Discontinuity stress: making parts fit

Put the same pressure in a thin cylinder and in a thick, stiff ring such as a flange. The thin cylinder swells outward noticeably; the stiff ring hardly moves. Less obviously, even two thin parts of the same thickness grow by different amounts if their shapes differ. For a cylinder and a hemisphere of the same mean radius R and thickness t, thin-shell theory gives the free radial growth as:

With ν = 0.3 the hemisphere grows 0.35 / 0.85 ≈ 0.41 of what the cylinder does, less than half. The sphere carries pressure with half the hoop stress of the cylinder, so it stretches less.

Welded together, the two parts cannot separate. At the joint they must share the same radial displacement and the same slope. To force that, a shear force Q₀ and usually a bending moment M₀ appear at the joint, per unit length of circumference. They pull the more flexible part in and push the stiffer part out until the two edges meet.

Finding Q₀ and M₀ is a compatibility calculation: write the edge displacement and rotation of each part under pressure plus the unknown edge forces, set them equal, and solve. For a cylinder joined to a hemisphere of the same thickness, the classical solution needs almost only the shear force, because the two edges are about equally flexible and the moment that each side would need cancels out.

The joint forces bend the shell close to the joint. The bending adds to the pressure stress on one face and subtracts from it on the other. The pressure is still carried as before; the bending exists only to keep the parts joined, so design codes class it as secondary. The local membrane stress at the joint, which the pressure needs for equilibrium, stays primary.

In this classic case the bending raises the largest longitudinal stress in the cylinder by close to 30 %, a short way from the joint rather than at it. A flat head or a thick flange forces a much larger mismatch to be closed, and the bending can then be several times the membrane stress.

The disturbance runs along the shell as a wave that shrinks fast. The deflection and the moment both follow an envelope e^(−βx), with β ≈ 1.285/√(R·t) for steel, as at a nozzle. In the illustration √(R·t) = 100 mm, so the bending is largely gone within a couple of hundred millimetres of the joint.

Beyond that zone the shell carries plain membrane stress again. The practical consequence is to keep openings, attachments and further changes of section out of the zone where you can, so their own local stresses do not land on top of the joint bending.

Any change of shape or stiffness along the shell does the same thing: a head joined to a cylinder, a cone joined to a cylinder, a change in plate thickness, a stiffening ring, a thick tubesheet or flange. In general, the more abrupt the change, the larger the joint forces and the bending they cause.

Design rules respond in two ways. They shape joints to reduce the mismatch, for example by requiring a gradual taper where plates of different thickness are butt welded; and design-by-rule formulas for heads and cones already allow for the typical discontinuity stress at their junctions. Where an unusual joint falls outside those rules, the compatibility forces can be found by the classical method above or with a model of the shell. The exact taper and junction rules are code-specific, so check your governing code.