Yielding one way, and then the other

Ductility appears in none of the equations and decides a great deal, and the reason it is hard to put in an equation is that it is a property of a history rather than of a state. The simplest history that shows it is one reversal.

The yield surface slides rather than grows #

The number 550 is 2 × 275, and that is not a coincidence about mild steel.

Think of the yield condition as a surface in stress space — the criterion that says when — and ask what plastic straining does to it. Two idealisations are available.

Isotropic hardening grows the surface: yielding in tension raises the yield stress in both directions equally. Under that model a bar taken to 1.20 per cent strain would yield in reverse at a stress larger in magnitude than the original yield, and the elastic range would be more than 2f_y.

Kinematic hardening translates the surface: the elastic range keeps its size and moves. Under that model the reverse yield happens 2f_y below the forward one, whatever the forward stress reached.

Real metals do the second, very nearly, and the effect is named after Bauschinger, who measured it in 1881. The consequence is that a material’s forward and reverse yield stresses are linked — raise one and lower the other — rather than being independent properties.

Which free body produced the number #

The free body is a cube of material and the argument is about what is inside it.

Plastic deformation in a metal is dislocations moving through a crystal lattice. Pushing them one way piles them up against obstacles — grain boundaries, precipitates, other dislocations — and leaves behind a back stress: a field of internal stress, self-equilibrating over the volume, opposing further motion in that direction.

Reverse the load and the back stress now assists the motion. So the material yields earlier in reverse by exactly the back stress, which is the same amount by which it was hardened forwards. The surface translates because the back stress is a vector, and its magnitude is what a monotonic test reports as hardening.

That is the mechanical content of the effect and it explains why the elastic range is preserved: nothing about the obstacles has changed, only the direction they are being pushed against.

The structural analogue is exact. A back stress is a residual stress field at the scale of a grain — self-equilibrating, produced by yielding, and modifying the response to the next load — which is the same object a shaken-down section carries at the scale of a member.

There is a third idealisation worth naming because it is what a hand calculation usually assumes, and it is neither of the two above.

Elastic–perfectly plastic has no hardening at all: the surface neither grows nor moves, and the reverse yield is at $-f_y$ exactly. Under that model the elastic range after a reversal is $2f_y$ as well — which is the same answer kinematic hardening gives, and is why the simplest idealisation happens to be right about the thing that matters most.

That coincidence is worth knowing because it says when the distinction does matter. It does not matter for the size of the elastic range, and it does not matter for the shape of a stable loop. It matters for the force the member actually reaches, which under kinematic hardening keeps rising and under perfect plasticity does not.

For capacity design that difference is the whole question: the member protecting a ductile one has to be stronger than the ductile one’s real strength, which includes the hardening. Perfect plasticity is right for the deformation and wrong for the force, which is why codes use one idealisation for the analysis and an overstrength factor for the protection.

What a permanent set is #

The other half of a reversal is what is left when the load comes off.

Unloading is elastic, always, and at the original modulus. That is the property that makes a permanent set computable: the strain recovered is $\sigma_{\max}/E$ and everything else stays.

What is left when the load comes off is therefore not a damaged material — it is the same material at a different place — and its stiffness on reloading is the original one. That is why a structure that has yielded is stiff again afterwards, and why a proof load is a legitimate operation rather than a destructive one.

The loop, and what its area is #

Reverse repeatedly and the two effects combine into the object the whole subject is about.

Two properties of that closure carry the whole of seismic design.

It is stable. Because the hardening is kinematic, the loop settles into the same shape after a cycle or two and stays there. A material with strongly isotropic hardening would keep raising its yield stress, the loop would shrink, and the energy dissipated per cycle would fall away.

And its area is energy. Work done per unit volume per cycle is the enclosed area, and it leaves the material as heat. That is the only mechanism by which a structure permanently removes energy from an earthquake without a device, and it is the reason a ductile structure survives a load its strength cannot carry.

How much energy, against what #

It is worth putting the loop’s area beside the energy a structure has to lose, because the comparison is what makes ductility a strategy rather than a virtue.

A steel member cycled to a plastic strain of one per cent dissipates roughly $f_y \times \Delta\varepsilon_p \times 4$ per unit volume per full cycle — about 275 × 0.01 × 4 = 11 MJ per cubic metre. A brace of 3,000 mm² core over 6 m is 0.018 m³, so it sheds about 200 kJ per cycle at that strain.

Set that against what an earthquake delivers. A ten-storey frame of 3,000 tonnes at a spectral displacement of 100 mm holds of the order of a few hundred kilojoules of energy at its peak, and an earthquake delivers several times that over its duration.

So a handful of braces at a plastic strain of one per cent, over a dozen cycles, is the right order of magnitude — which is why the strategy works at all, and why it needs the members to be numerous rather than strong. Doubling a brace’s strength does not double its dissipation; doubling its volume of yielding material does.

That is the whole design consequence of the loop’s area being $f_y \Delta\varepsilon_p$ per unit volume. Energy dissipation is bought by volume and by strain, not by strength, and a stronger structure that stays elastic dissipates nothing at all.

Where the loop is not available #

The stable loop is a property of the material. A member can lose it entirely.

The material in both braces is identical and has the same stable loop. What differs is that one of them has a second failure mode — buckling — that the material knows nothing about, and the mode degrades with each cycle because a plastic hinge does not straighten itself.

A brace that yields both ways is a member built so that the material’s own loop is the member’s loop, by carrying the axial force in a core and the flexural stiffness in a casing that carries no axial force at all. The whole device exists to decouple two properties the material never coupled — and its casing has to satisfy one inequality, which is a buckling check on a member carrying nothing.

Two consequences for a design that reverses #

The translation of the yield surface has two direct effects on how a reversing member behaves, and both are counter-intuitive.

A member preloaded in one direction is weaker in the other. A brace stressed into tension during erection, a bar bent to shape and then straightened, a section rolled with residual stress: each has a back stress and yields earlier against it. That is not damage and it is not covered by any material certificate, since a tensile test reports the forward yield only.

And prestraining does not raise the reverse capacity. A designer who counts on strain hardening to give a reserve is counting on it in the direction of the strain, and getting the opposite in the other direction. For a member that will be cycled that is a net loss: the forward gain is available once and the reverse loss is available every cycle.

The practical rule that comes out of it is short. For a monotonic member, hardening is a bonus; for a cycling one, treat the yield stress as the whole capacity in both directions. Codes do exactly that, by ignoring hardening in every capacity check and then adding an overstrength factor where a member’s actual strength matters — which is a way of using the forward gain for capacity design and refusing it for capacity.

What limits the number of cycles #

Nothing above says how many circuits a material has, and the loop’s own area is what consumes them.

Each cycle imposes a plastic strain range, and the material’s life at a given plastic strain range is governed by the Coffin–Manson relation — a power law with an exponent near −0.5, so halving the strain range multiplies the life by about four.

That regime is low-cycle fatigue, and it is a different subject from the stress-range fatigue of a welded detail: the lives are hundreds or thousands rather than millions, the controlling variable is a strain rather than a stress, and no S–N category applies.

A member designed to dissipate energy is therefore designed to a number of cycles, and the number is small. A seismic brace is expected to survive perhaps twenty to fifty cycles at its design displacement, which is a specification rather than an incidental — and the test that qualifies it is a cyclic protocol rather than a monotonic pull.

What it changes about a section #

At the level of a cross-section the same argument produces the moment-curvature curve.

The two shapes differ enormously in how much curvature they need to reach their plastic moment, and that is the section’s contribution to ductility. A rectangle has material near the neutral axis that yields last and has to be strained a long way; an I-section’s material is nearly all at the extreme fibres and yields almost at once.

High shape factor means a long journey to the plastic moment, which is a demand on the material’s ductility rather than a gift. The I-section reaches its capacity easily and has less reserve afterwards; the rectangle has to be strained four times as far and has more.

That is worth knowing because it inverts the usual reading of a shape factor as a bonus: it is a ratio of capacities and simultaneously a demand for curvature, and the second is what decides whether the first is available.

What to carry away #

The elastic range after a reversal is 2f_y, because the yield surface translates rather than growing. Forward and reverse yield stresses are linked.

The loop closes and its area is energy. That stability is what makes cyclic dissipation possible, and it is a consequence of the kinematic hardening rather than a separate fact.

A member can lose the material’s loop. Buckling is the usual way, and it degrades with cycles because a plastic hinge does not straighten.

And the number of cycles is finite and small. Dissipation is paid for in low-cycle fatigue at a life governed by the plastic strain range.

The one place it is a nuisance #

Every use above treats the Bauschinger effect as a fact to be worked with. There is one place where it is straightforwardly unwelcome, and it explains a fabrication rule.

A plate that is cold-formed — rolled into a tube, brake-pressed into a channel, bent round a former — has been strained plastically in one direction at every corner. It is hardened there against further straining the same way and softened against straining the other way, and the corner carries a large residual stress field to go with it.

That has two consequences a designer meets. A cold-formed section is stronger at its corners than its flat parts, by 10 to 20 per cent, and codes let that be counted. And the same section is less ductile at its corners, so a cold-formed member asked to hinge does so away from the corner or not at all.

The rule that follows is a limit on how tightly a plate may be bent, expressed as a ratio of bend radius to thickness and varying with the steel’s own ductility. It is a rule about the reverse yield rather than the forward one, since a corner that has to unbend is being asked for exactly the strain the forming used up.

Straightening a bent member is worse than bending it was, which is the practical form of that and is why a distorted member is replaced rather than pressed back.

Where the model stops #

Hardening is treated as purely kinematic. Real metals do a mixture, and the isotropic part is what makes the first two or three loops differ from the stable one.

Cyclic softening is absent. Some steels and most aluminium alloys soften under cycling, so the stable loop is smaller than the first — which reduces the dissipation and is measured rather than predicted.

Rate effects are left out. A material loaded in a millisecond has a higher yield stress than one loaded slowly, and a seismic strain rate is somewhere between.

Nothing here is a temperature statement. The energy in the loop becomes heat, and a member dissipating a great deal of it in a short time gets hot enough to change the properties the loop was drawn with.

And the section curves assume the material has the ductility they need. A rectangle at 4.3 times its yield curvature is at a strain the material must be able to reach, and a brittle material simply fails on the way there.

One last connection, because it is the reason this page sits in materials rather than in dynamics. Everything above is a property of a cube of steel, measured in a testing machine, with no structure anywhere in it. The stable loop, the doubled elastic range, the permanent set and the finite number of cycles are all there before any member is made — and every structural strategy that depends on ductility is spending a material property that was fixed by a rolling mill.

That is worth stating because the profession’s vocabulary hides it. A “ductile frame” is not made ductile by its detailing; its detailing arranges for the material’s ductility to be reachable, by keeping every brittle mode — buckling, fracture, bond failure, shear — above the yielding one. The ductility was always there. The design decides whether the structure gets to use it.

The ductility spent here is the same quantity three other essays are about spending. A section that yields from the outside in is where the demand sits inside a member; the map with three regions is where repeated cycles put a structure on a diagram; and the property that appears in none of the equations is the awkward fact that the quantity every one of these arguments rests on is not in any of them.

The ladder from here #

Later rungs on this anchor: rotation capacity computed properly, with hinge length and moment gradient, against the demands redistribution generates. The upper- and lower-bound theorems proved, and exactly which step of each consumes ductility. Section classification as a rotation-capacity requirement rather than a table. Low-cycle fatigue and the Coffin–Manson relation, worked as a life. Robustness and tying, which is a ductility argument at the scale of a building. Ductility in reinforced concrete, where the design philosophy is arranged so the steel yields before the concrete crushes. And capacity design, the seismic idea of choosing where the ductility will be demanded and making everything else stronger than the chosen place.

Bauschinger measured the effect in 1881 on wrought iron, in a laboratory built to test the materials of the railway boom, and it stayed a curiosity of metallurgy for eighty years. What made it structural was earthquake engineering: a design philosophy that expects a building to yield needs to know what the material does on the way back, and the answer had been sitting in a German test report since before the profession existed.