Primary stress is the stress a vessel needs to balance its applied loads, such as pressure and weight; secondary stress exists only so that joined parts can keep fitting together while they try to deform differently. The difference matters because yielding does not relieve primary stress but does relieve secondary stress, so design codes give primary stress tighter limits and judge secondary stress by its range over a load cycle. Peak stress, a third class, is too local to distort the part and is counted for fatigue.

The split comes from the design-by-analysis approach of ASME Section VIII and is used, in one form or another, by most pressure vessel codes. This article explains the reasoning behind each class rather than the code rules themselves.

Primary stress: needed for equilibrium

Cut a ring of cylindrical shell in half along its length. Pressure P pushes outward on the inside of the half-ring, over a projected width equal to the diameter D. The only thing holding that half on is the hoop stress in the two cut walls, each of thickness t. Balancing the two gives the hoop stress, and a cut across the cylinder gives the longitudinal stress, which is half as large:

σθ = P·D / 2t σx = P·D / 4tthin wall, D = diameter, t = thickness

Nothing else can carry that load, so as long as the pressure is there the stress has to be there. That is the test for a primary stress: remove it and the part is no longer in equilibrium with its loads. Primary stress can be membrane, like the hoop stress, or bending, like the bending at the centre of a flat cover under pressure, which the cover needs in order to carry the pressure to its edge.

Why primary stress gets the tightest limits

Suppose the pressure in the illustration were raised to 5 MPa, so the hoop stress reached yield right through the wall. Yielding would not lower the stress, because the pressure would still be pushing and the wall would still have to carry it. The wall would keep stretching with no new resistance, apart from what strain hardening gives, until the part distorted grossly or burst.

Figure 2. With the load held, nothing stops the strain once the section has yielded through.

Primary stress is therefore not self-limiting, and it gets the tightest limits, set with a margin against gross yielding and collapse. General primary membrane stress, the average through the wall away from joints and openings, is held to the basic allowable stress. Local primary membrane stress and primary bending get somewhat higher limits, for reasons covered in the article on stress categories.

Secondary stress: needed for compatibility

Secondary stress has a different origin. It comes from parts that must stay joined while they try to move by different amounts. A hot part joined to a cooler one is the classic case: the hot part wants to grow, the cool part holds it back, and the hot part ends up squeezed while the cool part is stretched. A thin shell joined to a thick flange under pressure is another: the shell wants to swell, the flange hardly moves, and the joint bends both.

Figure 3. A hot bar joined along its length to a cool bar. Free, the hot bar would grow to the dashed outline; joined, the two share the difference.

No outside load needs this stress for balance. The forces in the hot bar and the cool bar are equal and opposite and cancel inside the part. The stress exists only so that the shapes still fit. That is the test for a secondary stress: it is set by a deformation that must be matched, not by a load that must be carried.

Why secondary stress is judged by its range

Because secondary stress is set by a deformation, a little local yielding is enough to let the parts fit, and then the stress stops rising. One application of a secondary stress, even well above yield on paper, does not cause collapse. Secondary stress is self-limiting.

The risk lies in repetition. If the part is heated and cooled, or pressurised and emptied, many times, the region that yielded can be pushed past yield in both directions on every cycle. That alternating plasticity leads to fatigue cracking in a small number of cycles, and combined with a steady primary stress it can lead to ratcheting. So secondary stress is judged by its range over a cycle, with a limit chosen so that the part settles to elastic behaviour after the first few cycles. The article on ratcheting and shakedown explains why.

Peak stress: very local, counted for fatigue

Peak stress is the extra stress that crowds into a notch, a weld toe, a small fillet radius or a sharp corner. It is confined to a region much smaller than the wall thickness, so it cannot cause any noticeable distortion, and it plays no part in the collapse checks. It is, however, exactly where fatigue cracks start, so it is added in when the fatigue life is assessed.

Figure 4. Lines of stress crowd together where the fillet welds meet the plate: the weld toes, where peak stress sits.

Where each category appears around a nozzle

A nozzle in a pressurised shell shows all of them in a few centimetres. Far from the nozzle the shell carries general primary membrane stress. Near the junction the membrane stress rises, because the opening removes material that would otherwise carry load; that local membrane stress is still needed for equilibrium, so it stays primary. The bending that appears at the junction because the nozzle and shell must deform together is secondary. At the weld toes sits the peak.

Figure 5. One junction, four categories: general membrane away from the joint, local membrane and secondary bending next to it, and peak at the weld toes.

Sorting real stresses is not always this clean. The usual cautious rule is that a stress which cannot be shown to be self-limiting is treated as primary. When classification is genuinely unclear, an elastic-plastic analysis, which needs no classification, is the alternative route.

Key points

Primary stress balances applied loads and does not relax when the metal yields, so it gets the tightest limits.
Secondary stress makes joined parts fit; it relaxes once the metal yields a little, so it is judged by its range over a cycle.
Peak stress is very local, causes no distortion and is counted only for fatigue.
Local membrane stress at a junction or opening remains primary; the bending that keeps the parts together there is secondary.
When in doubt, a stress that cannot be shown to be self-limiting is treated as primary.

Where this matters in practice

Nozzle-to-shell junctions, where local primary membrane, secondary bending and peak stress all sit close together.
Head-to-shell junctions and cone junctions, where the shapes must match under pressure.
Lugs, clips and supports, where the carried load gives primary stress in the shell next to the attachment.
Thickness transitions and thick flanges joined to thin shells.
Weld toes at attachments, where peak stress governs fatigue life.

Junctra reports the local stresses at a nozzle or clip as membrane and combined values and compares each with its limit, built from the allowable stress S.

See the attachment module

Next principle: Load-controlled and displacement-controlled loads

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

Primary and secondary stress