What is Stress Concentration?
Every time a part changes cross-section — a shoulder on a shaft, a hole in a plate, a groove for a retaining ring — stress multiplies at the transition. The stress concentration factor (Kt) tells you how much higher the local stress is compared to the nominal stress.
Actual peak stress = Kt × nominal stress
Shoulder Fillet — Stepped Shaft Under Bending
| r/d ratio | D/d = 1.1 | D/d = 1.5 | D/d = 2.0 | D/d = 3.0 |
|---|---|---|---|---|
| 0.01 | 2.5 | 3.3 | 3.8 | 4.3 |
| 0.02 | 2.2 | 2.8 | 3.2 | 3.5 |
| 0.05 | 1.8 | 2.2 | 2.4 | 2.7 |
| 0.10 | 1.5 | 1.8 | 2.0 | 2.2 |
| 0.15 | 1.35 | 1.6 | 1.7 | 1.9 |
| 0.20 | 1.25 | 1.45 | 1.55 | 1.65 |
| 0.30 | 1.15 | 1.3 | 1.35 | 1.4 |
r = fillet radius, d = smaller diameter, D = larger diameter
Shoulder Fillet — Stepped Shaft Under Torsion
| r/d ratio | D/d = 1.1 | D/d = 1.5 | D/d = 2.0 |
|---|---|---|---|
| 0.01 | 2.0 | 2.5 | 2.8 |
| 0.02 | 1.8 | 2.2 | 2.5 |
| 0.05 | 1.5 | 1.7 | 1.9 |
| 0.10 | 1.3 | 1.45 | 1.6 |
| 0.20 | 1.15 | 1.25 | 1.35 |
Hole in a Flat Plate (Tension)
| d/W ratio | Kt (net section) |
|---|---|
| 0.0 (theoretical) | 3.00 |
| 0.1 | 2.85 |
| 0.2 | 2.60 |
| 0.3 | 2.35 |
| 0.4 | 2.12 |
| 0.5 | 2.00 |
d = hole diameter, W = plate width. For a small hole in a wide plate, Kt ≈ 3.0 (the classic result).
Retaining Ring Groove — Shaft Under Bending
| Groove Depth/d | Kt (bending) | Kt (torsion) |
|---|---|---|
| 0.02 | 2.8 | 2.2 |
| 0.03 | 3.2 | 2.5 |
| 0.05 | 3.8 | 2.9 |
Retaining ring grooves are brutal stress risers. Always check fatigue life at groove locations.
How to Use Kt in Design
- Static loading (ductile material): Kt can often be ignored — local yielding redistributes stress. Use nominal stress with a safety factor.
- Static loading (brittle material): Apply full Kt — brittle materials crack without redistributing. σ_max = Kt × σ_nom.
- Fatigue loading: Use the fatigue notch factor Kf, which is related to Kt but accounts for the notch sensitivity of the material:
Kf = 1 + q(Kt – 1)
where q = notch sensitivity factor (0 to 1, from Peterson’s charts based on material and notch radius) - Design for low Kt: The cheapest way to reduce stress concentration is to increase the fillet radius. Going from r/d = 0.02 to r/d = 0.10 can cut Kt nearly in half.
Rules of Thumb for Fillet Radii
- Minimum fillet radius for fatigue-critical shafts: r ≥ 0.02d (but larger is always better)
- Preferred: r/d = 0.10 to 0.15 for a good balance of stress and shoulder function
- If space allows: r/d = 0.20+ virtually eliminates the stress concentration
- Undercut grooves can provide a large effective radius in a small axial space
Related: Shaft Design | Tolerance Stack-Up | GD&T Reference