A stress concentration factor, written Kt, is the ratio of the peak stress at a geometric discontinuity to the nominal stress the section would carry without it:
Kt = σmax ÷ σnom
Holes, fillets, notches, grooves, keyways and abrupt changes of section all disturb the flow of stress through a part, and the stress at the edge of the disturbance rises above the average. Kt is called the theoretical or elastic stress concentration factor because it assumes the material stays linear elastic. On that assumption it depends only on the geometry and the type of loading, whether tension, bending or torsion, and not on the material.
Failure starts at the peak stress, so the average across a section can be misleading. A plate carrying a nominal 100 MPa with a small circular hole in it sees about 300 MPa at the edge of the hole. A ductile part under a single static load may tolerate that by yielding locally. Under repeated load, fatigue cracks start there, and in a brittle material, fracture does. Knowing Kt tells you where a part is most likely to fail first and how much a change of geometry will help.
Defining the nominal stress
A Kt value only has meaning alongside the nominal stress it was calculated from. For a plate in tension, the nominal stress may be based on the gross section, ignoring the hole, or on the net section, the width that remains once the hole is removed. For a shaft, the nominal bending stress is usually calculated on the smaller diameter at a shoulder or on the root diameter at a groove. Published charts state which definition they use, and mixing them up gives errors that approach a factor of two when the hole is half the plate width. For torsion and shear, the shear form Kts = τmax ÷ τnom is used in the same way.
Classic examples
A circular hole in a wide plate under tension
The best-known result: a circular hole in a plate that is wide compared with the hole, loaded in uniaxial tension, has Kt = 3. The peak occurs at the two edges of the hole on the line perpendicular to the load. At the two points in line with the load, the stress is compressive and equal in size to the nominal stress. The concentration is also very local: one radius away from the edge of the hole, the stress has already fallen to about 1.22 times nominal.
The value changes with the loading. Under equal tension in two directions, Kt is 2. Under pure shear, equivalent to tension in one direction and equal compression in the other, it is 4.
Elliptical holes and the effect of sharpness
For an elliptical hole in a wide plate, with semi-axis a perpendicular to the load and semi-axis b parallel to it:
Kt = 1 + 2a ÷ b
An ellipse three times as long across the load as along it gives Kt = 7. Turned through 90°, with its long axis along the load, the same ellipse gives about 1.7. As b shrinks towards zero the ellipse becomes a crack and Kt grows without limit, which is why crack-like features are so damaging, and why cracks themselves are assessed with fracture mechanics rather than with Kt.
Shoulder fillets, grooves and keyways in shafts
For a stepped shaft, Kt depends mainly on the fillet radius relative to the smaller diameter, r/d, and to a lesser degree on the ratio of the diameters, D/d. Machine design texts give first estimates such as these for a shaft with D/d of around 1.5, to be refined once the dimensions are known:
| Feature | Kt in bending | Kts in torsion | Kt in tension |
|---|---|---|---|
| Sharp shoulder fillet, r/d ≈ 0.02 | 2.7 | 2.2 | 3.0 |
| Well-rounded shoulder fillet, r/d ≈ 0.1 | 1.7 | 1.5 | 1.9 |
| End-milled keyseat, r/d ≈ 0.02 | 2.1 | 3.0 | not listed |
| Sled-runner keyseat | 1.7 | not listed | not listed |
| Retaining ring groove | 5.0 | 3.0 | 5.0 |
The retaining ring groove stands out. A circlip groove is a narrow notch with sharp corners, and placing one in a highly stressed length of shaft is a frequent source of fatigue trouble. A sled-runner keyseat, cut with a side-milling cutter so that it runs out gradually, has a lower bending factor than an end-milled pocket keyseat with abrupt ends.
Other common stress raisers
Threads concentrate stress at the root, and in a bolted joint the first engaged thread carries a disproportionate share of the load. Cross holes for lubrication or pins, weld toes, press-fitted hubs, the bases of bosses, and abrupt changes of wall thickness in castings and mouldings all act in the same way. When two features sit close together, their stress fields interact and the combined peak can exceed that of either alone, a case that charts for isolated features do not cover.
Why sharp corners matter
The elliptical hole result can be rewritten in terms of the depth of a notch, a, and the radius at its root, ρ:
Kt ≈ 1 + 2√(a ÷ ρ)
The concentration depends on the square root of depth over radius, so cutting the root radius to a quarter of its value doubles the amount by which Kt exceeds 1. A perfectly sharp internal corner, with zero radius, has an unbounded elastic stress in theory. Real parts always have some radius, left by the cutter, the casting or the mould, but if the drawing does not specify one, the part will be made with whatever radius the process happens to leave.
That is why a minimum internal radius belongs on the drawing for every corner in a highly stressed region, and why a sharp corner in a CAD model is worth questioning wherever the stress is high.
Kt and the fatigue notch factor Kf
Real materials are not fully sensitive to notches. Fatigue tests on notched and smooth specimens show that a notch usually reduces fatigue strength by less than Kt would predict. The fatigue notch factor captures this:
Kf = fatigue strength of a smooth specimen ÷ fatigue strength of a notched specimen
and the notch sensitivity, q, relates the two factors:
q = (Kf − 1) ÷ (Kt − 1)
q runs from 0, where the notch has no effect on fatigue strength, to 1, where Kf equals Kt. In general terms, three things affect it:
- Material strength. Higher-strength steels are more notch-sensitive than softer ones, so upgrading the material gains less fatigue strength at a notch than smooth-specimen data suggests.
- Notch radius. Very sharp notches have steep stress gradients, so only a tiny volume of material sees the peak, and q tends to be lower. Larger radii bring q closer to 1.
- Microstructure. Grey cast iron, already full of graphite flakes that act as internal notches, is relatively insensitive to added geometric notches.
As an example, a fillet with Kt = 2.5 in a steel with q = 0.8 gives Kf = 1 + 0.8 × 1.5 = 2.2. Under static load in ductile metals, local yielding redistributes the stress around a notch, and Kt is often not applied to static strength at all. For brittle materials, the full Kt generally applies. Our guide to fatigue analysis covers how notch effects feed into life prediction.
Design remedies
- Increase the radius. Usually the most effective change. Even a modest increase in a sharp fillet radius lowers Kt noticeably.
- Smooth the transition. Taper or blend changes of section instead of stepping them, and avoid abrupt changes of wall thickness in castings and mouldings.
- Move the feature. Place holes near the neutral axis of a beam in bending, and keep grooves, cross holes and keyway ends away from the most highly stressed sections.
- Align it with the load. Slots and elongated holes cause far less concentration when their long axis runs along the load direction.
- Use relief features. Where a shoulder must be sharp to locate a bearing, an undercut with a larger radius, or a relief groove beside it, can ease the flow of stress past the corner.
- Choose the gentler variant. Sled-runner keyseats instead of end-milled pockets; rolled threads instead of cut threads in fatigue-loaded fasteners.
- Treat the surface. Shot peening and fillet rolling introduce compressive residual stress that improves fatigue life at a notch. Kt stays the same; what changes is how the material responds to it.
Where stress analysis helps
Charts give Kt for isolated, idealised features under simple loads. Real parts combine several features, carry bending, torsion and axial load together, and have geometry that no chart describes. Finite element stress analysis calculates the peak stress at the actual geometry directly, so the concentration is part of the result rather than an estimate added afterwards. Beyond the peak value, the useful output is where it sits and how quickly it falls away, which shows whether a larger radius, a relocated hole or a different transition will solve the problem.
When to bring in a specialist
If a part has cracked at a corner, or a new design has to carry repeated load through holes, fillets or keyways, a focused study shows where the stress concentrates and which change of geometry removes it. SixGate™ combines stress analysis with fatigue analysis to give you peak stresses, predicted life and the design changes that improve both, reported against the code or standard you specify. The same team can then take those changes through detail design into the manufacturing drawings and production, so the radius that fixes the problem is the radius the part is made with.