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Ratcheting and shakedown

What happens when a steady stress meets one that cycles: shakedown, plastic cycling or ratcheting.

Principle 6 of 10 · 6 min read ·

Ratcheting and shakedown: A steady stress and a cycling oneStress against time: a flat line for the steady pressure stress, with a thermal stress rising and falling on top of it on each cycle and passing the yield line.TimeStressYieldSteady pressure stressThermal stress, every start-upStrainStressYieldyields a littleLocked-in residual stressCyclesPlastic strainShakedown: it stops growingRatcheting: it grows every cycleshakedown, for comparison
Figure 1. A steady stress and a cycling one.
Contents

Shakedown is when a part that yields a little on its first few load cycles settles into purely elastic behaviour afterwards, because the yielding leaves residual stresses that keep later cycles below yield. Ratcheting is the opposite outcome: a steady stress combined with a large cycling stress makes the metal yield on every cycle in the same direction, so plastic strain keeps building up until the part distorts or fails. Design codes guard against ratcheting by limiting the range of primary plus secondary stress, typically to about 3 S, or by an explicit cyclic analysis.

This article explains both outcomes with a steady pressure stress and a cycling thermal stress, the textbook pairing, and shows the four ways a cycled metal can end up.

A steady stress and a cycling one

Picture a shell wall with a steady membrane stress from pressure, plus a bending stress through the wall that comes and goes with every start-up and shut-down, for example from a temperature difference across the wall. The pressure stress pushes one way all the time. The cycling stress swings up and down on top of it, so on one face the sum rises above the steady value and falls back on every cycle.

The thermal stress is used here only because it is the classic example of a cycling secondary stress; any cycling stress that is self-limiting behaves the same way. The question is what the metal does over many cycles once the sum passes yield.

The first cycle yields a little

On the first cycle the sum passes yield in a thin layer at the surface, and that layer yields. When the cycling stress goes away, the layer cannot return to its original length, because it has been stretched plastically, while the elastic metal around it can. The surrounding metal pulls it back, leaving a locked-in residual stress of the opposite sign.

Ratcheting and shakedown: The first cycle yields a littleStress against strain for the first cycle: up to yield, a short plastic stretch, then straight back down, ending below zero with a residual stress.TimeStressYieldSteady pressure stressThermal stress, every start-upStrainStressYieldyields a littleLocked-in residual stressCyclesPlastic strainShakedown: it stops growingRatcheting: it grows every cycleshakedown, for comparison
Figure 2. The first cycle: up to yield, a short plastic stretch, then straight back down, ending below zero with a residual stress.

That residual stress changes the next cycle. The layer now starts from a stress below zero, so it has more elastic range to rise through before it reaches yield again.

Shakedown: elastic from then on

If the stress range on each cycle is no more than twice the yield stress, the residual stress can shift the whole cycle into the elastic range: the layer runs from about minus yield to plus yield and back without yielding again. Plastic strain stops growing after the first cycle or the first few. The part has shaken down. This holds for an elastic-perfectly-plastic metal when the steady stress is modest; real steels harden, which helps a little.

Ratcheting and shakedown: Shakedown: elastic from then onPlastic strain against the number of cycles: a small rise over the first cycles, then flat.TimeStressYieldSteady pressure stressThermal stress, every start-upStrainStressYieldyields a littleLocked-in residual stressCyclesPlastic strainShakedown: it stops growingRatcheting: it grows every cycleshakedown, for comparison
Figure 3. Plastic strain against the number of cycles: a small rise over the first cycles, then flat.

This is what the typical 3 S range check aims for. With the allowable S at about two-thirds of yield, 3 S is about twice yield, the shakedown limit for a cycling stress.

Four ways a cycled metal can end up

Below yield the metal stays elastic throughout. A little above, it shakes down. If the range is more than twice yield, the metal yields in tension and in compression on every cycle: plastic cycling, also called alternating plasticity. The loop closes each cycle, so the part does not grow, but the repeated plastic strain uses up fatigue life quickly. The fourth outcome is ratcheting.

Three ways a cycled metal can respondThree small stress-strain sketches. Shakedown: after a first yield the later cycles run up and down one straight elastic line. Plastic cycling: a closed loop that yields in both directions every cycle. Ratcheting: loops that yield every cycle and step further along the strain axis each time.Shakedownstressstrainelastic from cycle 2Plastic cyclingstressstrainsame loop every cycleRatchetingstressstrainstrain grows each cycle
Figure 4. Shakedown, plastic cycling and ratcheting as stress-strain paths. Only ratcheting moves along the strain axis cycle after cycle.

Ratcheting: strain grows every cycle

If the cycling range is large while the steady stress keeps pushing, the metal yields on every cycle and the steady stress makes each yield go further one way than the other. Plastic strain builds up cycle after cycle, a step each time, and the wall grows or thins until it fails. That is ratcheting, and it is the outcome codes most want to rule out, because a part can ratchet to failure in a modest number of cycles without any single cycle looking alarming.

Ratcheting and shakedown: Ratcheting: strain grows every cyclePlastic strain against the number of cycles: a staircase that keeps rising, beside the flat shakedown line for comparison.TimeStressYieldSteady pressure stressThermal stress, every start-upStrainStressYieldyields a littleLocked-in residual stressCyclesPlastic strainShakedown: it stops growingRatcheting: it grows every cycleshakedown, for comparison
Figure 5. Plastic strain against cycles: a staircase that keeps rising, beside the flat shakedown line.

A classic analysis of a thin cylinder under steady pressure and a cycling linear temperature difference through the wall maps where each outcome occurs. The chart below follows its idealised form for an elastic-perfectly-plastic metal. Its region boundaries depend on the shape of the cycling stress and on the material model, so treat it as a picture of the idea.

Steady stress against cyclic stress range: four outcomesA schematic interaction chart. Across, the steady stress divided by yield from 0 to 1; up, the cyclic stress range divided by yield from 0 to 4. A triangle near the origin is elastic. Above it, up to twice yield, is shakedown. Above twice yield at low steady stress is plastic cycling. Upper right, where both are large, is ratcheting. Point A sits in shakedown and point B in ratcheting.12340.51ElasticShakedownPlasticcyclingRatchetingABCyclic stress range ÷ yieldSteady stress ÷ yield
Figure 6. Schematic chart of steady stress against cyclic range, both divided by yield, for the idealised classic case. Points A and B are the two cases in the illustration below.

How design rules guard against ratcheting

  • A range limit: the primary plus secondary stress range is held to about 3 S, close to twice yield, so the part shakes down.
  • A separate thermal stress ratcheting check in some codes, which lowers the allowable cyclic range as the steady primary stress rises, in the spirit of the chart above.
  • An elastic-plastic analysis that applies, removes and reapplies the operating loads for several cycles and confirms that the plastic strain stops growing.

In a cyclic analysis, codes typically accept the result when the plastic strain no longer grows between cycles, when an elastic core remains through the wall, or when the overall dimensions stop changing. The criteria, the number of cycles and the material model to use are code-specific, so check the governing code. Fatigue is assessed separately in every case.

Key points

  • A steady stress plus a cycling stress can yield the metal on the first cycle.
  • If the cycling range stays within about twice yield, residual stress makes later cycles elastic: shakedown.
  • Above that range the metal yields both ways every cycle, using up fatigue life quickly.
  • With a large steady stress as well, plastic strain builds up every cycle: ratcheting.
  • The typical 3 S range limit is about twice yield, the shakedown limit.

Where this matters in practice

  • Nozzles and attachments on vessels that start up and shut down often, where the local stress range is checked.
  • Head-to-shell and cone junctions that see the same cycles as the shell.
  • Supports and lugs on equipment that carries a cycling load on top of a steady one.
  • Any attachment where the primary plus secondary range comes close to its limit.

Junctra's local check at a nozzle or clip compares the combined local stress with a limit of 3 S by default. It does not run a cycle-by-cycle analysis.

See the attachment module

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Educational article. It explains principles and is not design guidance; follow your governing code and edition.