Stress Concentration Factor (Kt) Calculator
Look up the theoretical stress concentration factor Kt for three classic cases — a hole in a plate, a shoulder fillet (tension & bending), and a notch / groove — then convert Kt to the fatigue factor Kf with Neuber notch sensitivity. Every formula is documented below, with its validity range.
Hole in a plate under tension
Kt = 3 − 3.13·(d/W) + 3.66·(d/W)² − 1.53·(d/W)³ (Heywood, d/W ≤ 0.7)σnom = P / ((W − d)·t) · σmax = Kt · σnomShoulder fillet — axial tension or bending
Kt = fit(D/d, r/d) (least-squares polynomial fit of Peterson's charts: Fig. A.15.1 tension, Fig. A.15.5 bending)σnom = P/A, A = πd²/4 (tension) · σnom = 32M/(πd³) (bending) · σmax = Kt · σnomNotch / groove (Inglis)
Kt = 1 + 2·√(t/ρ) (Inglis, narrow notch; t = depth, ρ = root radius)σmax = Kt · σnom — enter σnom for your own reference section.Fatigue: Kt → Kf via notch sensitivity
q = 1 / (1 + a/√ρ) (Neuber) · Kf = 1 + q·(Kt − 1)a = 0.246 − 3.08×10⁻³·Sut + 1.51×10⁻⁵·Sut² − 2.67×10⁻⁸·Sut³ (Sut in kpsi, a in √in — curve-fit of Shigley's Fig. 6-20)Methods & sources
Every number on this page comes from a documented closed-form formula — no hidden tables, no server calls. The four models:
| Case | Formula used | Source |
|---|---|---|
| Hole in plate | Kt = 3 − 3.13(d/W) + 3.66(d/W)² − 1.53(d/W)³ | Heywood finite-width correction; gives exactly 3.0 (Kirsch solution) as d/W → 0. Valid for d/W ≤ 0.7. |
| Shoulder fillet, tension | Kt = 1 + Σ c[k]·(D/d)i·(r/d)j | Least-squares polynomial fit (12 coefficients) of Peterson's Stress Concentration Factors, Fig. A.15.1, fitted to 28 chart points. Check point: D/d = 1.33, r/d = 0.10 → Kt ≈ 2.10. |
| Shoulder fillet, bending | Kt = 1 + Σ c[k]·(D/d)i·(r/d)j | Same method on Peterson's Fig. A.15.5. Check point: D/d = 1.09, r/d = 0.10 → Kt ≈ 1.32. |
| Notch / groove | Kt = 1 + 2√(t/ρ) | Inglis solution for an elliptical notch; approximation for narrow notches small versus the section. |
| Fatigue Kf | Kf = 1 + q(Kt−1), q = 1/(1 + a/√ρ) | Neuber's rule; Neuber constant a(Sut) from the published curve-fit of Shigley's Fig. 6-20 (bending/axial), valid Sut 30–220 kpsi. |
The Peterson fits are approximations of printed charts, not exact values: maximum deviation about 0.05 from the 28 chart readings each polynomial was fitted to, valid only for 1.09 ≤ D/d ≤ 1.50 and 0.025 ≤ r/d ≤ 0.30. Inputs outside any validity range are computed anyway but flagged as extrapolation — treat those numbers as rough estimates. Nominal stresses are always stated per tab: net-section P/((W−d)·t) for the hole, P/A on the small diameter for fillet tension, 32M/(πd³) for fillet bending, and your own reference section for the notch tab.
Scope boundary — what this tool is and is not
- Is: a quick-lookup approximation of classical stress-concentration charts for isolated holes, shoulder fillets, and notches in linear-elastic, isotropic material — plus a Neuber-based fatigue estimate.
- Is not: finite element analysis. It cannot see interacting stress raisers, plasticity, residual stress, welds, threads, surface finish, size effects, multiaxial loading, contact, or buckling.
- Kt vs Kf: Kt is the full theoretical elastic peak (use for static checks against yield). Kf ≤ Kt is the reduced factor that actually applies in fatigue (use against fatigue strength). This page computes both and explains the difference.
- Not for: certification, safety-critical final sizing, or any design where failure could injure people — confirm those with FEA and testing by a qualified engineer.
Engineering disclaimer: This calculator is an educational and scoping aid, not professional engineering advice. Formulas are approximations of classical references with stated validity limits. The author accepts no liability for designs based on these results — have safety-relevant work reviewed by a qualified engineer.
Frequently asked questions
What is the stress concentration factor Kt?
The stress concentration factor Kt is the ratio of the peak local stress at a geometric discontinuity (a hole, fillet, or notch) to the nominal stress computed from elementary strength-of-materials formulas at the same section: Kt = σmax / σnom, so σmax = Kt · σnom. It is a purely geometric factor for linear-elastic material behavior — it does not depend on the material, only on the shape and the loading type.
What is the stress concentration factor for a small hole in a plate?
For a small circular hole in a wide plate under tension, Kt = 3.0 exactly (the Kirsch solution): the hoop stress at the edge of the hole is three times the remote stress. This calculator uses Heywood's finite-width formula Kt = 3 − 3.13(d/W) + 3.66(d/W)² − 1.53(d/W)³, which returns exactly 3.0 as d/W approaches zero and corrects Kt downward as the hole becomes large relative to the plate width.
How do you calculate Kt for a shoulder fillet?
Enter the large diameter D, the small diameter d, and the fillet radius r, then choose axial (tension) or bending loading. The calculator evaluates a least-squares polynomial curve-fit of Peterson's stress-concentration charts (Fig. A.15.1 for tension, Fig. A.15.5 for bending) in the variables D/d and r/d, then multiplies the nominal stress on the small section — P/A for tension, 32M/(πd³) for bending — by Kt to get the peak stress.
Are the shoulder-fillet Kt values exact?
No. They are approximations of Peterson's published charts, not exact values and not FEA results. Each polynomial was fitted to 28 points read from the charts with a maximum deviation of about 0.05 from those readings, and the fit is only valid for D/d between 1.09 and 1.50 and r/d between 0.025 and 0.30. The calculator flags any input outside that range as extrapolation. For certification or safety-critical sizing, confirm with FEA or testing.
What is the Inglis formula for a notch?
For a narrow elliptical notch of depth t with root radius ρ, Inglis derived Kt ≈ 1 + 2√(t/ρ). A semicircular notch (t = ρ) gives Kt = 3, and a sharp notch (ρ → 0) drives Kt toward infinity — which is why fatigue analysis uses the reduced factor Kf instead of Kt for very sharp notches. The formula is an approximation for notches that are small compared with the surrounding section.
What is the difference between Kt and Kf?
Kt is the theoretical (geometric) stress concentration factor from elasticity theory — the full local stress peak. Kf is the fatigue stress concentration factor: the factor by which a notch actually reduces fatigue strength under cyclic loading. Real materials are less sensitive to notches in fatigue than theory predicts, so Kf always lies between 1 and Kt. They are linked by the notch sensitivity q: Kf = 1 + q(Kt − 1).
What is notch sensitivity q?
Notch sensitivity q measures how much of the theoretical stress concentration actually materializes in fatigue: q = (Kf − 1)/(Kt − 1). It ranges from 0 (no notch sensitivity — Kf = 1, typical of soft steels with blunt notches) to 1 (full sensitivity — Kf = Kt, typical of high-strength steels and sharp notches). q increases with material strength and decreases as the notch root radius gets smaller.
How is notch sensitivity q calculated?
This calculator uses Neuber's rule q = 1 / (1 + a/√ρ), where ρ is the notch root radius and a is the Neuber material constant, obtained from the ultimate tensile strength Sut through a published curve-fit of Shigley's Fig. 6-20: a = 0.246 − 3.08×10⁻³·Sut + 1.51×10⁻⁵·Sut² − 2.67×10⁻⁸·Sut³ (Sut in kpsi, a in √in). The calculation is unit-consistent in metric and imperial modes.
When should I use Kt and when Kf?
Use Kt for the static yield check: the peak elastic stress Kt·σnom must stay below yield (with your safety factor), because a ductile material yields locally and redistributes the peak. Use Kf for fatigue and endurance-limit checks: compare Kf·σnom against the material's fatigue strength, since fatigue cracks initiate at the notch and feel the reduced concentration.
Which nominal stress does this calculator use?
It depends on the case, and each tab shows it. Hole in a plate: net-section stress P/((W−d)·t). Shoulder fillet in tension: P/A on the small diameter, A = πd²/4. Shoulder fillet in bending: 32M/(πd³) on the small diameter. Notch tab: you enter the nominal stress directly, because the reference section depends on your geometry. Kt is meaningless without its matching nominal-stress definition, so never mix a Kt from one definition with a σnom from another.
What are the validity limits of this calculator?
Hole: d/W ≤ 0.7 (Heywood formula range). Shoulder fillet: 1.09 ≤ D/d ≤ 1.50 and 0.025 ≤ r/d ≤ 0.30 (Peterson chart-fit range). Notch (Inglis): best for narrow notches with depth small versus the section width. Neuber q: Sut between 30 and 220 kpsi (about 207–1517 MPa). Inputs outside these ranges are still computed but flagged as extrapolation — treat those numbers as rough estimates.
Why can't this replace FEA?
These are closed-form approximations of classical charts for simple, isolated features in linear-elastic, isotropic material. They ignore interacting stress raisers, plasticity, residual stress, surface finish, size effects, multiaxial loading, and real boundary conditions. Use this calculator for scoping, homework, and sanity-checking FEA — not for final sizing of anything whose failure could hurt people or cost serious money.
Does Kt apply to ductile materials under static load?
For static loading of ductile materials, local yielding at the notch blunts the peak, so designers often design to the nominal stress against yield and effectively take Kf ≈ 1 for the static check. Kt still matters for ductile materials in fatigue, and it always matters for brittle materials (cast iron, hardened steels, composites), where there is no local yielding to relieve the peak.
How accurate is the Peterson chart curve-fit?
The fit reproduces the two documented check points almost exactly — Kt ≈ 2.10 for axial loading at D/d = 1.33, r/d = 0.10, and Kt ≈ 1.32 for bending at D/d = 1.09, r/d = 0.10 — and deviates at most about 0.05 from the 28 chart points each polynomial was fitted to. Reading a printed Peterson chart by eye carries similar uncertainty, so the fit is as good as manual chart lookup inside its validity range, and worse than nothing outside it.
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