Stress concentration and the weld toe

Picture the stress in a plate under tension as lines of force running along it. Where nothing interrupts them they are evenly spaced. A hole or notch forces them to crowd round it, and where they crowd the stress rises. The ratio of the highest stress to the nominal one is the stress concentration factor:

Kt depends on the shape of the feature, not its size. A small hole and a large hole of the same shape give the same factor. Size decides how far the raised stress reaches: about one or two hole radii, beyond which the plate is back to its nominal stress.

The classic case is a small round hole of radius a in a wide plate under a remote tension σ. Along the line through the hole's centre, square to the load, elasticity gives the stress along the load direction at a distance r from the centre:

Sharper features give higher factors. For an elliptical hole with half-width a across the load and half-height b along it, the edge factor is 1 + 2a/b. A long, thin slot across the load, the shape of a crack, gives a very large value.

In a ductile metal the peak at a concentration reaches yield long before the part as a whole does. The small zone that yields stops taking extra stress, and further load moves to the elastic metal round it. The part still fails only when the whole section yields, so for one application of load the concentration hardly lowers the strength.

That is why design-by-analysis rules treat the concentrated part of a stress as peak stress. It does not enter the strength checks. It is kept for the fatigue check, where it does matter. Brittle materials, and steels at temperatures where they lose toughness, are the exception: there a concentration can start a fracture under one load.

The toe is where the surface of a weld meets the plate. Several things come together there. The surface turns through an angle, often 30° to 60°, with a very small radius. There may be undercut, a small groove left by the welding arc. Welding leaves residual tension near yield in the metal close by. And the toe holds small flaws from the welding process that act as tiny cracks from the start.

Through the wall, the stress at a toe has three layers. The structure's membrane and bending stress is the straight-line part. On top of it the toe's notch adds a sharp rise confined to a thin layer at the surface. That rise is the peak part, and it is what makes the toe the worst spot for fatigue.

A model with a sharp corner at the toe has no finite peak there: the finer the mesh, the higher the reading. So rules that judge welds rarely use the raw peak. They use a structural stress, also called a hot-spot stress, which leaves the notch out. It is either the membrane plus bending stress through the wall at the toe, found by linearisation, or a value extrapolated to the toe from surface points a short way from it. One common surface rule reads the stress at 0.4·t and 1.0·t from the toe:

Under cycling load the toe sees the largest stress range in the joint, in metal already in residual tension, with flaws that are already sharp. A smooth part spends most of its fatigue life starting a crack. A welded toe skips most of that stage, and the crack grows from the toe into the plate from early on. This is why welded joints have their own, lower fatigue curves.

• Keep welds away from the highest stress where the layout allows, for example off the crotch of a nozzle.
• Use full-penetration welds where a fillet weld root would be loaded across.
• Ask for a smooth weld profile; grinding or dressing the toe lowers the notch effect.
• Check the toe on both faces: the worst one is not always the outside.