Structural Engineering & Statics
where the load goes
Concept

Low-cycle fatigue — where it appears

Failure after hundreds or thousands of cycles in which a material yields every time, rather than the millions of elastic cycles of ordinary fatigue. It is what alternating plasticity leads to: a section yielding one way and back again on every cycle of load.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

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 The loads a portal can carry for ever. For a fixed-base portal 10 m wide and 5 m high, plastic moment 200 kNm in the columns and 200 in the beam, in the plane of the gravity load's peak V·L/Mp — the load comes and goes — and the wind's H·h/Mp — it reverses: the first-yield boundary (dotted), the shakedown boundary (solid), and the collapse boundary under the same two loads applied once (dashed). With little wind the shakedown boundary is the beam's own mechanism, which the wind does no work in, falling from 8 to 6.0 as the wind grows to H·h/Mp = 2.7; beyond that it is alternating plasticity at a column base, nearly level. The dot is 160 kN of gravity with 60 kN of wind and the ring its shakedown point, a factor of 0.88: the gravity load alone is at the beam's collapse load, and the wind takes 12% off what the pair can carry without doing any work in the beam's mechanism.

The beam the wind walks without pushing

A fixed-base portal whose beam carries exactly its collapse load is safe under that load applied once. Add a wind that comes from either side — a wind that does no work at all in the beam's collapse mechanism — and the beam can no longer carry the load for ever: 12 per cent comes off, and above that the midspan gains fifteen milliradians every cycle while the frame does not sway. Each knee takes its worst moment from a different wind, and the beam assembles a mechanism no single storm drives.

materials · Shakedown Pinned bases: weaker, and nothing lost to repetition while gravity leads. For a portal 10 m wide and 5 m high, plastic moment 200 kNm in the columns and 200 in the beam, in the plane of the gravity load's peak V·L/Mp and the reversing wind's H·h/Mp: the shakedown boundary (solid) and the collapse boundary under the loads applied once (dashed), on fixed bases and on pinned ones. On pinned bases the two coincide wherever gravity leads, along the beam and combined mechanisms, until the wind reaches about H·h/Mp = 1.6; past that a knee alternates and shakedown falls inside collapse, meeting it again at H·h/Mp = 2 with the wind alone. On fixed bases collapse reaches 8 and 4 and shakedown sits inside it except where the beam's mechanism meets the gravity axis. The dot is 160 kN of gravity at midspan that comes and goes with 60 kN of wind that reverses: its shakedown factor is 0.88 on fixed bases and 0.73 on pinned ones, where it is also the collapse factor.

The base that sends the wind to the knees

A fixed-base portal loses 12 per cent of its load to a wind that reverses, because each knee takes its worst moment from a different storm. Pin the bases and the loss disappears — not because the pinned frame is better, but because its collapse load already contains it. What the bases decide is where the moment goes, and a base that is merely weaker than its column can cost nothing at all, or be made into the part that yields every storm so that nothing else does.

materials · Shakedown Past yield, the notch trades stress for strain. The root of a notch of elastic factor 3.0 in a mild steel whose cyclic curve reaches 0.2 per cent strain at 252 N/mm², under a nominal stress of 200 N/mm²: the cyclic stress–strain curve (solid), the elastic line the root would follow if it stayed elastic (thin), and Neuber's hyperbola σε = (KₜS)²/E (dashed), which the curve crosses at the root's state. Elastic, the root would carry 600 N/mm² at 0.30 per cent. By Neuber it carries 293 at 0.61 per cent; by Glinka's energy rule 272 at 0.44; by the linear strain rule 246 at 0.30.

The notch that trades stress for strain

A notch with an elastic factor of three triples the stress at its root only while the root stays elastic, which in a mild steel is up to a nominal stress of about 84 N/mm². Past that the root's stress and strain part company: at a nominal 200 the root carries 293 N/mm², not 600, and strains 0.61 per cent, twice what the elastic factor implies. Neuber's rule and Glinka's bracket the truth a factor apart, and in the low-cycle range that factor is a factor of two in life. The trade has two gifts in it: one overload leaves the root in compression, and a mean stress mostly yields itself away.

materials · Stress concentration Spread over the span, the rotation gathers at the supports every drop shares. The plastic rotation each station of a 6 m beam fixed at both ends (EI 42,000 kN·m², plastic moment 200 kN·m) struck 20 times by a weight of 4.9 kJ has gathered by the last drop, for drops always at mid-span, at 0.3 and 0.7 of the span in turn, and at random. Always at mid-span, the worst station holds 0.193 radians, at 3.0 m, and 3 stations more than a milliradian; at 0.3 and 0.7 in turn, the worst station holds 0.104 radians, at a support, and 4 stations more than a milliradian; at random, the worst station holds 0.128 radians, at a support, and 15 stations more than a milliradian.

The weight that lands somewhere else

A beam that yields under a falling weight takes a permanent set from every drop, and if the weight always lands in the same place the set grows by the same amount every time. Let the weight land somewhere different each time — rocks on a canopy, crates on a deck — and the set still grows: the weight's energy fixes how much plastic work every drop does, wherever it lands. What the landing point decides is where the work goes. Half of every drop's hinge rotation goes under the weight and half to the supports, so wandering drops spread the first half along the span and leave the second half where it always was: the worst hinge turns half as fast, and it is at a support.

dynamics · Impact factor

Named alongside it

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

Residual stressShakedownMechanismPlastic hingeRatchetingCyclic loadingDuctilityPortal frameSelf-stressBase plateBauschinger effectBuckling-restrained

All concepts