Local loads on pressure vessel shells

Pressure acts on the whole shell at once. A local load is different: it arrives at one spot. Whatever is connected to a nozzle, or supported from a clip, lug or bracket, can push, pull, bend and twist that attachment, and the attachment passes those actions into the shell around it. The engineer usually gives them as static forces and moments at the face where the attachment meets the shell.

At that face there are six of them. A radial force acts along the attachment, pushing it into the shell or pulling it out. Two shear forces act across it, one along the vessel axis and one around the circumference. Two bending moments try to tilt it, again one in each direction, and a torsional moment tries to twist it about its own axis.

However large or small the attachment, all of this load has to cross into the shell through the patch of wall it is welded to. That is why local stress can be high even when the loads look modest beside the pressure load on the whole vessel.

The shell answers in two ways at once. It stretches like a drum skin, with the load spread evenly through the wall: that is membrane action, described by a membrane force per unit length, N. It also bends locally, one face stretched and the other squeezed, described by a bending moment per unit length, M. For a wall of thickness t the stress at either surface is the sum of the two:

The ± is the bending. It adds to the membrane stress on one face and subtracts from it on the other, so a local stress check looks at both the outside and the inside surface, and at several points round the attachment, because a moment raises the stress on one side and lowers it on the other.

Pulled outward, the shell lifts next to the attachment and settles back to its original line a short way off. That short, sharp curvature is the local bending, and it is why the bending part is often a large share of the local stress right at the edge.

Local bending in a shell does not spread far. Moving away from the edge of the attachment it oscillates and dies out, and thin-shell theory gives the rate. The bending follows an envelope that falls off as e^(−βx), where x is the distance from the edge and β depends on the shell radius R, the wall thickness t and Poisson's ratio ν:

So √(R·t) is the natural length scale. At one √(R·t) from the edge the envelope is down to about 28 % of its starting value; at two, about 8 %; at three, about 2 %. Most of the local bending has faded within one such length and nearly all within a few.

Because the decay length is short, an attachment a long way from any other can usually be treated on its own. When two attachments sit within a few times √(R·t) of each other, their stressed patches overlap and the effects add. Methods that treat one attachment on a uniform shell no longer describe that case well.

The same thinking applies to a joint in the shell itself. An attachment close to a head-to-shell junction, a cone junction or a change in thickness sits in a region that is already bending, so the two effects combine. The governing code or a company standard may set a minimum spacing; where it does not, a model of the shell that includes both features is the usual way to check the combined effect.

Cut the wall at the joint and plot the stress across its thickness. The average value is the membrane part. The straight-line part, equal and opposite on the two faces and zero in the middle, is the bending part. What is left, a sharp rise at the surface where a weld toe or a notch crowds the stress together, is the peak part.

The split matters because each part is judged differently. Membrane stress spread over a large region can lead to gross yielding; bending can be carried well past first yield; peak stress is too local to distort anything but is where fatigue cracks start. The articles on primary and secondary stress and on stress categories take this further.

• A thinner shell, or a larger radius for the same wall, is more flexible, so it bends more under the same load.
• A larger attachment footprint spreads the load over more wall, which lowers the local stress for the same load.
• A reinforcing pad adds thickness under the attachment. It lowers the stress under the pad, and a second check is then needed at the pad edge, where the load steps back onto the bare shell.
• Pressure stress is already present. The local stress adds to the membrane stress from pressure, so both are combined before any limit is applied.
• Moments usually matter more than forces of similar size, because a moment loads one side of the attachment in tension and the other in compression over a short lever arm.