Why does a small hole cause Kt ≈ 3?

For an infinite plate with a circular hole under uniaxial tension, the exact elastic solution gives Kt = 3.0 at the hole edge. The stress must flow around the hole, concentrating at the narrowest cross-section.

What's the difference between Kt and Kf?

Kt is the theoretical (elastic) stress concentration factor from geometry alone. Kf is the fatigue notch factor that accounts for material sensitivity — some materials are less affected by sharp notches.

How can I reduce stress concentrations?

Increase fillet radii, use gradual transitions, add relief notches, or reduce the size of holes relative to the part width. Even small increases in fillet radius dramatically reduce Kt.

Does Kt apply to dynamic (fatigue) loading?

For fatigue, use Kf instead of Kt. Materials with low notch sensitivity (like cast iron) have Kf < Kt, while high-strength steels may have Kf ≈ Kt.

What is notch sensitivity?

Notch sensitivity q ranges from 0 (not sensitive, Kf = 1) to 1 (fully sensitive, Kf = Kt). It depends on material, notch radius, and loading type. Harder materials tend to have higher q.

Can Kt be less than 1?

No — Kt ≥ 1 by definition. A value of 1 means no stress concentration (uniform stress field). Sharp corners can produce Kt → ∞ in theory, which is why sharp internal corners should always be avoided.

Stress Concentration Factor Calculator

Calculate stress concentration factor Kt for holes, fillets, notches, and grooves. Compare nominal vs. peak stress with fatigue notch factor.

Common Scenarios

Geometry Type

Nominal Stress (σ_nom)

MPa

Plate Width (W)

Hole Diameter (d)

Plate Thickness

Stress Concentration Factor (Kt)

2.722

Kt ≈ 3.0 − 3.13(d/W) + 3.66(d/W)² − 1.53(d/W)³

Peak Stress

136.1 MPa

σ_max = Kt × σ_nom = 2.722 × 50

Nominal Stress

50.0 MPa

Applied uniform stress

Fatigue Factor (Kf)

2.550

Kf = 1 + q(Kt − 1), q = 0.9 assumed

Fatigue-Effective Stress

127.5 MPa

Stress for fatigue life calculations

Stress Amplification

172.2%

Increase over nominal stress

Stress Distribution (Conceptual)

Far fieldFeature centerFar field

Kt vs. Geometry Parameter

d/W ratio	Kt	Peak Stress (MPa)
0.05	2.852	142.6
0.10	2.722	136.1
0.15	2.608	130.4
0.20	2.508	125.4
0.25	2.422	121.1
0.30	2.349	117.5
0.35	2.287	114.4
0.40	2.236	111.8
0.45	2.193	109.7
0.50	2.159	107.9

Planning notes, formulas, and examples

About the Stress Concentration Factor Calculator

Geometric discontinuities such as holes, notches, fillets, and grooves cause local stress to rise above the nominal applied stress. The stress concentration factor Kt describes that amplification with the simple relationship sigma_max = Kt x sigma_nom.

This calculator covers several common geometries and estimates both the peak stress at the feature and the fatigue notch factor Kf for cyclic loading. That makes it useful when you want a quick estimate of how much a shape change increases local stress before moving to a more detailed analysis.

The result is most valuable when you are comparing designs, because small geometric changes can make a large difference in the peak stress around a discontinuity.

When This Page Helps

Stress concentrations are often the hidden reason a part fails sooner than expected. Having Kt, peak stress, and fatigue notch factor together makes it easier to judge whether a hole, notch, or fillet is still acceptable or whether the geometry needs to be softened.

How to Use the Inputs

Select the geometry type — hole in plate, shoulder fillet, U-notch, or circumferential groove.
Enter the nominal (far-field) stress applied to the part.
Enter the geometric dimensions specific to your selected feature.
Review the calculated Kt, peak stress, and fatigue notch factor.
Use the parameter variation table to see how changing the geometry affects Kt.
Check the stress distribution visualization for intuitive understanding.
Click preset buttons for common engineering scenarios.

Formula used

Hole in Plate: Kt ≈ 3.0 − 3.13(d/W) + 3.66(d/W)² − 1.53(d/W)³
U-Notch: Kt ≈ 1 + 2√(t/r)
Fatigue Notch Factor: Kf = 1 + q(Kt − 1)

Where:
• Kt = theoretical stress concentration factor
• Kf = fatigue notch factor
• q = notch sensitivity (0 to 1, material dependent)
• d = hole diameter, W = plate width
• t = notch depth, r = notch root radius

Example Calculation

Result: Kt = 2.73, peak stress = 136.5 MPa

For d/W = 10/100 = 0.1: Kt = 3.0 − 3.13(0.1) + 3.66(0.01) − 1.53(0.001) = 2.73. Peak stress = 2.73 × 50 = 136.5 MPa at the hole edge.

Tips & Best Practices

Increasing fillet radius is the most effective way to reduce Kt in stepped shafts — even small radius increases yield large Kt reductions.
For holes near the edge of a plate, Kt increases significantly — maintain at least 3 diameters of edge distance.
Shot peening can introduce compressive residual stresses at notch roots, improving fatigue life without changing geometry.
Multiple small holes have lower Kt than one large hole of equivalent total area.
For accurate Kf values, use material-specific notch sensitivity data from experimental handbooks rather than the q = 0.9 estimate.
FEA (Finite Element Analysis) can verify Kt for complex geometries not covered by handbook solutions.

Peterson's Stress Concentration Factors

The definitive reference for Kt values is "Peterson's Stress Concentration Factors" by Walter Pilkey and Deborah Pilkey. This handbook contains fitted equations and charts for hundreds of geometric configurations derived from elastic solutions and finite element analyses. The formulas in this calculator are simplified versions suitable for preliminary design.

Fatigue Life Implications

A stress concentration factor of Kt = 3 doesn't simply reduce fatigue life by a factor of 3. The relationship between Kt and fatigue life is nonlinear and depends on the S-N curve slope, mean stress, and material behavior. A part with Kt = 3 operating at half the nominal endurance limit may still fail in fatigue — always apply proper fatigue analysis methods.

Design Strategies for Stress Management

When stress concentrations cannot be eliminated, they can be managed. Relief grooves, generous radii, surface treatments (peening, nitriding), and load redistribution through design changes all help. The goal is to either reduce Kt geometrically or improve the local material resistance to fatigue crack initiation.

Sources & Methodology

Last updated: January 15, 2025

Frequently Asked Questions

For an infinite plate with a circular hole under uniaxial tension, the exact elastic solution gives Kt = 3.0 at the hole edge. The stress must flow around the hole, concentrating at the narrowest cross-section.