior column will eventually shed is handed out in 45 cycles. Floors eight times as stiff do it in 6, and the same column loads spread over twelve floors do it in 22, because each floor need lift only half as much. Three floors carrying the same column load take 89.

Within an earthquake’s ten or twenty cycles only the stiffest floors on the tallest frames equalise their columns. Everything else carries most of the difference into the next earthquake. Ratcheting does not reset between events, and neither does the differential: the column is as short after the shaking as it was during the last cycle, and the floors are bent by what it left. A building’s lifetime of earthquakes is one long sequence of cycles with pauses in it, and the hand-over continues where it left off.

Together, or not at all #

The long run has two endings, and the threshold decides which. Under 25 kN/m and 20 kN/m the average compression lies above the threshold, so the columns end at the average and all three ratchet together, the frame descending evenly and its floors bent by a fixed amount. Under 15 kN/m the average, 0.12, lies below it. The interior column starts at 0.22, sheds load until it reaches 0.15 and stops ratcheting there; the exterior columns, at 0.10, never start. The whole line has shaken down, and the floors carry the shed load as a permanent bending.

A column line ratchets together or stops together, and which it does is decided by its average load against the section’s threshold. No individual column’s load decides it, and that is the useful finding: a designer who checks the most heavily loaded column against the ratchet is checking a column the frame will rescue, eventually, if the line’s average is low enough, and missing a line that will ratchet as a whole if it is not. It is the map with three regions read for a frame rather than a member — elastic, shakedown or ratchet — with the column line’s average as the coordinate. The answer to the earlier essay’s second possibility is that the load handed out does push the neighbours into ratcheting, but only to the average and never past it.

What a designer controls #

The finding moves the check from the column to the column line. A designer has three levers, and they act on different numbers.

The first is the threshold. A stockier section, with more of its area near its centroid, settles under a larger compression, as the section that yields from the outside in found for the reserve beyond first yield: the material that carries load without yielding is the material that lets a section shake down. A column line whose average load sits under its threshold stops ratcheting once its floors have handed out the excess, whatever its most heavily loaded column does on the way.

The second is the curvature. The threshold falls as the hinges are bent harder — 0.22 at two first-yield curvatures, 0.15 at three, 0.11 at four — so a frame whose drift is limited keeps its columns on the right side of the line for longer. That is a stiffness requirement on the whole frame, not on the column.

The third is the floor, and it cuts both ways. A stiffer floor hands the load out sooner and keeps the differential smaller in the cycles that matter, but it carries the same total force in bending when it has finished, because the force is the column’s excess and the floor’s stiffness only decides how quickly it arrives. A floor stiff enough to equalise the columns inside one earthquake has to be strong enough to carry the whole hand-over, and that is the check a column-by-column design never makes. The hinge a squeezed section has to work for found that compression costs a hinge its curvature; here it costs the floors above it their moment.

The hand-over, by hand #

Each floor carries $1.25 \times 25 \times 8 = 250$ kN to the interior column and $0.375 \times 25 \times 8 = 75$ kN to each exterior one; six floors make 1,500 and 450 kN, or 0.36 and 0.11 of 4,125 kN.

The interior column on its own ratchets at about $13.4 \times (0.36 - 0.15) = 2.9$ yield strains a cycle, or $2.9 \times 0.80 = 2.3$ mm. Twenty cycles of that is 46 mm.

The floors lift $6 \times 210{,}000 \times 294 \times 10^6/8{,}000^3 = 0.72$ kN per millimetre on each floor, 4.3 on six. Equalising all three columns at the average, $(1{,}500 + 2 \times 450)/3 = 800$ kN, takes 700 kN off the interior column, so the floors must drop it $700/4.3 = 161$ mm. Each floor then lifts $700/6 = 117$ kN off its middle support, which turns its moment there by $117 \times 8/2 = 467$ kN·m, from 200 hogging to 267 sagging.

What the column line leaves out #

Only the ground storey ratchets. Every hinge is in the ground storey’s columns, and every storey above is elastic and moves with it. In a real frame the storeys above have their own hinges at lower compression, and with the threshold this section has, most of them would not ratchet at all.

The floors stay elastic. At 25 kN/m they reach 355 kN·m against 405 at the end. A floor that yielded would cap the hand-over at its own plastic capacity and leave the interior column ratcheting at a higher load, which is the same shape as the moment that was shed has to land, with a column in the place of a beam.

Every column is cycled to the same curvature. In a frame the drift is common to every column, but the curvature it puts into each hinge depends on the column’s stiffness and its joints, and an exterior column with a weaker beam framing in may be bent less. Since the threshold falls as the curvature rises, a column bent harder starts to ratchet at a lower load.

The overturning is left out. An earthquake that rocks the frame swings the exterior columns’ loads up and down by more than the interior’s. The earlier essay found that swing barely moves the ratchet, and the mean load decides it; here the mean is what the floors redistribute.

The steel does not harden. Strain hardening would close the ratchet slowly in every column, as it slowed the hinge squeezed while it turns, and would make each shortening finite rather than unbounded. It would not change which column ratchets fastest or how slowly the floors hand the difference out.

Still open: the floor that yields before the columns agree #

Every floor here was just strong enough to carry the difference elastically until the columns had equalised. A floor that is not reaches its plastic moment over the interior column first — sagging, where it was designed to hog — and from then on it cannot lift any more load off that column. The column goes on ratcheting at whatever load it was left with, the floor goes on rotating at its hinge to follow it, and the difference grows without the bound the elastic floors gave it. Whether the floor’s hinge over the column then ratchets too, turning a little more with every cycle of the column’s shortening, and whether the frame’s floors or its columns are the first to run out of rotation, is the question the structure that settles down leaves for a structure whose two ratchets are in series.