The map with three regions

A structure that settles down has yielded, left a residual stress field behind, and then gone on responding elastically for ever. A structure that does not settle down keeps going, and the difference between the two is not a load — it is a region on a map.

Three fates, and only one is a failure load #

Read the three curves as three different kinds of behaviour rather than as three sizes of the same one.

Elastic. The stresses never reach yield anywhere, the response is the same on every cycle, and nothing accumulates. This is what an elastic analysis assumes and it is a genuine region rather than an idealisation.

Shakedown. The member yields on the first cycle or two, which leaves behind a self-equilibrating residual stress field. That field then subtracts from the applied stresses on subsequent cycles, so the total stays below yield and every cycle after the first few is elastic. The deformation stops growing.

Ratcheting. No residual field exists that makes the response elastic, so each cycle adds a little more plastic strain. The member does not collapse — at no point in any cycle is the section at its plastic moment — and it does not return. It simply arrives somewhere further round every time.

The third has no failure load. There is no cycle at which anything breaks, so a collapse calculation on any single cycle returns a comfortable answer, and the failure is a deformation reached after some number of cycles that nothing in the calculation counts.

Which free body produced the number #

The free body is the section, and what makes shakedown a theorem rather than an observation is what is being looked for on it.

The applied load produces an elastic stress distribution $\sigma_e$ that varies through the cycle. Melan’s theorem says: if there exists any time-independent residual stress field $\bar\sigma$ — self-equilibrating, so in equilibrium with no external load — such that $\sigma_e + \bar\sigma$ nowhere exceeds yield at any point in the cycle, then the structure shakes down.

Two things about that statement are worth pressing.

It is an existence theorem. It does not say what the residual field is or how the structure finds it; it says that if one can be exhibited, shakedown occurs. That makes it a lower-bound theorem of exactly the kind the collapse theorems are, with the residual field playing the part the equilibrium stress field plays there.

And the residual field is a self-stress state. It is in equilibrium with no load at all, which is the same object a settled support and a prestressed tendon produce. The structure’s own first-cycle yielding is what installs it.

So the whole of the subject is: does a self-stress state exist that makes the rest of the history elastic? If yes, shakedown; if no, ratcheting.

There is a fourth possibility that belongs on the same map and does not appear on these figures, and it is worth naming so that the three are not mistaken for all of them.

Alternating plasticity happens when the secondary action alone is large enough to yield the section in both directions on every cycle. The plastic strain reverses rather than accumulating, so nothing moves — the curvature returns to where it started each cycle — and the material is being worked back and forth at a plastic strain amplitude.

That is not a deformation failure and it is not a collapse. It is low-cycle fatigue: the material exhausts its ductility after some hundreds or thousands of cycles and cracks, at a life governed by the plastic strain range rather than by any stress.

So the map has four regions and the two failures on it are of entirely different kinds — one a displacement that grows without limit and one a crack that arrives after a countable number of cycles. Neither is a load, which is the property this whole subject has and every other check in this collection does not.

The map has two axes #

The boundary between the regions is not a curve in one variable, and that is the practical content.

Those two figures are the axes of the Bree diagram: primary load across, secondary load up, with the three regions separating out.

The asymmetry between them is the finding. A secondary load — a temperature, a settlement, an imposed strain — is self-limiting: once the section has yielded it relaxes, and no amount of it alone will produce ratcheting. A primary load — a moment from gravity, a pressure — is not: it has to be carried, it does not relax, and it is what pushes the section round the loop.

Ratcheting needs both. A structure with only a primary load either stands or collapses; one with only a secondary load either stays elastic or shakes down. It is the combination that produces a mechanism with no failure load.

The material moves the boundaries too #

Everything above holds the material constant, and the map depends on it in a way that is worth separating from the geometry.

Two mechanisms are at work in those figures and they point opposite ways.

A higher yield stress raises the boundary in the primary direction, because the section reaches its plastic moment later. It does not change the thermal stress at all.

A lower modulus lowers the thermal stress, because a temperature profile imposes a strain and the stress it produces is $E\alpha\Delta T$. So a soft material is better against secondary actions and worse against primary ones.

The axes of the map are therefore not measured in the same currency, and a material change moves a structure diagonally rather than along either axis — which is why a substitution that improves the collapse load can move a member into the ratcheting region.

Why a collapse calculation cannot see it #

The three regimes are all below the collapse load, and that is not an accident.

A collapse calculation asks whether a mechanism exists at the current load. In every case on these figures the answer is no — the section is nowhere near its plastic moment under the combined action at any instant, because the thermal stress relaxes as soon as it yields.

Ratcheting is a failure of a different kind: an accumulation rather than a mechanism. The quantity that grows is a displacement, the number of cycles is the independent variable, and there is no single state of the structure that is unacceptable.

That has an awkward consequence for how it is checked. A ratcheting limit is a serviceability limit with no service load in it — the load is the ultimate combination, the criterion is a deformation, and the number of cycles is a lifetime rather than a load case. No standard limit-state framework has a place for it, which is why it is checked, when it is checked at all, by a special provision.

Where it turns up #

The classic case is a pressure vessel wall — Bree’s own problem, and where the diagram is named from — but the mechanism needs no temperature at all.

A bridge bearing walks. A deck expands and contracts daily, the friction at the bearing resists both ways, and if there is any bias — a slope, a braking force, a difference between the two ends — the deck arrives a little further along each year. A structure whose thermal cycle is not symmetric is ratcheting in translation rather than in curvature, and bridge bearings have been found metres from where they were installed.

An integral abutment ratchets against its backfill. The deck pushes the soil each summer, the soil settles into the gap each winter, and the earth pressure rises year on year toward the passive value.

A pavement ratchets under traffic. Each axle produces a small permanent strain in the subgrade, the material has no yield stress and no shakedown limit in the classical sense, and the rut deepens as the number of passes rather than as the load.

And a frame ratchets sideways under repeated lateral load with gravity present. The gravity load is the primary action, the lateral cycle the secondary, and the frame leans a little further each time — which is exactly the seismic ratcheting that governs a structure with a bias in its strength.

What the residual field actually is #

The theorem asserts a self-stress field into existence, and it is worth seeing what one looks like on a section, because it makes shakedown feel mechanical rather than abstract.

Take the rectangle under a moment plus a thermal gradient. On the first cycle the thermal stress adds to the bending stress at one face and the section yields there — a thin layer of material at that face is taken past its elastic limit and does not come back.

When the temperature returns to zero, that layer is left with a residual stress of the opposite sign to the one that yielded it, held in equilibrium by the rest of the section. There is no external load associated with it: it is a self-equilibrating distribution through the depth, and it is exactly the field Melan’s theorem is about.

On the next cycle the thermal stress arrives again and adds to the sum of the bending stress and this residual. If the residual is large enough, the sum stays below yield and the cycle is elastic. The structure has manufactured its own protection out of its first excursion.

Ratcheting is what happens when the residual field it can manufacture is not enough — when the layer that yields on one side is undone by yielding on the other before it can stabilise, so each cycle installs and destroys a residual field rather than accumulating one.

The whole difference between the two regimes is whether a self-stress state can survive the cycle, which is why the theorem is an existence statement and why it is a lower bound: exhibiting one field is enough.

What sets the boundary #

The two boundaries have simple forms for a rectangle, and both are worth carrying.

The elastic limit is where the sum of the primary and secondary elastic stresses first reaches yield. It is a straight line on the map, and inside it nothing ever yields.

The shakedown limit is where no residual field can keep the response elastic. For a rectangle under a moment plus a linear thermal gradient the boundary is a hyperbola: at low primary load a very large thermal range shakes down, and at high primary load almost none does.

The shakedown load is between the elastic limit and the collapse load, always, and closer to the second when the secondary load is small. That ordering is what makes the concept useful: a structure designed to its collapse load may ratchet, and one designed to its elastic limit is wasting most of its capacity.

How it is actually kept out of a design #

Given that no ordinary check finds it, the practical defences are three, and each is a design decision rather than a calculation.

Keep the primary load below about half the plastic capacity. The ratcheting boundary is furthest from the origin at low primary load, and a member at 30 per cent of its plastic moment will shake down under almost any secondary action. That is a generous limit for most members and a binding one for a member sized by strength.

Let the secondary action out. A movement joint, a sliding bearing, a release: each converts an imposed strain into a movement and removes the secondary axis from the map. Where a structure is allowed to move is the most direct answer available, and it is a detailing decision made at the start.

Or accept it and count the cycles. A member that ratchets at 0.4 per cent of first-yield curvature per cycle, under a hundred cycles in its life, has gained 40 per cent of a first-yield curvature and nobody will notice. The same rate under a hundred thousand cycles is a demolition. The rate is only half of the criterion and the count is the other half, and the count is a use rather than a load.

That third route is the one a pavement takes, a bearing takes and a soil takes, because for all three the alternatives are unavailable — and it is the reason each of those subjects has a permanent-deformation limit stated in cycles rather than a strength check.

What to carry away #

Three regions, and the third has no failure load. A ratcheting member never collapses; it accumulates.

The map has two axes and they are not interchangeable. Secondary loads relax and primary ones do not, so ratcheting needs both.

The theorem is an existence statement about a self-stress field, which makes it a lower bound of the same family as the collapse theorems.

And nothing in a strength check will find it. The load is safe on every cycle and the failure is a displacement after some number of them.

Where the model stops #

The material is elastic–perfectly plastic. Real steel hardens, which raises the shakedown boundary, and softens cyclically after enough reversals, which lowers it. The two act at different numbers of cycles.

The section is the only free body. Shakedown of a structure is a different and harder problem: the residual field has to be self-equilibrating over the whole frame, and the load factor it gives is not the sum of the sections’.

Alternating plasticity is not covered. There is a fourth regime in which the plastic strain reverses each cycle rather than accumulating, so nothing moves and the material fails by low-cycle fatigue instead. It occupies a corner of the same map.

Creep is absent. At temperature a structure ratchets and creeps at once, and the two cannot be separated — which is the case the original vessel work was actually about.

And the number of cycles is not in the criterion. A member ratcheting at 4.7 per cent of first-yield curvature per cycle is unacceptable after two hundred cycles and irrelevant after five, and nothing on this page distinguishes them.

A last observation about where the concept sits. Shakedown is bounded below by the elastic limit and above by the collapse load, so it is a third load factor between two that structural engineering already computes. Nothing in an ordinary design finds it, because a design computes the two ends and assumes the interval between them is safe — and for a structure carrying only primary loads it is. The moment a cycling secondary action appears, the interval acquires a boundary inside it, and the boundary is the one that governs.

The map’s three regions each have an essay of their own elsewhere in this field. Elastic shakedown is the structure that settles down; alternating plasticity is yielding one way and then the other, which is where the ductility is spent; and the monotonic boundary is what happens after the first yield. The map’s contribution is to say that the three are regions of one diagram rather than three separate checks.

The ladder from here #

Later rungs on this anchor: Melan’s static theorem and Koiter’s kinematic one, proved, with the residual field they assert into existence. The Bree diagram drawn as a map with its regions and their boundaries derived. Shakedown of frames rather than sections, and the load factor it produces against the collapse factor. Alternating plasticity as the fourth regime, where the strain reverses and the failure is low-cycle fatigue. Deck walking and bearing design, which is ratcheting in translation. Ratcheting in soils under repeated traffic, which is the same mechanism in a material with no yield stress. And the interaction with creep, where a structure at temperature does both at once.

Bree’s paper is from 1967 and is about a nuclear fuel can: a thin cylinder with an internal pressure and a thermal gradient through its wall, cycling with the reactor’s power. The map that came out of it has been redrawn for frames, pavements, bearings and abutments since, and the reason it travels is that its two axes are not a material property or a geometry — they are a load that has to be carried and a strain that is imposed, which every structure has.