Stress categories and their limits

The total stress at a point mixes several things: membrane and bending parts, from loads that must be carried and from parts that must fit, plus any concentration at a notch. Comparing that total with one number would be either unsafe for some parts or wasteful for others. So the stress is first cut along a line through the wall and split into a uniform membrane part, a linear bending part and the remaining peak part.

Each part is then classed by its cause, as explained in the article on primary and secondary stress. Because the stress at a point has several components, each check first combines its components into one equivalent stress, using the yield criterion the code specifies, before comparing it with the limit.

General primary membrane stress is the average stress through the wall away from any joint or opening, such as the hoop stress P·D/2t in a plain cylinder. It is held to the basic allowable stress, S. It gets the tightest limit because it acts over a large region: if it reaches yield, the whole shell yields, and nothing around it can help.

S itself comes from the material tables of the governing code. It is set as the lower of fractions of the tensile strength and the yield strength at temperature, and for most steels at moderate temperature it is no more than about two-thirds of yield. The fractions differ between codes and between divisions of the same code.

Local primary membrane stress, near a nozzle, a support or a junction, is allowed about 1.5 S. A local zone that starts to yield is held in place by the elastic shell around it, so it cannot run away the way a whole shell can. The governing code defines how small a zone must be, and how far it must be from other such zones, to count as local.

Membrane plus primary bending is also allowed about 1.5 S, for a different reason: a section in bending has a reserve beyond first yield. When the outer faces of a rectangular wall section first reach yield, the middle of the section is still elastic. Bending can increase until the whole section has yielded, and for a rectangle that fully plastic moment is exactly 1.5 times the first-yield moment.

Once secondary stress is added, the question changes. Secondary stress cannot cause collapse on its own, but if it cycles it can make a region yield back and forth. So this check is made on the range of primary plus secondary stress over an operating cycle, typically against about 3 S.

With S at about two-thirds of yield, 3 S comes to about twice yield. That is not an accident. A region whose stress range stays below twice yield can yield on the first cycle, lock in a residual stress, and then behave elastically on every later cycle: it shakes down. Above twice yield it yields in both directions every cycle. Some codes allow a different value for this limit in certain cases, so check the governing code.

Add the peak stress and the question becomes fatigue life. The alternating stress, half the range of the total stress including peak, is read against a fatigue curve to find the number of cycles the part can take, and that is compared with the number of cycles expected. Where several kinds of cycle occur, their damage is added. Whether a fatigue check is needed at all depends on the expected cycles and on screening rules in the governing code.

Note what the illustration does not do: it never compares the 380 MPa total with 140 MPa. That comparison would reject a sound design, because most of that stress is bending that a section can carry and secondary stress that relaxes once the metal yields a little.