Stress Concentration Factor Calculator (Kt, Kf, q)
Calculate the theoretical stress concentration factor (Kt) for four classic geometry cases: a rectangular plate with a central circular hole under axial load or bending, and a rectangular bar with a shoulder fillet under axial load or bending. Enter material strength to get notch sensitivity (q) and the fatigue notch factor (Kf) from Neuber-Peterson theory. The nominal and maximum stresses update as you type, with a full show-your-work panel.
Formula
Worked example
A steel plate (Sut = 600 MPa) is 60 mm wide, 10 mm thick, with a 20 mm central hole under 15 kN axial load. d/w = 20/60 = 0.333. Kt = 3 - 3.13(0.333) + 3.66(0.333)^2 - 1.53(0.333)^3 = 2.47. Net area = (60 - 20)*10 = 400 mm^2; sigma_nom = 15 000/400 = 37.5 MPa; sigma_max = 2.47 * 37.5 = 92.6 MPa. Peterson a = 0.0254*(2070/600)^1.8 = 0.198 mm; q = 1/(1 + 0.198/10) = 0.981; Kf = 1 + 0.981*(2.47 - 1) = 2.44.
What is a stress concentration factor?
Any geometric discontinuity in a loaded part, such as a hole, notch, groove, fillet or shoulder, causes the local stress to rise well above the average stress you would calculate from simple beam or axial formulas. The ratio of the peak local stress to that average (nominal) stress is the theoretical stress concentration factor, Kt. A circular hole in a large plate under uniform tension has Kt = 3, meaning the stress at the hole edge is three times the far-field stress. This factor depends only on geometry: loading type and material do not change Kt.
Kt vs. Kf: theoretical versus fatigue concentration
For fatigue loading, the full theoretical factor Kt is rarely mobilized because real materials are not infinitely sensitive to sharp notches at the microscopic scale. The effective factor for fatigue design is Kf, the fatigue notch factor, which is always between 1.0 and Kt. Kf is related to Kt through the notch sensitivity index q: Kf = 1 + q*(Kt - 1). When q = 0 the material ignores the notch; when q = 1 the full Kt applies. High-strength steels typically have q close to 1, while grey cast iron can have q near 0. The most widely used engineering formula for q is the Neuber-Peterson equation: q = 1 / (1 + a/r), where a is a material length constant that depends on ultimate tensile strength and r is the notch radius.
Peterson polynomial curve fits used in this calculator
This calculator uses the closed-form polynomial fits from Peterson's "Stress Concentration Factors" (2nd and 3rd editions, as revised by Pilkey and Pilkey). For a rectangular plate with a central circular hole under axial tension, the fit is Kt = 3 - 3.13(d/w) + 3.66(d/w)^2 - 1.53(d/w)^3, valid for d/w up to 0.8. For bending the coefficients change slightly: Kt = 3 - 3.66(d/w) + 4.44(d/w)^2 - 2.08(d/w)^3. For shoulder fillets, the calculator applies a segmented power-law fit parameterized by both r/d and D/d, which approximates the curves in Peterson's Table 2-7 for axial and bending loading of flat bars. All fits are valid for the geometric ratios shown; the calculator outputs NaN for values outside the supported range.
How to use the results in a fatigue design
In the modified Goodman or Morrow fatigue criterion, you divide the endurance limit Se by Kf (or multiply the alternating stress amplitude by Kf) before comparing to the material limit. The fully corrected endurance limit becomes Se' = Se / (Ka * Kb * Kc * ... * Kf), where Ka through the other K factors account for surface, size and reliability. Using Kt instead of Kf for fatigue is conservative and often acceptable for preliminary design, but using the correct Kf gives a more realistic and often more economical estimate of fatigue life. For static overload in ductile materials, stress concentrations are largely relieved by local yielding, and Kt is often set to 1 in static analysis. For brittle materials, apply Kt even for static loads.
Typical Kt ranges for common geometry cases
| Geometry | Loading | Typical Kt range | Severity |
|---|---|---|---|
| Plate, central circular hole | Axial | 2.5 - 3.0 (small hole) | Moderate-High |
| Plate, central circular hole | Bending | 2.0 - 3.0 (small hole) | Moderate-High |
| Stepped bar, shoulder fillet | Axial | 1.4 - 3.0 (sharp fillet) | Low-High |
| Stepped bar, shoulder fillet | Bending | 1.3 - 2.5 (sharp fillet) | Low-High |
| Circular shaft, U-groove | Torsion | 1.2 - 2.0 | Low-Moderate |
| Circular shaft, shoulder fillet | Torsion | 1.1 - 1.9 | Low-Moderate |
Indicative Peterson / Pilkey values; actual results depend on the exact geometry ratios.
Frequently asked questions
What does a Kt of 3 mean for a circular hole?
For a small circular hole in a very wide plate under uniform tension, the stress at the edge of the hole is exactly 3 times the remote tensile stress. This is the classic Kirsch (1898) result and is a limiting case: as the hole grows relative to the plate width, Kt drops below 3 because less load-carrying material remains outside the hole. For d/w = 0.5 in axial loading, for example, Kt is closer to 2.3.
Why is Kf always less than or equal to Kt?
Kf accounts for the fact that a real material cannot fully "see" the sharp stress peak at a notch root at the fatigue-relevant scale. Below the notch root, over a small material volume, the stress gradient is steep and the highly stressed region may not be large enough to initiate and propagate a crack at the full theoretical stress. This grain-scale effect is captured by the notch sensitivity q, which is less than 1 for most structural metals, making Kf = 1 + q*(Kt - 1) less than Kt.
How does increasing the fillet radius reduce Kt?
A sharp fillet (small r) concentrates the flow of stress more severely than a generous radius because the curved load path must change direction more abruptly. As r/d increases, the stress path through the shoulder becomes less abrupt and peak stresses drop. A common design rule is to aim for r/d at least 0.1, and preferably above 0.2, to keep Kt below about 2.0 for axial loading.
Does Kt apply to both tension and compression?
Yes, the same geometric factor applies to compressive loads, producing a stress concentration in compression at the same location and with the same magnitude as under tension. For static loading in ductile materials, compressive stress concentrations are usually not a concern because local yielding redistributes stress harmlessly. For fatigue, compressive stress cycles do reduce crack growth rates but the concentration still matters when the cycle includes tension.
What material properties does this calculator need?
The Kt calculation is purely geometric and needs no material input. To compute the fatigue notch factor Kf you also need the ultimate tensile strength (Sut) in MPa, which is used to evaluate the Peterson material constant a. Most steels have Sut between 400 and 2000 MPa; the higher the strength, the more sensitive the material to notches (higher q) and the closer Kf is to Kt.
Can I use this for round shafts?
This calculator covers rectangular flat bars and plates. The Kt polynomial coefficients differ for circular cross-sections under torsion, bending and axial load. Peterson's charts for round shafts with grooves, shoulder fillets and keyways are distinct from the flat-bar curves implemented here. For round shafts, consult the shaft-specific tables in Peterson's "Stress Concentration Factors" or in Shigley's "Mechanical Engineering Design" Appendix A.