Structural Engineering & Statics
where the load goes
Concept

Shakedown — where it appears

The state a structure reaches when repeated loading stops producing new permanent deformation. The residual stresses left by the first few cycles make everything afterwards elastic, so the structure settles down; its failure is ratcheting, in which each cycle adds a further increment and the deformation accumulates without limit.

Named by 7 essays across 4 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 30% of the plastic moment with a 20°C profile it never yields at all; At 60% of the plastic moment with a 120°C profile it shakes down; At 85% of the plastic moment with a 200°C profile it ratchets, at 1.9% 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 structure that settles down, and the one that walks

A load that is safe applied once may not be safe applied ten thousand times. Nothing about that is fatigue — the structure never breaks, it simply arrives somewhere slightly further round every cycle, until it has arrived somewhere unusable.

materials · Shakedown The summer that is worse than the one before it. The earth pressure behind an integral abutment, summer by summer, as a multiple of the at-rest value it started at. A 60 m deck expands by 10.8 mm at each end and pushes the abutment into the backfill. Granular soil under cyclic strain densifies, so the same movement next year needs a higher pressure to achieve, and K climbs from 0.38 toward 0.96 — a factor of 2.49 on the force, reached after about a century. The design load on an integral abutment describes the bridge's whole life rather than a load case, and it is the only load in this collection that gets larger because time has passed rather than because something was added.

The summer that is worse than the last

An expansion joint is a hole in a deck that leaks salt water onto the bearings underneath it. Remove it and the thermal movement does not go away — it goes into the soil behind the abutment, twice a day for a hundred and twenty years, and granular soil under cyclic strain gets denser.

structures · Integral bridge 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 Same deck, same load, and two pier forces. A deck bearing on a pier of 20 kN/mm with μ = 0.03, taken to the same final state two ways: the deck moves 4 mm over the pier, and the bearing's load rises from 2000 to 4000 kN. Moved first, while the bearing carries 2000 kN, the pier force reaches the limit of 60 kN and the bearing slides for the rest of the movement; the load arriving afterwards raises the limit and changes nothing, and the pier is left carrying 60 kN. Loaded first, the limit is 120 kN before the deck moves, the bearing grips throughout, and the pier carries 80 kN. Both states are at the same displacement under the same load, and both satisfy equilibrium and the friction bound; the order is the only difference, and it appears in neither.

The order the loads arrived in

Statics allows a contact with friction a whole range of forces and has no way to choose between them. A real structure does choose, and what it chooses by is the order in which things happened to it — so the force in a pier under a sliding bearing is a record of its history, not a function of its loads.

equilibrium · Friction Settling down or walking away, cycle by cycle. The total plastic hinge rotation of the beam after each cycle of loading — span 1, both, span 2, neither — with the midspan load at 0.98, 1.02, 1.05, 1.10 times the shakedown load of 126.3 kN. At 0.98 it stops at 0.59 mrad. At 1.02 it grows 4.57 mrad a cycle. At 1.05 it grows 11.43 mrad a cycle. At 1.10 it grows 22.86 mrad a cycle. Nothing collapses in any single cycle; above the shakedown load the beam walks.

The load it can carry once

A two-span beam whose loads come and go span by span collapses at 150 kN under any one arrangement, and walks at 127. Between the two it can carry every arrangement once and none of them forever: each cycle leaves a few more milliradians of rotation at the support and a midspan fifteen millimetres lower. Melan's theorem finds the limit as the last residual moment line that fits, Koiter's as a mechanism no single load state can drive, and a cycle-by-cycle calculation walks exactly where both say it will.

materials · Shakedown The force climbs back to the elastic one. The peak force of each of 40 drops of a 500 kg weight dropped 1.0 m onto a structure whose stiffness gives it 2.0 mm of static deflection under the weight, onto a structure yielding at 80 kN — half the 160 kN an elastic structure takes (dotted). Perfectly plastic, every drop peaks at 80 kN and the saving holds, at the price of the ratchet. Hardening at 2 per cent, the peak climbs drop by drop: 82 kN at the first, 91 at the fifth, 131 at the 40th; hardening at 5 per cent, the peak climbs drop by drop: 86 kN at the first, 104 at the fifth, 152 at the 40th. A hardening structure ends up carrying almost the elastic force it was made weaker to avoid.

The same weight, dropped again

A structure allowed to yield under a falling weight needs only a fraction of the strength an elastic one needs — half of it for a ductility of two and a half — because the energy goes into bending the steel rather than into force. That is the whole saving, and it is spent on the first drop. Drop the same weight again and the structure yields again: a perfectly plastic one adds the same permanent set every time and never stops, and a hardening one stops only when it has stiffened itself back up to nearly the elastic force it was made weaker to avoid, after moving more than half a metre.

dynamics · Impact factor 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

Named alongside it

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

RatchetingResidual stressSelf-stressCollapseImposed deformationPlastic hingeRestraintServiceabilityThermal gradientThermal movementAbutmentArticulation

All concepts