Discontinuity stresses at a shell-to-head junction

Cut the vessel at the joint. On the cylinder's edge, and equal and opposite on the head's edge, act a shear force Q₀ and a bending moment M₀, both per unit length of circumference. Each part, loaded by pressure and by these edge forces, must end up with the same radial movement and the same slope at the joint. Two conditions, two unknowns.

For a long cylinder the edge response to each force is classical. With D the wall's bending stiffness and β the decay constant:

A head near its edge behaves much like a short shell of the same radius, so similar expressions apply to it. Add the free pressure growth of each part, set the edge movements and slopes equal, and solve for Q₀ and M₀.

The edge forces are driven by how far apart the free edges would be. For a cylinder and a head of the same mean radius a and thickness t, the membrane formulas give the free radial growth at the joint:

The 2:1 head's equator moves inward under internal pressure, because its hoop stress there is compressive. The mismatch with the cylinder is then 2·p·a²/(E·t), four times that of a hemisphere, whatever ν is.

A formed head is not a sphere. Its meridian is flatter at the crown and curves sharply near the edge, in the knuckle. For a shell of revolution with meridional radius of curvature R₁ and second radius R₂ (measured along the normal to the axis), the membrane stresses under internal pressure are:

Where the meridian curves more tightly than half of R₂, the bracket goes negative and the hoop stress is compressive. On a 2:1 ellipsoidal head the hoop stress falls from +p·a/t at the crown, through zero where the normal to the head is at about 35° to the axis, to −p·a/t at the equator. That compression is as large as the cylinder's hoop tension.

A torispherical head has a spherical crown and a small knuckle radius, so the change of curvature is sharper and the hoop compression is concentrated in a narrow band. On thin heads of large diameter that band can wrinkle or buckle even under internal pressure, and the knuckle is also where such heads first yield. Design rules for heads limit the knuckle radius and thickness for this reason; the details are code-specific.

A flat head carries pressure by bending, and its edge rotates as it dishes. The cylinder welded to it must follow that rotation, so the joint carries a large moment. For a circular plate of radius a, the two ideal edge conditions bracket the real head:

A real flat head sits between the two: the shell partly holds its edge, and in doing so takes a moment that bends the shell hard next to the joint. Flat-head formulas in design codes carry a factor for the attachment detail that allows for this.

A cone under internal pressure carries a meridional force along its slope. At the joint with a cylinder that force has a radial part with nothing on the cylinder side to balance it, so the junction itself must carry it as a ring load:

At the large end it pulls the junction inward and puts a short length of shell in hoop compression; at the small end it pushes outward. For R = 1000 mm, p = 1 MPa and α = 30°, the ring load is about 289 N/mm. Rules require a reinforcing ring or a thicker junction once α passes a code-specific value.

The bending that the edge forces cause in the shell is there to keep the parts joined, so it is classed as secondary and judged by its range, typically against about 3 times the allowable stress. The membrane stresses in the head carry the pressure and stay primary. Two exceptions need care: the bending of a flat head across its span carries the pressure, so it is primary bending; and the compressive hoop stress in a knuckle is primary and can govern by buckling. Check your governing code for the exact rules.