Stress Concentration Factor Calculator

The stress concentration factor (Kt) is a dimensionless ratio that quantifies how much the local stress at a geometric discontinuity — such as a hole, notch, or fillet — exceeds the nominal (average) stress in the cross section. It is defined as Kt = σ_max / σ_nom. A Kt of 2.5 means the peak stress is 2.5 times greater than the nominal stress, which is critical for fatigue and fracture assessments.

Once Kt is known, the maximum (peak) stress at the discontinuity is simply σ_max = Kt × σ_nom, where σ_nom is the nominal stress calculated on the net cross-sectional area (for axial loads: F / A_net) or using standard beam bending formulas (Mc / I) for bending loads. This peak stress is what drives fatigue crack initiation.

This calculator covers the most common cases from Roark's Formulas and Peterson's Stress Concentration Factors: rectangular bars with a central circular hole (axial and bending), rectangular bars with U-shaped notches (axial and bending), and rectangular bars with shoulder fillets (axial and bending). Each configuration uses validated curve-fit formulas derived from experimental data.

The Kt values are computed using well-established polynomial curve-fit equations (e.g., from Roark's Formulas for Stress and Strain and Neuber/Peterson's references) that approximate experimental and FEA-validated data. Accuracy is generally within a few percent for geometries within the valid range of dimension ratios. As with all stress concentration estimates, empirically determined values from physical testing will always be more precise.

The h/r ratio strongly influences Kt for notched geometries. A small notch radius relative to its depth (high h/r) creates a sharper discontinuity and results in a much higher Kt. The curve-fit formulas used here are typically split into two ranges (e.g., 0.1 ≤ h/r ≤ 2.0 and 2.0 ≤ h/r ≤ 50.0) to maintain accuracy across the full geometric spectrum.

Kt is the theoretical (geometric) stress concentration factor based purely on geometry, assuming a perfectly elastic material. Kf is the fatigue stress concentration factor, which accounts for the material's notch sensitivity (q). The relationship is Kf = 1 + q(Kt − 1), where q ranges from 0 (fully notch-insensitive) to 1 (fully notch-sensitive). For fatigue design, Kf is the more relevant quantity.

The most effective strategies are: (1) increasing the fillet or notch radius — even a small increase dramatically lowers Kt; (2) reducing the abruptness of section changes by adding relief grooves or tapered transitions; (3) minimizing the size of holes or notches relative to the section width; and (4) using surface treatments like shot peening to introduce compressive residual stresses that counteract the peak tensile stress.

The Kt values from this calculator are based on linear-elastic theory and are independent of material — they apply to any isotropic, homogeneous material (metals, plastics, etc.) as long as the nominal stresses remain in the elastic range. For ductile materials under static loading, local yielding redistributes stress and the effect of Kt may be less critical, but for fatigue, brittle materials, or dynamic loading, Kt must always be considered.