Hong Life CalculatorsUnderstanding the stress concentration calculator — concept and principle
A hole in a loaded plate makes the stress flow around it, and right at the edge of the hole the stress rises far above the average. This calculator estimates the theoretical stress concentration factor Kt for a central circular hole in a flat plate under tension, the net-section nominal stress, and the resulting peak stress. It is used for bolt holes, rivet holes, lightening holes and fatigue checks of brackets and straps.
How it is calculated
- Diameter ratio: d/W, hole diameter over plate width.
- Kt (net-section basis): Kt ≈ 3 − 3.14(d/W) + 3.667(d/W)² − 1.527(d/W)³, a curve fit published in Peterson's stress concentration charts for a finite-width plate.
- Net nominal stress: σ_net = F ÷ ((W − d) × t), with F in kN converted to N, giving MPa.
- Peak stress: σ_max = Kt × σ_net.
The principle
In an infinitely wide plate, Kirsch's elasticity solution (1898) shows that a circular hole raises the stress at its edge to exactly three times the remote stress. Load lines that would have passed through the hole are forced to crowd around it. In a finite plate, the net section gets narrower as the hole grows, so the nominal stress is already higher; measured on that net-section basis, Kt falls from 3 toward 2 as d/W increases. The polynomial captures this trend.
Worked example
- Inputs: F = 20 kN = 20,000 N, W = 60 mm, d = 12 mm, t = 5 mm.
- d/W = 12 ÷ 60 = 0.2.
- Kt = 3 − 0.628 + 0.14668 − 0.012216 = 2.506464.
- Net area: (60 − 12) × 5 = 240 mm²; σ_net = 20,000 ÷ 240 = 83.33333 MPa.
- Peak: 2.506464 × 83.33333 = 208.872 MPa.
| Plate width (mm) | Kt (Peterson/Heywood approx.) |
|---|---|
| 15 | 2.053056 |
| 30 | 2.232992 |
| 45 | 2.394474667 |
| 60 | 2.506464 |
| 75 | 2.585220608 |
| 90 | 2.642904889 |
| 120 | 2.721143 |
How to read the result
For static loading of ductile metals, the peak is local; the material yields slightly at the hole edge and redistributes load, so the net-section stress is what usually matters for strength. For fatigue or brittle materials the peak matters directly. Fatigue designers use Kf = 1 + q(Kt − 1), where q (0 to 1) is the notch sensitivity of the material.
Common mistakes
- Multiplying Kt by the gross stress F/(Wt) when the fit is defined on the net section; here the gross stress would be 66.7 MPa and give the wrong peak.
- Applying this Kt to a hole near an edge, or one carrying a bolt (pin-loaded holes have much higher factors).
- Treating Kt as material-dependent; it depends on geometry only.
Frequently asked questions
Why is Kt = 3 for a small hole?
It is the classical result for a circular hole in an infinite plate under uniaxial tension.
Does a bigger hole mean a bigger Kt?
On the net-section basis Kt decreases, but the net stress increases, so the peak stress still rises.
How can I reduce stress concentration?
Use generous radii, avoid sharp corners, place holes away from high-stress regions, and consider cold expansion or smooth finishing for holes in fatigue-loaded parts.
Sources: W. D. Pilkey and D. F. Pilkey, Peterson's Stress Concentration Factors, Wiley · G. Kirsch, "Die Theorie der Elastizität und die Bedürfnisse der Festigkeitslehre", Zeitschrift des VDI 42 (1898)