Q: What is the Inglis formula for stress concentration at a U-notch?

For a U-shaped notch with depth a and tip radius r, the Inglis approximation gives Kt ≈ 1 + 2√(a/r). For example, with a = 5 mm and r = 0.5 mm: Kt ≈ 1 + 2√10 ≈ 7.3. As the tip radius approaches zero, Kt theoretically approaches infinity, but plastic deformation in real materials limits the actual peak stress. A finite-width correction factor F(a/W) is multiplied for plates of finite width W.

Question 1

Why is the stress concentration factor Kt = 3.0 for a circular hole in an infinite plate?

Accepted Answer

Kirsch's exact elasticity solution (1898) shows that for a circular hole in an infinite plate under uniaxial tension, the tangential (hoop) stress at the hole edge perpendicular to the load is σ_θ = 3σ_nom, giving Kt = 3.0 regardless of hole size. At the hole edge parallel to the load direction, σ_θ = −σ_nom (compressive). For a finite-width plate, the correction formula Kt ≈ 3 − 3.13(d/W) + 3.66(d/W)² − 1.53(d/W)³ applies.

Question 2

How does fillet radius affect Kt for a stepped shaft?

Accepted Answer

For a stepped shaft shoulder, Kt decreases sharply as the fillet radius ratio r/d increases (r = fillet radius, d = small-end diameter). For D/d = 2 (large/small diameter ratio): at r/d = 0.02, Kt ≈ 4–5; at r/d = 0.1, Kt ≈ 2; at r/d = 0.2, Kt ≈ 1.5. The fundamental design guideline is to maximise the fillet radius to reduce stress concentration.

Question 3

What is the difference between the theoretical stress concentration factor Kt and the fatigue notch factor Kf?

Accepted Answer

Kt is the elastic (theoretical) stress concentration factor based solely on geometry. Kf is the fatigue notch factor accounting for actual fatigue sensitivity: Kf = 1 + q(Kt − 1), where q is the notch sensitivity index (0 ≤ q ≤ 1). High-strength steels have q close to 1 (Kf ≈ Kt), while ductile low-carbon steels have q < 1. For conservative design, Kf = Kt is often assumed.

Question 4

What is the Inglis formula for stress concentration at a U-notch?

Accepted Answer

For a U-shaped notch with depth a and tip radius r, the Inglis approximation gives Kt ≈ 1 + 2√(a/r). For example, with a = 5 mm and r = 0.5 mm: Kt ≈ 1 + 2√10 ≈ 7.3. As the tip radius approaches zero, Kt theoretically approaches infinity, but plastic deformation in real materials limits the actual peak stress. A finite-width correction factor F(a/W) is multiplied for plates of finite width W.

Stress Concentration Factor K_t Calculator

Theoretical Background

Stress Concentration Factor Kt Calculator

Theoretical Background

Stress Concentration Factor K_t Calculator