The column that shortens on every cycle

The hinge squeezed while it turns bent a column section while its compression rose with the overturning of a swaying frame, and found that its hinge forms early, at the load it started with, and then rides down its plastic moment as the load grows. The overturning there grew in one direction. An earthquake reverses it every half cycle: the frame leans one way and then the other, each column’s hinge is bent one way and then the other, and the column carries its share of the building’s weight throughout.

The essay ended on what that does to the column’s length. A section bent and unbent while it carries an axial load accumulates plastic strain in the fibres that yield in compression each time. Whether a column hinge under cyclic bending shortens without limit — ratcheting down, storey by storey, until the floor above it has sagged by more than its own drift — or settles into a fixed cycle once its fibres have found their range was the question.

Shorter after every cycle #

The sections are the earlier essay’s: an I-section and a rectangle, each of 3,000 mm² and 200 mm deep, in a steel yielding at 275 N/mm² and taken as elastic–perfectly plastic, cut into 200 fibres through their depth. Each carries a constant compression and is bent back and forth to a curvature amplitude stated as a multiple of its unloaded first-yield curvature. At every step the axial strain at the centroid is found that makes the fibres carry the load, and every fibre that reaches yield keeps its plastic strain.

Neither settles. The I-section comes back from every cycle shorter, by 8.14 yield strains — 1.1 per cent strain at the hinge, every cycle, at exactly the same load. After twelve cycles it has shortened by 97 yield strains and the next cycle will take another 8.14. The rectangle shortens too, by 1.60 a cycle, a fifth as fast and just as steadily.

This is ratcheting, and the earlier essays on shakedown have drawn it. The map with three regions put a member under a constant load and a cycling temperature and found three fates: it stays elastic, it yields and then stops, or it gains a little more deformation every cycle for ever. A column in an earthquake is the same map with the drift in place of the temperature. The weight is the constant, primary load; the curvature the sway imposes on the hinge is the cycling, secondary one; and a hinge that yields both ways under a constant compression walks shorter on every cycle, with no collapse load and no point at which it stops.

Each half-cycle takes its own step #

The staircase shows where the shortening happens. Each half-cycle, as the curvature sweeps from one extreme to the other, the axial strain steps down by about four yield strains and then holds while the curvature returns through the elastic part of its range. Two steps a cycle, one for each direction of bending, and none of them ever reversed.

The mechanism is clearest in an I-section reduced to its two flanges. With the column carrying a compression $n$ times its squash load, each flange carries $n f_y$ when the section is straight. Bend it far enough one way and the flange on the inside of the bend yields in compression at $-f_y$. The other flange must then carry what is left of the column’s load: the two flanges’ stresses add to $-2nf_y$, so it sits at $-2nf_y + f_y = (1 - 2n)f_y$ in tension — below its yield stress whenever there is any compression at all. It stays elastic. All the strain the bending asks for beyond the flanges’ elastic range goes into the compressed flange as plastic shortening.

Bend the section back, and the roles swap. The flange that yielded is now on the outside of the bend and is pulled — but only to $(1-2n)f_y$, never to $+f_y$, because the other flange has just yielded in compression and the column’s load still has to be carried. It stretches elastically and keeps its plastic shortening. Each flange in turn is shortened and never lengthened back, and the column between them shortens by the sum. The tension flange never yields, so nothing ever pulls the column long again. The weight is what holds the tension flange below yield, and in perfectly plastic steel it does not take much: the rate below hardly depends on how large the compression is, only that there is one. With no compression at all both flanges yield at once, one each way, and a perfectly plastic section has no preferred direction to drift in; the slightest compression, or the slightest hardening, decides it.

The ratchet starts as soon as the hinge yields both ways #

The rate grows almost in proportion to the curvature amplitude once the threshold is passed, and its slope is the section’s depth: a flange’s strain swing is the curvature’s swing times its distance from the centroid, and everything beyond the elastic range becomes shortening. The compression sets where the threshold is and adds a little to the rate — 6.69 yield strains a cycle at a tenth of the squash load and 9.02 at half, at ±3 first-yield curvatures — but even a tenth of the squash load, a light column by any standard, ratchets as soon as its flanges yield both ways.

The rectangle is far more resistant. At ±1.5 first-yield curvatures it settles at every compression drawn; at ±3 it ratchets at 1.60 a cycle against the I-section’s 8.14. A rectangle has a core of material near its centroid that the bending barely strains, and that core stays elastic and carries the column’s load while the outer fibres cycle, so the outer fibres on the tension side are free to yield and pull back. An I-section has almost no core: its web was already the first part to yield under compression, and its flanges carry the load and the bending together. The section shape that makes an I-section efficient in bending is the shape that makes its hinge ratchet.

Where the I-section settles #

On the whole plane of compression and curvature amplitude the I-section settles in a sliver: seven cells of 96, all at amplitudes of about one first-yield curvature or less, where the section barely yields at all. Above about one first-yield curvature it ratchets at every compression from 15 per cent upward, and at 5 per cent from 1.5. The first-yield curvature here is the unloaded section’s; a column under compression yields earlier, which is why the boundary moves to smaller amplitudes as the load rises.

A frame designed for ductility asks its column bases to turn through several first-yield curvatures. That is the right half of the map, where the I-section shortens by several yield strains a cycle at every load a column carries.

The overturning’s swing does not drive it #

The question the earlier essay asked was about the overturning: the leeward column is squeezed as the frame leans and relieved as it leans back, so its compression cycles with its curvature. It is natural to suppose that this cycling load is what shortens it.

It is not. With no swing at all the I-section shortens 8.14 yield strains a cycle; swinging its load by ±30 per cent with the curvature takes that down to 6.36. The rectangle’s rate hardly moves. The swing does change which flange works hardest in which half-cycle — the squeezed half-cycle shortens the compressed flange more, the relieved one less — but the two effects nearly cancel over a cycle, and what is left is a small reduction, because at the relieved extreme the tension flange comes closer to yielding back. The ratchet is made by the mean load and the curvature. An interior column whose load hardly swings at all shortens as fast as a corner column whose load swings most.

For a doubly symmetric section the timing does not matter either: a column squeezed as it bends one way and one squeezed as it bends the other are mirror images, and ratchet alike.

What a column’s two hinges add up to #

Turned into a length, the strain is a column shortening by millimetres per cycle. Each hinge is taken as yielding over a length equal to the section’s depth, 200 mm, which is about what a plastic hinge in a column with a moment gradient spreads over, and a column bent in double curvature has two. At ±3 first-yield curvatures it shortens by 4.2 mm a cycle: 42 mm after ten cycles, 85 after twenty. At ±1.5 first-yield curvatures, 13 mm after ten.

An earthquake that drives a frame’s column bases through several first-yield curvatures does so for a handful of large cycles and many smaller ones, and every cycle that yields both flanges adds its step. So the column shortens, and the floor it carries comes down with it. A column next to it carrying less load shortens less — slower on the map — and the beam between them is bent by the difference, the problem the columns that are shorter than the core found from creep, arriving here in a few minutes from plasticity. The floor above a ratcheting column can sag by more than the storey’s own drift, which is the outcome the earlier essay feared.

Why the rectangle can settle and the I-section cannot #

The difference between the two sections is the difference between the two conditions under which a cycling structure stops. A structure settles down when some residual stress, locked in by its first cycles, keeps every later state of loading inside yield — Melan’s condition, which the load it can carry once set out for beams. A section can lock in a residual stress only if it has fibres that stay elastic while others yield, so that the elastic ones can hold the yielded ones in place.

A rectangle has them. Its fibres near the centroid see a small fraction of the curvature’s strain, stay elastic through moderate cycles and carry the column’s load, and once the outer fibres have yielded a little each way they find a locked-in stress that keeps them inside yield from then on. That is why the rectangle settles at every compression up to ±1.5 first-yield curvatures and, at ±3, ratchets only slowly: by then its elastic core is small, and only part of the axial load can be carried there.

The I-section has almost no core. Its web is thin, it carries little of the load, and under compression it yields first; the flanges carry the axial load and the bending between them, and a flange that has to carry its share of the column’s weight cannot also be held in a residual state that keeps it inside yield both ways. No residual field exists that satisfies Melan’s condition once the curvature’s range exceeds what two flanges can absorb elastically, and the two-flange arithmetic below says exactly when that is.

Where a design meets the ratchet #

A frame designed for an earthquake is designed to yield somewhere, and the usual rule is that the beams yield and the columns do not — strong columns, weak beams — so that the displacement the earthquake asks for is supplied by beam hinges that carry no axial load and so cannot ratchet. The one place the rule cannot keep a column elastic is its base. A frame pushed over until it will not stand forms its mechanism with hinges at the beam ends and at the column bases, and the base hinges carry the whole weight of the building above them.

So the ratchet is a base-hinge problem, and a heavy-column problem: the columns of a tall frame’s lowest storey carry the largest compressions and the largest base rotations. And because the rate is set by the mean load rather than the swing, it is the interior columns, which carry the most gravity, rather than the corner columns, which carry the most overturning, that shorten fastest. A design that checks a base hinge’s rotation capacity — how far it can turn before it loses strength — checks something real, but the column can shorten by centimetres while every one of its hinges is still well inside its rotation capacity. The property that appears in none of the equations, ductility, is here spent on length rather than on rotation.

The two flanges by hand #

Take the I-section as two flanges a distance $h$ apart, carrying a compression $n$ of the squash load, cycled to a curvature $\pm k\kappa_y$, where $\kappa_y = 2\varepsilon_y/h$ is the curvature at which a flange first yields with no load. The flanges’ strains are the centroid’s, $\varepsilon_0$, plus and minus $\kappa h/2$.

In the half-cycle from $-k\kappa_y$ to $+k\kappa_y$ the curvature term changes by $2k\varepsilon_y$ in each flange, upward in the top flange and downward in the bottom one. The bottom flange is the one pushed into compression; it ends at $-f_y$, yielding. The top flange, which ended the previous half-cycle yielded at $-f_y$, is pulled only to $(1 - 2n)f_y$ and stays elastic: its strain changes by exactly $(2 - 2n)\varepsilon_y$. Since its strain is $\varepsilon_0 + \kappa h/2$, the centroid must have moved by

$$\Delta\varepsilon_0 = (2 - 2n)\varepsilon_y - 2k\varepsilon_y = -(2k - 2 + 2n)\,\varepsilon_y$$

in that half-cycle — a shortening — and by the same in the next, with the flanges’ roles exchanged. A whole cycle shortens the centroid by $2(2k - 2 + 2n)\varepsilon_y$.

At $k = 3$ and $n = 0.3$ that is 9.2 yield strains a cycle for a section that is nothing but flanges; the fibre model gives 9.08 for two thin flanges and 8.14 for the real I-section, whose web, partly elastic near the centroid, holds a little of the shortening back. The same arithmetic gives the threshold: no ratchet while $k < 1 - n$, below which the elastic flange’s swing can absorb the whole curvature. At 30 per cent of the squash load that is 0.7 first-yield curvatures, and the map’s first ratcheting cell at that load is at 1.0.

Steel that hardens, and a hinge with a length #

The steel is elastic–perfectly plastic. Real structural steel hardens after it yields, and under cycling it hardens kinematically: its yield surface moves with its plastic strain, so a flange that has been shortened plastically finds it easier to be pulled back. That slows a ratchet, sometimes to a stop after a number of cycles, and the rates here are the perfectly plastic ones — the upper end of what a real hinge does, not the expected value. Tests on steel columns under constant axial load and cyclic drift do shorten progressively, and by more at higher load and larger drift, which is the shape of these figures if not their size.

The section does not buckle locally. An I-section’s flanges cycled through several yield strains of compression are the flanges that buckle locally, and a buckled flange carries less of the load and passes more of it to the web; the shortening then concentrates in the buckle. Ratcheting and local buckling feed each other in real columns, and neither is in this calculation.

The hinge has a fixed length. The yielded length of a real hinge grows as the curvature grows and spreads as the column shortens; the millimetres are the strain times an assumed length, and the strain is the result.

The column is not in a frame. A real column that shortens hands part of its load to its neighbours through the beams, which lowers its compression and slows its ratchet. The column here keeps its load whatever its length.

The weight is the primary load #

The earlier essay’s question had two answers, and the figures choose the worse one. A column hinge under cyclic bending and a constant compression does not settle into a fixed cycle; it ratchets, every cycle the same step, for as long as the cycles yield both of its sides. What drives it is not the swing of the overturning but the weight the column carries the whole time — the primary load of the shakedown map — and the curvature the sway imposes, the secondary one. An I-section, efficient in bending because its material is far from its centroid, is efficient at ratcheting for the same reason.

Still open: the column that hands its load away #

Every column here keeps its compression whatever happens to its length. A column in a frame does not: as it shortens, the beams it supports bend down towards it and lift its share of the floor onto the columns beside it, so its load falls and its ratchet slows, while its neighbours’ loads rise and theirs speed up. Whether a frame whose columns ratchet at different rates equalises their shortening — the faster ones shedding load until all shorten together — or whether the redistribution concentrates load on the stiffer, less yielded columns until they too start to ratchet, is the question a single column under a fixed load cannot ask and every frame in an earthquake answers.