What is left when the load comes off

Loading is complicated and unloading is simple, and the asymmetry is the most useful thing in this part of the subject. It is also the reason that a structure’s state after an overload is computable at all: without it, every question about what a bent beam is now carrying would require simulating the whole history that bent it.

A material being loaded past yield is doing something that no linear relation describes: its stiffness depends on how far it has come, its response depends on which direction it is going, and superposition — the property that makes almost every method on this site work — has stopped applying. Reverse the direction and all of that goes away at once. The material unloads down a straight line of slope $E$, exactly the slope it started with, and it does so whatever plastic strain it has accumulated.

Why the unloading is straight #

The reason is worth being clear about because it licenses everything below. Yielding is the movement of dislocations, and dislocations move in one direction under a stress that exceeds a threshold. Reduce the stress below that threshold and they stop moving; there is no mechanism by which the accumulated plastic strain reverses itself, and what remains is the elastic strain of the lattice, which relaxes exactly as it would in a bar that had never yielded.

So the unloading path is $\Delta\sigma = E\,\Delta\varepsilon$, and the total state at any point of the unloading is the plastic state at the reversal plus an elastic solution. That is a superposition, and it is available even though the loading that produced the state was not superposable at all.

The practical consequence is that the residual state of a structure after an overload can be computed in two steps, neither of which needs an incremental analysis. Solve the plastic problem at the peak. Solve the elastic problem for the load being removed. Add them. Whatever comes out is what is left.

What a section keeps #

Apply that to a rectangle that has been bent past first yield and then released.

At the peak, the stress diagram is flat-topped: yield stress over the outer parts, a linear core in the middle. Removing the moment means adding an elastic stress distribution of the opposite sign, and an elastic distribution is a straight line through the neutral axis — so the removal subtracts most from the extreme fibres and least from the middle. Subtracting a straight line from a flat-topped block does not give zero.

What survives is compression where there was tension and tension where there was compression, in a pattern that sums to no force and no moment because the applied moment has gone. The section is left carrying a self-equilibrating stress field of exactly the kind a rolling mill leaves in it, and by the same test: it is invisible to statics, because every free body is in equilibrium with or without it.

The section is now better than it was #

Here is the part that reads as a paradox and is not. Reload the released section in the same direction. It behaves elastically all the way back to the moment it reached before, because the residual field left by unloading is precisely the field that cancels the overshoot.

So a section overloaded once and released has a larger elastic range in that direction than it started with. Its first-yield moment has risen to the moment it was taken to. Nothing about the material changed — the yield stress is the same, the modulus is the same — and the improvement is entirely geometric bookkeeping.

The improvement is not free and its price is paid in the other direction. The residual field that helps on reloading hinders on reversal: the fibres that now carry residual compression reach yield in compression sooner. The elastic range has been shifted, not widened, and shifting it has cost exactly as much on one side as it gained on the other.

Which free body produced the number #

The permanent strain in the hero figure is the free body at its smallest: a point of material with one stress and one strain. The state is carried as three numbers — the current stress, the accumulated plastic strain, and the centre of the yield interval — and the rule is that the stress may not leave the interval.

Given a strain increment, the stress is advanced elastically to a trial value. If the trial is inside the interval, that is the answer and nothing plastic happened. If it is outside, the excess is converted to plastic strain at a rate $1/(E+H)$ and the interval’s centre is dragged along behind. Sweeping a strain path through that rule produces the loops above with nothing else supplied.

The Bauschinger figure’s 550 is the check that the rule is doing what it claims. Nowhere in the implementation is there an instruction to yield in compression at $2f_y$ below the tensile peak; there is only an interval of half-width $f_y$ whose centre moves. That the reversal comes out at exactly twice the yield stress is a consequence, and it is the consequence that distinguishes this rule from the alternative one — isotropic hardening, in which the interval grows instead of sliding, and reversal would come at the same stress as before.

The permanent strain is checked a second way. The specimen was taken to 0.6% and the elastic recovery is $\sigma/E = 275/210000 = 0.131\%$, so the permanent set has to be $0.6 - 0.131 = 0.469\%$, which is what the figure reports. Two routes, one from the integration and one from a division.

Where the model stops #

Real materials do not have a sharp corner on reversal. The measured Bauschinger effect is a gradual softening rather than a clean elastic range followed by a clean yield, and the effective elastic range on reversal is typically somewhat less than $2f_y$. Kinematic hardening is the simplest model that gets the direction and the order of magnitude right, and its sharpness is an idealisation.

Neither pure kinematic nor pure isotropic hardening describes anything. Real cyclic behaviour is a mixture, and which mixture depends on the material and on how many cycles have been applied. Steels that are initially soft harden under cycling; steels that have been cold-worked soften. Predicting either requires a model with more state in it than three numbers.

The residual field a section keeps is bounded, and the bound is useful. Unloading elastically from the fully plastic state subtracts a straight line whose extreme value is the elastic moment $M_p/Z$, and the block it is subtracted from tops out at $f_y$. So the residual stress a section can retain from bending is at most $f_y(\text{shape factor} - 1)$ at the extreme fibre — half the yield stress for a rectangle, and only 9% of it for a compact I-section. A section with a small shape factor cannot store much, which is the same fact as its having had little in reserve, seen once more from a third direction.

And unloading is only elastic if nothing else changed while it was loaded. A section that buckled locally at the peak does not unload down a straight line, because part of its geometry has gone. A cracked concrete section does not either, because the cracks close progressively and the stiffness recovers over the unloading rather than being constant through it.

What a full cycle looks like #

Taking the specimen round and round rather than once and back draws the object the whole of cyclic plasticity is about.

Two features of that loop are worth naming because both are consequences of the sliding interval rather than inputs to it. Its width along the stress axis at any strain is $2f_y$, because that is the width of the interval. And it is closed, meaning the material has no long-term memory of how many cycles it has been through — which is a property of perfect kinematic hardening and an idealisation, since real metals harden or soften over the first few dozen cycles before settling into a stable loop.

The enclosed area is the design quantity for anything meant to absorb energy. A seismic damper, a crash barrier, a yielding brace: each is a device arranged so that its loop is as fat as possible for the displacement available, and the calculation of how much energy it can dissipate per cycle is the calculation of that area.

The generalisation #

The pattern is that plasticity is a one-way accumulation and elasticity is a two-way spring, so any load history can be decomposed into an irreversible record and a reversible response to the present.

That decomposition is what makes plastic analysis tractable. It is why the collapse load of a structure does not depend on the order in which the loads were applied, why a residual state can be computed by subtraction, and why shakedown is a question with a yes-or-no answer rather than a simulation.

It is also why an overload is not simply damage. A structure that has been overloaded and survived is in a different state, not merely a worse one, and the difference is a stress field that helps against a repeat of the same load and hinders against its reverse. Whether that is an improvement depends entirely on what happens next.

It is worth noticing which of this site’s methods survive the loss of superposition and which do not. Influence lines are superposition made visible and stop meaning anything once a section has yielded. Maxwell’s reciprocal theorem is a theorem about elastic structures and is false for plastic ones. But the collapse load survives, because the work equation is written at the mechanism rather than along the path, and the residual-state calculation on this page survives too, because unloading put the elasticity back.

A surprising place this turns up #

Almost every deliberate use of this effect is the same trick: overload something once, in a controlled direction, so that the residual field it keeps is the field that will help.

Autofrettage. A thick-walled cylinder — a gun barrel, a high-pressure vessel — is pressurised past yield at the bore once, at the factory. Releasing the pressure leaves the bore in compression, so in service the pressure has to overcome that compression before it produces any tension at all. The vessel’s elastic pressure capacity is substantially higher than it was, from one application of the thing that would have destroyed it if repeated in the other direction.

Shot peening. The surface of a component is hammered with small hard particles, yielding a thin layer in tension so that on release it is held in compression by the material below. Fatigue cracks start at surfaces in tension, and a surface that only reaches tension after the compression is used up starts them much later.

Cold forming. A cold-rolled section’s corners have been bent past yield, which raises their yield stress locally by strain hardening and leaves them holding a residual field. Design codes for cold-formed steel allow the corner strength to be counted, which is the one place the profession routinely takes credit for it. It also explains why a cold-formed section classifies differently from a hot-rolled one of the same proportions: the residual pattern has the opposite sign at the corners, and the corners are what hold the plates.

Cold straightening. A member that has arrived on site bent is straightened by loading it past yield the other way and releasing it, which is this page’s calculation run backwards to find the load that leaves the required permanent set. It works, it is routine, and it leaves behind a residual field and a locally reduced ductility that nothing subsequently records.

And prestressing. The largest instance of all is an entire structural material designed around it: concrete has essentially no tensile capacity, so a compression is put into it before the load arrives and the load’s tension is spent cancelling it. The mechanism is different — the compression comes from a tendon rather than from unloading — but the strategy is identical: arrange for the stress that will matter to start from the far side of zero.

Where the ladder goes next #

Later rungs on this anchor: springback in forming, which is the same calculation run to find a die shape rather than a residual stress. Autofrettage worked through, with the optimum overpressure. The residual stress field after a plastic hinge has formed and unloaded, and what it does to the beam’s subsequent elastic range. Cyclic hardening and softening, and the stabilised loop that most metals reach after a few dozen cycles. Isotropic against kinematic hardening as competing models, and the experiments that separate them. Ratcheting under combined steady and cyclic load. And the energy dissipated per loop, which is the enclosed area and is the quantity a seismic damper is designed against.

Historically the effect is named for Johann Bauschinger, who measured it in Munich in the 1880s while running one of the first laboratories dedicated to testing structural materials rather than to physics. He was looking for the elastic limit and found that it depended on what the specimen had already been through — an inconvenience for anyone hoping the elastic limit was a constant, and the first clear evidence that a metal’s state includes its history.