Ratcheting and the Bree diagram

Take a long, thin cylinder far from its ends, so that nothing but pressure and temperature acts on the wall. The metal is idealised as elastic-perfectly plastic: it follows a straight line up to yield and then flows at constant stress. Yield is taken as the same hot and cold.

Inside, a steady pressure P sets up a membrane stress that is the same through the whole wall and never goes away. On top of it, each operating cycle heats the inside face more than the outside, with a temperature difference ΔT that varies linearly through the wall. At shut-down the difference goes back to zero, while the pressure stays.

The hot inside wants to grow more than the cooler outside. The wall cannot curl, because the cylinder is closed round on itself, so the gradient is held flat. The result is a bending stress through the wall: compression on the hot face, tension on the cool face, zero at mid-thickness. It is self-limiting, so it is a secondary stress.

The chart divides both by the yield stress. The horizontal scale is X = σp / Sy and the vertical scale is Y = σt / Sy, the range of the cycling thermal stress at the surface. A point on the chart is one pair of pressure and temperature cycle, and its region tells what the wall does after many cycles.

On the hot half of the cycle the two stresses add on the cool face and subtract on the hot one. On the cold half only the pressure stress is left. So the surface stress swings by σt on every cycle around a steady value σp.

• E, elastic: X + Y ≤ 1. The largest surface stress σp + σt never reaches yield, so nothing yields at all.
• S, shakedown: the first cycle yields a surface layer, the residual stress it leaves shifts later cycles into the elastic range, and the wall then cycles elastically. This needs Y ≤ 2, because an elastic swing can be at most twice yield.
• P, plastic cycling: Y above 2 but below 1/X. Both faces yield on every cycle, first one way and then the other, with no net growth. The wall does not distort, but each cycle spends low-cycle fatigue life.
• R1 and R2, ratcheting: to the right of Y = 4(1 − X) for X from 0.5 to 1, or above Y = 1/X for X up to 0.5. The wall stretches a little further on every cycle. In R1 an elastic core remains and the faces shake down; in R2 the faces also cycle plastically.

At X = 1 the pressure stress alone reaches yield across the wall. That is plastic collapse of the cylinder, with no thermal stress needed.

Ratcheting needs both stresses. When the cool face yields in tension on the hot half of the cycle, that layer can carry no more stress. The pressure still needs the same total force across the wall, so the elastic part has to carry more, and to do so it must stretch. On the cold half the thermal stress is removed, so its change runs the other way, the layer at the hot face yields in turn, and the same shift happens again. Each half cycle adds a little hoop strain, in the same direction every time, because the pressure always pulls the same way.

With a small pressure stress the elastic core is wide enough to carry the shifted load without yielding, which is why a large thermal range on its own only gives plastic cycling. As X rises, a smaller thermal range is enough to start the growth, so the boundary falls from Y = 2 toward zero at X = 1.

The example also shows how steep the change is. The temperature difference grows by less than a factor of two between case A and case B, but the outcome goes from a wall that settles after one cycle to one that grows on every cycle.

Design rules take two ideas from the chart. The first is the Y = 2 line: the range of primary plus secondary stress is limited to about twice yield, which in allowable-stress terms is often written as about 3 S. Keeping below it keeps the wall in the elastic or shakedown regions.

The second is a thermal stress ratcheting check that lowers the allowed thermal range as the steady stress rises, following Y = 1/X for X up to 0.5 and Y = 4(1 − X) above it. Design-by-analysis rules, including ASME Section VIII, Division 2, have a check of this kind. The stresses that go in, the yield value at which temperature, and the cases it applies to are code-specific, so check the governing code.

The chart has limits of its own. It assumes a thin wall, a linear temperature gradient, the same yield hot and cold, and no hardening. At a nozzle or a junction the stresses are not this simple, and an elastic-plastic analysis over several cycles is the more direct check.