The structure that settles down, and the one that walks

Every collapse calculation on this site asks the same question: how large can the load get. None of them asks how many times it arrives.

For a structure loaded once that is the right question, and the answer is a collapse load below which nothing happens. For a structure loaded repeatedly it is not, and the reason has nothing to do with cracks or with metal fatigue. A structure below its collapse load can yield a little on every cycle, in the same direction every time, and accumulate deformation without ever failing in the sense the collapse calculation means. It never breaks. It walks.

The two loads have to be of different kinds #

This is the condition the whole subject turns on, and getting it wrong is the fastest way to conclude that ratcheting does not exist.

A moment cycled between two values is force-controlled at both ends of the cycle, in the sense this site means by a load: something outside the structure insists on it, and equilibrium has to be found with it. The section either carries the peak moment or it does not; if it does, it may yield on the way there, but the residual field it forms is available to help every subsequent time, and after a few cycles it is elastic. There is no arrangement of two force-controlled loads that ratchets.

What ratchets is a steady load that has to be carried alongside a cycled deformation that does not. A moment applied by gravity cannot be relieved by yielding — the weight is still there and something has to hold it. A temperature difference can: yield a bit and the thermal strain is accommodated, the stress it was causing disappears, and the section is left slightly further round than it was. Next cycle, the same. The steady load provides a direction and the cycled deformation provides the increments.

The classical arrangement is a pressurised tube with a cyclic temperature gradient through its wall, and it is called the Bree problem after the analysis published in 1967. The version here is its bending equivalent: a beam under a constant moment, with the sun on it.

The temperature profile has to be curved #

There is a second condition, and the calculation that produced these figures reported every case as elastic until it was noticed.

A linear temperature gradient across a section that is free to bend produces exactly no stress. The material at the top wants to be longer, the material at the bottom wants to be shorter, and the amount each wants varies linearly with height — which is precisely the shape a section can accommodate by curving. Plane sections stay plane; a linear thermal strain field is a plane; the section simply bends to suit, and nothing is restrained.

What produces stress is the part of the profile that a straight line cannot follow. A real bridge deck heated by the sun is hot in the top few tens of millimetres and near-ambient below, a profile that falls away far too quickly for a straight line to track. The section takes up the best straight-line fit as an average expansion and a curvature, and everything the profile does beyond that is locked in.

That is the same class of object as a residual stress: a self-equilibrating field, invisible to statics, generated by an imposed strain rather than by an applied force. What is different here is that it comes and goes with the weather.

The magnitude is worth a moment. Thirty-five N/mm² is not a large stress against a yield of 275, and a thirty-degree profile is an ordinary summer day on a dark deck. Scale both up — a fifty-degree profile on a deeper section — and the self-stress reaches a third of yield, generated by weather, present in a structure that no static calculation can find it in, and cycling once a day for the life of the bridge. It is the standard explanation for a class of cracking in concrete decks that appears without any traceable overload, and it is the same mechanism as a settled support with the sun in place of the ground.

The three regimes #

Elastic. Neither load is enough to yield anything. The section returns exactly to where it started every cycle and continues indefinitely. This is where design normally intends to be, and where an ordinary serviceability check assumes it is.

Shakedown. The first few cycles yield. That yielding leaves a residual stress field behind, and — this is the mechanism — the residual field is exactly the field that makes the subsequent cycles elastic. The structure has manufactured its own protection out of the damage of the first few cycles, and it is elastic ever afterwards. Melan’s theorem says that if any self-equilibrating residual field exists that would make the whole cycle elastic, then the structure will find it.

Ratcheting. No such field exists. There is no residual state from which the whole cycle can be run elastically, so plastic strain is added on every cycle, in the same direction, without limit.

What separates the second from the third is not the size of either load on its own. It is whether a residual field exists that can serve both, and that is a question about the combination — which is why shakedown limits are drawn as a region on a plane with one axis per load rather than as a number.

Which free body produced the number #

The free body is the cross-section, cut, with the stresses on the face — the same one as everywhere else in this field — and it is now carrying memory. Each of the seventy fibres holds three numbers: its current stress, its accumulated plastic strain, and the centre of its own yield interval.

Two equations are enforced at every step. The axial force on the face equals zero, and the moment equals the applied one. The unknowns are the section’s two kinematic quantities, the reference strain and the curvature. Solving them at each step of a slowly ramped load history gives the whole record, and the ramping is not optional: a plastic step is path-dependent, so a single large increment lands somewhere a real history never visits.

The mechanical strain a fibre feels is its total strain minus its free thermal strain. That subtraction is the entire content of “an imposed deformation is not a load”, and it is why the temperature enters here and not on the right-hand side of an equilibrium equation.

Two checks. The self-stress field in the figure above integrates to $-1.6 \times 10^{-8}$ N of axial force and $-2.5 \times 10^{-6}$ Nmm of moment on a section carrying stresses of tens of N/mm² — zero to the precision of the arithmetic, which it has to be, since nothing is being applied. And the classification of each case is taken from the last third of the record rather than the first, because a shakedown’s opening cycles are indistinguishable from a ratchet’s; a classifier reading the first three cycles calls everything a ratchet.

Where the model stops #

This is a section, and shakedown is a structural phenomenon. The real theorems are about whole structures, where the residual field that has to exist is a set of self-equilibrating member forces rather than a stress distribution across one cut. A redundant structure carries such fields routinely — that is exactly what a settlement moment is — and the section-level calculation here is the simplest system in which the effect can be seen at all.

The shakedown limit is a load, and this figure does not find it. What the plot shows is which side of the limit three particular cases fall on. Locating the boundary requires either a bisection over many runs or the static shakedown theorem applied directly, which asks whether a suitable residual field exists rather than simulating until it appears.

A section that ratchets in this calculation may buckle first in reality. The curvature accumulates, the compression flange goes further into compression on each cycle, and local buckling ends the process before the deflection does. Which failure arrives first is not decided here.

Kinematic hardening flatters the result. With any hardening at all, a ratchet decelerates: each cycle’s plastic strain raises the local yield stress and the next increment is smaller. Perfect plasticity, used here, is the honest lower bound and the reason the drift is a straight line rather than a curve flattening out.

And nothing here has any cracks in it. A structure that ratchets is accumulating plastic strain, and plastic strain accumulated cyclically is what low-cycle fatigue counts. In practice the two failure modes arrive together and this calculation sees only one of them.

What the picture cannot show #

A curvature is not a deflection. The vertical axis counts curvature gained, in multiples of the curvature at first yield, and a beam’s sag is that curvature integrated twice along its length. The conversion depends on how much of the span is ratcheting, which depends on where the moment is high and where the sun falls, and neither is on the plot.

Sixteen cycles is not a service life. A bridge deck sees a thermal cycle every day and a few hundred larger ones a year over decades. A drift of 1.85% of the first-yield curvature per cycle is negligible at sixteen cycles and is a different matter entirely at ten thousand, and the plot’s flat-looking lower lines would stay flat while the upper one left the page many times over.

And the load history drawn is a triangle wave with one peak. A real thermal history has a daily cycle inside an annual one, with occasional extremes, and the order in which those arrive matters because plasticity is path-dependent. A single large excursion early can create a residual field that protects against everything afterwards; the same excursion late finds a section that has already accommodated itself to something else.

The generalisation #

The pattern is that a limit-state calculation answers the question it was asked, and the interesting failures are the ones nobody asked about.

The collapse load answers “how large”. Shakedown answers “how many times”. Serviceability answers “how far”. Fatigue answers “how many times, with cracks”. Each is a different question about the same structure and none of them is derivable from the others, and the reason ratcheting is the least familiar of the four is that it has no dramatic ending: a ratcheting structure is not approaching a collapse, it is approaching a deflection.

There is a second, sharper generalisation about what kind of load a load is. This site has largely treated a load as a force. An imposed deformation — a settlement, a temperature, a shrinkage, a fabrication error — is not a force, and behaves differently in three ways that keep recurring: it produces stress only where it is restrained, it is relieved by yielding rather than resisted by it, and it does not scale with anything a load factor multiplies. Ratcheting is what happens when a structure has one of each kind and cannot serve both.

The second generalisation has a practical edge worth stating on its own. A structure’s shakedown limit is generally below its collapse limit and above its elastic limit, so it sits in a band that neither of the two calculations normally performed will find. A design that checks elastic behaviour under working loads and collapse under factored loads has checked both ends of that band and neither point inside it. Whether the band is wide enough to matter depends on how much of the loading is an imposed deformation, which is why the question surfaces in pressure vessels and bridge decks and almost never in a building frame carrying gravity.

A surprising place this turns up #

The most familiar ratchet in ordinary construction is a bridge deck’s expansion joint, and the mechanism is the same one with friction in place of plasticity.

A deck expands and contracts daily. At a bearing there is a steady load — the deck’s weight, and any longitudinal grade — and a cycled deformation, the thermal movement. If the bearing’s friction is enough to prevent the movement entirely, nothing happens. If it is not, the deck moves a little further downhill on each cycle than it comes back, because the steady load biases which direction slips more easily. Over a few thousand cycles it has walked, and the joint that was supposed to open and close has closed permanently at one end and opened beyond its range at the other.

Deck walking is normally explained as a friction problem, and so it is. What makes it the same problem as this page is the shape of the mechanism: a steady load providing the direction, a cycled deformation providing the increments, and a per-cycle irreversibility that is individually negligible and cumulatively decisive. Nothing about it is visible in a calculation that applies the loads once.

Where the ladder goes next #

Later rungs on this anchor: Melan’s static shakedown theorem and Koiter’s kinematic one, proved, with the residual field they assert into existence. The Bree diagram itself, with its regions mapped on axes of primary and secondary load. Shakedown of frames rather than sections, and the load factor it produces against the collapse factor. Alternating plasticity as the third regime, where the plastic strain reverses each cycle and the failure is low-cycle fatigue rather than deformation. Deck walking and bearing design. Ratcheting in soils under repeated traffic loading, which is the same mechanism in a material with no yield stress. And the interaction with creep, where a structure at temperature ratchets and creeps at once and the two cannot be separated.

Historically the problem arrived from pressure vessels and nuclear plant rather than from buildings, because that is where a steady primary load and a large cycled thermal load reliably occur together. Bree’s 1967 paper on fuel cans is the standard reference and the diagram carries his name; the theorems are older, Melan’s dating from the 1930s and Koiter’s from the 1950s, and both sat largely unused until an industry appeared with the loading that needed them.