Stress intensification factor

A pipe bend, a tee or a branch connection fatigues earlier than straight pipe under the same bending moment. The beam model of a pipe stress analysis, however, knows these components only as beams. The stress intensification factor i closes the gap: the nominal bending stress from moment and section modulus is multiplied by i before it is compared with the allowable value.

The reference case is straight pipe with a girth butt weld, for which the factor is 1.0. For all other components it is equal to or greater than one.

The factors do not come from a stress calculation but from fatigue tests on piping components carried out by A. R. C. Markl and his colleagues in the 1940s and 1950s. The tests produced the equations that are still found in the codes today.

ASME B31.3 lists them in an appendix; recent editions refer to ASME B31J instead. EN 13480-3 contains its own compilation. The values of the codes are similar but not identical – an analysis uses those of the chosen code throughout.

The factor is a matter of geometry. For a bend it is governed by the flexibility characteristic h = t · R / r², formed from wall thickness t, bend radius R and mean pipe radius r: the thinner the wall and the tighter the bend, the larger the factor. In addition, loading in the plane of the bend differs from loading out of plane.

For branch connections, the ratio of branch to header diameter, the wall thicknesses and above all the type of construction matter. An unreinforced fabricated branch has a considerably higher factor than a forged welding tee of the same size.

Three quantities sound similar and mean something else. The stress concentration factor describes the theoretical elastic peak stress at a notch; the stress intensification factor, by contrast, is a test-based value referred to a girth butt weld and therefore smaller. The stress intensity factor belongs to fracture mechanics and describes the stress field at a crack tip. The flexibility factor, finally, acts on the stiffness of a bend in the model, not on the stress.

The equations apply to the geometries on which they were established. For very thin-walled pipe of large diameter, for lateral branches, trunnions and special components they are not validated, or only to a limited extent. The factor is then determined by finite element analysis or by test, or the location is justified locally.

A frequent modelling error is the wrong type of branch: entering a welding tee where a branch is stubbed in on site means calculating with a factor that is too small.