The stress concentration factor (Kt) is found using one of three main approaches: published charts for standard geometries, finite element analysis for complex shapes, or experimental techniques like photoelasticity. Which method you use depends on whether your geometry matches a known case or requires custom analysis. The core relationship is simple: the maximum stress at a geometric feature equals Kt multiplied by the nominal stress.
The Basic Formula
The stress concentration factor relates the peak stress at a geometric discontinuity to the average (nominal) stress in the part. For normal stress:
σ_max = Kt × σ_nom
For shear stress, the equivalent is:
τ_max = Kts × τ_nom
Kt is purely a function of geometry. It doesn’t depend on the material, only on the shape of the part and how the load is applied. A plate with a central hole under tension, for example, has a Kt of about 3.0 regardless of whether it’s made from steel or aluminum. This makes Kt a starting point for any stress analysis near a hole, fillet, groove, or notch.
Calculating Nominal Stress First
Before you can use Kt, you need the nominal stress, which is the stress calculated as if the geometric feature weren’t there (or, in some conventions, calculated at the reduced cross-section but without accounting for the stress spike). The formula depends on your loading type:
- Axial loading: σ_nom = F / A, where F is the applied force and A is the cross-sectional area at the section of interest.
- Bending: σ_nom = My / I, where M is the bending moment, y is the distance from the neutral axis, and I is the area moment of inertia.
- Torsion: τ_nom = Tr / J, where T is the torque, r is the radial distance, and J is the polar moment of inertia.
Getting the nominal stress right matters because Kt charts are built around a specific definition of nominal stress. If you use a different cross-section or reference point than the chart assumes, your answer will be wrong even if you read Kt correctly. Always check how the source defines nominal stress for that particular geometry.
Using Published Charts and Formulas
The most common way to find Kt for standard shapes is through reference charts. Peterson’s Stress Concentration Factors is the standard reference, now in its fourth edition, and it covers the geometries engineers encounter most often: notches, grooves, shoulder fillets, and holes. The book provides both graphical charts and curve-fit formulas, organized by geometry type and loading condition.
To use these charts, you identify your geometry (say, a stepped shaft with a shoulder fillet), then look up the relevant dimensionless ratios. For a fillet, those ratios are typically r/d (fillet radius to smaller diameter) and D/d (larger diameter to smaller diameter). You locate your ratios on the chart axes and read off Kt. A generous fillet radius gives a lower Kt, while a sharp transition drives it higher. For a simple circular hole in a wide plate under uniform tension, Kt is approximately 3.0. For a deep, sharp notch, Kt can exceed 5 or 6.
Many engineering textbooks reproduce a subset of Peterson’s charts for the most common cases. If your geometry is close to one of these standard shapes, this is the fastest and most reliable approach.
Finite Element Analysis for Complex Geometry
When your part doesn’t match a standard chart case, finite element analysis (FEA) is the go-to method. You build a model of the part, apply loads and boundary conditions, and read the peak stress directly from the results. Kt is then the ratio of that peak stress to the nominal stress you calculate by hand.
The challenge with FEA is getting accurate stress values near the concentration. Mesh density at the stress raiser is critical, and simply using small elements everywhere isn’t a reliable strategy. The formal approach is a mesh convergence study: you refine the mesh progressively and plot the peak stress against mesh density. When further refinement changes the result by a negligible amount, you’ve converged.
A few practical guidelines from NAFEMS (the international engineering simulation standards body) make this more manageable. First, the mesh only needs to be fine near the region of interest. Elements far from the stress raiser just transmit load and represent geometry, so they can be coarse. Second, transition zones from coarse to fine mesh should be at least three linear elements away from the area you care about. Third, internal fillets must be modeled with enough elements around the radius to capture the actual curved shape. A fillet represented by a single straight element edge cannot predict true stresses, even if the element is small. The actual radius from the engineering drawing must be represented faithfully.
Avoid using element size alone as your convergence metric. An element’s proximity to the stress concentration and the local load gradient matter more than its absolute size.
Experimental Methods
For parts where neither charts nor FEA are practical, or when you need physical validation, experimental techniques can measure Kt directly. Photoelasticity is the classic method. A transparent model of the part (typically made from a birefringent plastic like polycarbonate) is loaded and viewed under polarized light. Stress differences show up as colored fringe patterns, and the fringe order at any point relates to the stress at that location through a calibration curve.
On a free boundary (the edge of a hole or notch), one of the two principal stresses is zero. That simplifies things considerably: the remaining stress tangent to the boundary is determined directly from the fringe order. The stress concentration factor is then the ratio of that boundary stress to the nominal stress. This technique works in both the elastic and plastic ranges, thanks to non-linear stress-optic relationships that extend its usefulness beyond purely elastic loading.
Digital image correlation (DIC) is a more modern alternative that tracks surface deformation on actual parts using cameras and speckle patterns, giving full-field strain data without requiring a special model material. Both methods are most commonly used in research or for validating FEA models rather than for everyday design calculations.
From Kt to Fatigue: Notch Sensitivity
Kt represents the theoretical worst case, assuming the material “feels” the full stress spike. In practice, especially under cyclic loading, many materials don’t respond to the full theoretical concentration. This is captured by the fatigue stress concentration factor, Kf, which is always less than or equal to Kt.
The relationship between the two is expressed through notch sensitivity, q:
q = (Kf − 1) / (Kt − 1)
When q equals zero, the material is completely insensitive to the notch, and Kf equals 1, meaning no fatigue penalty. When q equals 1, the material feels the full geometric concentration, and Kf equals Kt. Most real materials fall somewhere in between, with q values typically available in reference tables based on the material’s ultimate strength and the notch radius. Softer, more ductile materials tend to have lower notch sensitivity, while hard, high-strength steels approach full sensitivity.
For static loading of ductile materials, stress concentrations are often less critical because local yielding redistributes stress. For fatigue loading or brittle materials, Kt (adjusted to Kf where appropriate) directly affects predicted life and is essential to get right.
Choosing the Right Method
For a standard notch, fillet, or hole, start with Peterson’s charts or the equivalent formulas in your textbook. This takes minutes and gives you a reliable Kt backed by decades of validated data. If your geometry is non-standard or involves multiple interacting features, FEA with a proper convergence study is the appropriate tool. Experimental methods are reserved for validation, unusual geometries, or situations where the loading is difficult to model analytically. In all cases, the accuracy of your result depends as much on correctly defining the nominal stress and matching the chart or model assumptions as it does on the method itself.