| name | stress-concentration |
| description | Stress concentration factors Kt and Kf — Peterson charts, notch sensitivity, Neuber rule for plastic correction, fatigue notch factor, common geometry tables. |
| metadata | {"priority":7,"promptSignals":{"phrases":["stress concentration","Kt factor","Kf factor","notch sensitivity","Peterson","stress raiser","notch radius"],"minScore":4}} |
Stress Concentration — Complete Skill
Theoretical Stress Concentration Factor Kt
Kt = σ_max / σ_nom (elastic, theoretical)
σ_nom = stress in reduced section assuming uniform distribution
Kt depends ONLY on geometry: notch shape, r/d ratio, D/d ratio
Kt does NOT depend on material (elastic analysis)
Fatigue Notch Factor Kf
Kf = Se_unnotched / Se_notched (reduction in fatigue strength)
Kf = 1 + q(Kt - 1)
q = notch sensitivity [0-1]; q=0: material insensitive; q=1: fully sensitive
Notch sensitivity q from Peterson:
q = 1/(1 + √a/√r)
a = Neuber constant [mm] — material property
r = notch radius [mm]
Neuber constant a (Peterson, Shigley Table 6-15):
| Sut [MPa] | a [mm] (steel, bending/axial) | a [mm] (steel, torsion) |
|---|
| 350 | 0.51 | 0.64 |
| 560 | 0.28 | 0.35 |
| 700 | 0.20 | 0.25 |
| 900 | 0.13 | 0.16 |
| 1200 | 0.08 | 0.10 |
| 1400 | 0.06 | 0.08 |
Common Kt Values
Flat Plate with Central Hole (tension)
D/W = hole diam / plate width
D/W = 0.1: Kt = 2.73
D/W = 0.2: Kt = 2.50
D/W = 0.3: Kt = 2.27
D/W = 0.5: Kt = 1.90
D/W → 0 (small hole): Kt → 3.0
Flat Plate with Edge Notch (bending/axial)
r/d = 0.05: Kt ≈ 2.5-3.5 (depends on D/d)
r/d = 0.10: Kt ≈ 2.1-2.8
r/d = 0.25: Kt ≈ 1.6-2.0
r/d = 0.50: Kt ≈ 1.4-1.6
Stepped Round Shaft — Bending (Shigley Fig. A-15-9 through A-15-11)
D/d = 1.1, r/d = 0.1: Kt ≈ 1.65
D/d = 1.5, r/d = 0.1: Kt ≈ 1.85
D/d = 2.0, r/d = 0.1: Kt ≈ 2.00
D/d = 2.0, r/d = 0.05: Kt ≈ 2.50
D/d = 2.0, r/d = 0.02: Kt ≈ 3.10
Stepped Round Shaft — Torsion
r/d = 0.1, D/d = 1.5: Kt ≈ 1.50
r/d = 0.05, D/d = 2.0: Kt ≈ 1.90
r/d = 0.02, D/d = 2.0: Kt ≈ 2.40
Keyway (shaft, end-milled)
Bending: Kt = 2.14, Kts = 2.62 (Shigley, Eq. 6-37)
Semicircular keyway — bending: Kt = 1.6, Kt(torsion) = 3.0
Press Fit (hub on shaft)
Kt (at press fit edge) = 1.95 (typical, from Shigley Table 7-1)
Grooves and Retaining Rings
Deep groove: Kt ≈ 5–6 (highly localized)
Retaining ring groove: Kt ≈ 3.5–4.5
Peterson Equation for Kt (Stepped Shaft)
From ESDU or Peterson "Stress Concentration Factors":
Kt = C₁ + C₂(2r/D) + C₃(2r/D)² + C₄(2r/D)³
Where coefficients C₁–C₄ depend on D/d and loading mode — tabulated in Peterson (3rd Ed.)
Neuber Rule (Plastic Correction)
When σ_nom × Kt > Sy — use Neuber's rule to get actual σ_max, ε_max:
Kt² × σ_nom × e_nom = σ_max × ε_max
Combine with σ-ε curve (Ramberg-Osgood):
ε = σ/E + (σ/K)^(1/n)
Iterate to find σ_max, ε_max at notch root
Effective stress concentration for fatigue: Kε = ε_max × E / σ_nom (strain-based)
Application in Fatigue (Shigley DE-Goodman)
Modified endurance limit: Se' = Se / (Kf)
Kf applied only to alternating stress σ_a — NOT to mean stress σ_m (ductile materials)
Combined loading: use Kf for bending, Kfs for torsion
Von Mises combination: σ'_a = √((Kf×σ_a)² + 3(Kfs×τ_a)²)
Design Guideline
- Maximize r: increasing r from 0.01d to 0.05d drops Kt by 30-40%
- Reduce D/d: adding intermediate step reduces stress concentration at final step
- Change loading: compression → lower effective Kt (compressive residual stress helps)
- Shot peening: induces compressive residual, offsets notch effect in fatigue
- Avoid Kt > 3 at fatigue-critical locations
- For d = 25mm shaft, target r ≥ 1-2mm
Output
Provide: Kt from chart/formula, q from Peterson, Kf = 1 + q(Kt-1), Se_notched = Se/Kf, fatigue margin.
