Structural Engineering & Statics
where the load goes
Concept

Cyclic loading — where it appears

Load applied repeatedly rather than once, so that a structure's response depends on its history as well as on the current value. It raises questions a monotonic check cannot answer — fatigue, shakedown, ratcheting and the degradation of stiffness with each cycle.

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

Which of them stops moving. Three load cases on the same rectangle, each a constant moment plus a temperature profile cycled from nothing to a peak and back, over sixteen cycles. At 20% of the plastic moment with a 40°C profile it never yields at all; At 50% of the plastic moment with a 150°C profile it shakes down; At 90% of the plastic moment with a 260°C profile it ratchets, at 51.0% of the first-yield curvature per cycle. The ratcheting case never collapses and never returns: it simply arrives somewhere further round every cycle, which is a serviceability failure that no collapse calculation contains.

The map with three regions

A structure carrying a constant load and a cycling temperature has three possible fates and only one of them is a collapse. It can stay elastic, it can yield once and then stop, or it can gain a little more deformation every cycle for ever — and the third has no failure load at all.

materials · Shakedown 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 base comes back to a different joint. Moment against rotation at the base of a 500 × 400 × 20 mm base plate with two M24 holding-down bolts in each of two rows, preloaded to 150 kN a bolt, through four cycles to ±170 kN·m. It is stiff until the bed lets go at 57 kN·m, softer past it, and near 170 kN·m the bolts on the tension side yield and stretch for good. Coming back through zero the base is not the base it was: its bolts have kept 83 per cent of their preload, and it lets go at 46.0 kN·m instead of 56.6. After the first full cycle the loops lie on top of one another.

The base that lets go and closes again

A column base whose holding-down bolts are preloaded is rigid until the moment on it lifts the plate off its bed, and then softer. Under a storm the moment goes one way and then the other, many times, and the question is whether the preload survives. It does, entirely, as long as no bolt yields — and on an ordinary base the bolts do not yield until the moment is three times what first lifts the plate. Past that the yielded bolts lose part of their preload once, and the loss stops within a cycle or two: the base settles rather than ratchets. What decides how much is lost is how close to its yield the bolt was tightened, so that past a point a larger preload leaves less after the storm.

connections · Base plate Retightened after every storm, the bolts stretch in every storm. The permanent stretch in the worse bolt row of a 500 × 400 × 20 mm base plate with two M24 holding-down bolts in each of two rows, accumulated over ten storms of four cycles to ±170 kN·m, with the nuts retightened after every storm to 200 kN, to the 150 kN they were first given, to the 124 kN the first storm left, or left alone. Left alone, the bolts stretch 0.12 mm in the first storm and not at all after it. Retightened to 150 kN they add 0.12 mm in every storm, 1.25 mm after ten; to 200 kN, 0.43 mm a storm. Retightened only to what the first storm left, they stretch no more than if they had been left alone.

The nut turned after the storm

A preloaded column base that has been through a storm has lost some of its preload, and the obvious repair is to retighten its bolts. Left alone, the bolts stretched once in the first storm and never again; the base settled. Retightened, they are given back the preload the storm could not live with, and the next storm takes it away again by stretching them again. Retightening after every storm turns a base that settles into one that ratchets, a tenth of a millimetre at a time, and the count of storms it has in it is set by the bolt's elongation.

connections · Base plate 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

Named alongside it

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

ShakedownResidual stressYieldBase plateFree bodyLow-cycle fatiguePreloadRatchetingAnchor boltBauschinger effectBuckling-restrainedCollapse

All concepts