Structural Engineering & Statics
where the load goes
Concept

Bauschinger effect — where it appears

The drop in yield stress a metal shows on reversal after it has been taken past yield in the opposite direction. It shortens the elastic range on the second half-cycle, rounds the stress-strain loop, and is the reason a cyclically loaded member dissipates energy long before it reaches its monotonic capacity.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

What is left when the load comes off. Mild steel taken to a strain of 0.60% and then unloaded to zero stress, at which point the strain has not returned to zero: 0.469% of it is permanent.

What is left when the load comes off

Unload a section that has yielded and it does not return to nothing. It returns to a self-equilibrating stress field it did not have before, a permanent set, and an elastic range wider than the one it started with.

materials · Unloading Yielding one way makes it easier to yield the other. Mild steel taken to a strain of 1.20% and then pushed back the other way. The stress falls by 550 N/mm² before it yields again, against a yield stress of 275 — the elastic range is twice the yield stress and not once it, which is the Bauschinger effect and is a consequence of the yield surface sliding rather than growing.

Yielding one way, and then the other

A material that has yielded in tension yields earlier in compression than it did the first time, and by an amount that is exactly what makes its elastic range twice its yield stress rather than once it. That is a property no monotonic test reports and every reversing structure depends on.

materials · Ductility Past a wall ratio of 2.22, a thicker tube gains nothing more. The most autofrettage can raise a thick cylinder's elastic pressure, as a multiple of its first-yield pressure, against its wall ratio b/a. Yielding the whole wall would buy σy ln k ÷ py (faint), which keeps rising; the release caps it at 1 + β, where β is the reverse yield strength over the forward one. With β = 1.0 the two meet at b/a = 2.22, and the gain is 2.00 for every thicker wall; with β = 0.7 the two meet at b/a = 1.80, and the gain is 1.70 for every thicker wall; with β = 0.5 the two meet at b/a = 1.55, and the gain is 1.50 for every thicker wall. Thinner walls than that can be yielded right through and gain less.

The overstrain the release gives back

A thick tube is pressurised past yield once, at the factory, so that releasing the pressure leaves its bore in compression and the working pressure must overcome that before it does any harm. The obvious rule is to overstrain as far as possible — right through the wall. For a tube whose outside is more than 2.22 times its bore that is wrong: the release itself reverses the bore past yield, and every newton of overstrain beyond twice the first-yield pressure is given back on the way down. The best overstrain stops part-way through the wall, and a steel that yields early in reverse stops it sooner.

materials · Unloading Two ways to put a bore in compression. For a tube 2.5 times its bore in a 900 N/mm² steel: the stress difference σθ − σr through the wall, as a share of the yield strength, at rest (dashed) and at the working pressure (solid) — for the wall shrunk together from two layers so that every layer's bore yields together at work, and for the same wall autofrettaged to its best overstrain (thin lines). The layered wall carries 540 N/mm², 1.43 times the first-yield pressure, and at rest its bore sits at −0.43 of yield; the autofrettaged wall carries 756, 2.00 times, its bore at rest at −1.00.

The ceiling the bore sets

A thick tube can be put into compression at its bore two ways: overstrain it past yield once and let it spring back, or build it from layers each shrunk on to the one inside. The second never yields the steel at all. Two layers carry 1.43 times a plain tube's pressure where an overstrain carries 2; four layers 1.75; and no number of layers, no overstrain and no combination of the two carries more than twice. The ceiling is not in either method. It is the range the bore's stress swings through, which belongs to the assembled wall, and the floor that range may reach, which belongs to the steel.

materials · Unloading The wire's strength buys the wall. The thinnest wall, as a multiple of the bore, on which a wound tube can carry the ceiling, against the wire's strength over the liner's. Wire as strong as the liner needs 3.01; 1.2 times as strong, 2.30; 1.5 times, 1.69; from about 2.00 times, any wall. Dotted, 2.22: the thinnest wall on which shrunk-on layers of the liner's own steel reach the ceiling, where σy·ln k is twice the first-yield pressure — thinner than the same steel wound, because a layer can carry compression and a wire cannot.

The wire that cannot push

A tube's liner wound with wire under tension is a compound tube of a hundred layers, each shrunk on by its own winding. It reaches the same ceiling, twice the first-yield pressure, and no further, because the bore's range belongs to the whole wall and its floor to the liner's steel. What the wire's strength decides is where the ceiling can be reached at all. A wire cannot be compressed, so a jacket of the liner's own steel needs a wall three times the bore, where shrunk-on layers of the same steel need 2.22; wire half as strong again needs 1.69; wire twice as strong reaches it on any wall. Every turn is wound harder than it will rest, by what all the turns after it take away.

materials · Unloading

Named alongside it

The objects these essays reach for when they reach for this one.

Residual stressAutofrettageThick cylinderTrescaUnloadingHoop tensionKinematic hardeningPrestressShakedownBuckling-restrainedConstruction sequenceCyclic loading

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