A crack in an aircraft window, a fracture starting at a bolt hole, a shaft that snaps at the shoulder where the diameter steps down. These failures share a cause that has nothing to do with the average load the part carries. The metal broke because a change in geometry concentrated stress into a tiny region, and that local peak — not the nominal stress — drove the crack.

This article explains the stress concentration factor: what it means, where the classic value of 3.0 comes from, how to apply it, and the mistakes that make engineers underestimate the danger of an innocent-looking hole.

Why this calculation matters

Real parts are not smooth bars. They have holes for fasteners, fillets where sections change, keyways, grooves, threads, and shoulders. Every one of those features disturbs the flow of stress through the material. Where the lines of force have to bend around an obstacle, they crowd together, and the local stress climbs well above the value you would compute from force divided by area.

The stress concentration factor, K_t, is the multiplier that captures this. It matters most for two failure modes. Under static loading of a brittle material, the peak stress can trigger fracture before the bulk of the section yields. Under cyclic loading, the concentrated stress is where fatigue cracks nucleate — and the vast majority of fatigue failures begin at a geometric discontinuity. If you size a part on nominal stress alone and ignore K_t, you have skipped the step where most failures are actually decided.

The core formula

The stress concentration factor is defined as a simple ratio:

K_t = sigma_max / sigma_nom

Here sigma_max is the true peak stress at the discontinuity and sigma_nom is the nominal stress computed from elementary mechanics. The subscript t means "theoretical" — K_t depends only on geometry and loading mode, not on the material. It comes from elasticity theory, finite element analysis, or experiment, and it assumes the material is still behaving elastically.

One subtlety trips people up: nominal stress can be defined two ways. The gross-section nominal stress uses the full, undisturbed width. The net-section nominal stress uses the reduced cross-section that remains after the hole or notch is removed. A published K_t value is tied to one definition or the other, so you must apply it consistently. In this article we use gross-section nominal stress throughout.

The most famous result in this field is the circular hole in a wide plate under uniaxial tension. Elasticity theory (the Kirsch solution, 1898) gives an exact answer:

K_t = 3.0   (small circular hole, wide plate, gross-section nominal stress)

The hole triples the stress at its edge, and remarkably, the result is independent of hole size as long as the hole is small relative to the plate width. Sharper features behave worse. A narrow elliptical hole, an oval, or a sharp notch can push K_t to five, ten, or higher. As a notch radius shrinks toward zero, K_t grows without bound — which is exactly why a crack tip is so dangerous and why sharp re-entrant corners are banned in good design.

A related quantity, the fatigue notch factor K_f, accounts for the fact that materials are not perfectly notch-sensitive. K_f is often smaller than K_t and is the value used in fatigue life estimates. K_t is always the conservative starting point.

A worked example

Consider a wide, thin steel plate in uniaxial tension with a small central circular hole. The plate is 60 mm wide and 5 mm thick, and it carries an axial force of 15 kN.

Step 1 — choose the stress concentration factor. For a small hole in a wide plate, the theoretical factor based on gross-section nominal stress is:

K_t = 3.0

Step 2 — nominal stress. Using the gross section (full width times thickness):

sigma_nom = F / (W * t)
sigma_nom = 15000 / (0.060 * 0.005)
sigma_nom = 15000 / 0.0003 = 50 MPa

Step 3 — peak stress at the hole. Multiply the nominal stress by K_t:

sigma_max = K_t * sigma_nom
sigma_max = 3.0 * 50 = 150 MPa

The average stress across the plate is a modest 50 MPa, but the metal right at the edge of the hole sees 150 MPa. The hole triples the local stress. A material check that compared 50 MPa against the yield strength would look comfortable; the honest comparison is 150 MPa against yield, and against the fatigue strength if the load cycles. Same plate, same hole — but the number you must design to is three times larger than the one a back-of-envelope estimate would give you.

Common mistakes

Comparing nominal stress to the strength limit. The nominal stress is a bookkeeping value, not the stress the material experiences at the critical point. Always carry K_t through to a peak stress before you check against yield or fatigue strength.

Mixing gross-section and net-section definitions. If a chart gives K_t referenced to net-section stress and you apply it to a gross-section nominal value (or vice versa), the answer can be off by a large margin. Read the fine print on every K_t source.

Assuming a hole is harmless because it is small. For a circular hole in a wide plate, K_t is 3.0 regardless of how small the hole is. Size does not save you; only geometry shape does.

Forgetting that sharper means worse. A generous fillet radius lowers K_t; a tight radius raises it steeply. Re-entrant corners with effectively zero radius are stress raisers that can approach crack-like severity. Round every internal corner you can.

Treating K_t as a fatigue factor. K_t is the elastic, theoretical value. Fatigue life calculations use the notch factor K_f, which is often a little lower and depends on material and notch size. Using K_t for fatigue is conservative but not always accurate; using K_f for a brittle static check is unsafe.

Try the interactive NovaSolver calculator

Looking up factors in a chart works, but it is slow when you are iterating on a fillet radius or a hole-to-width ratio. The Stress Concentration Factor Kt Calculator on NovaSolver computes K_t in real time for a circular hole in an infinite or finite-width plate, a stepped shaft with a shoulder fillet, and a semicircular notch. You enter the geometry ratios and the nominal stress, and it returns K_t, the maximum stress, the fatigue notch factor K_f, and plots of the cross-section stress and how K_t varies with the geometry parameter.

Related calculators

• Fatigue notch factor — converts the elastic K_t into the K_f that fatigue life estimates actually use, accounting for notch sensitivity.
• Mohr's circle — for resolving the multiaxial stress state around a discontinuity into principal stresses.
• Fracture mechanics — for when the notch has already become a crack and K_t no longer applies.

The full set is in the structural calculators hub.

Closing note

The stress concentration factor is a small idea with large consequences. It says that geometry, not just load, sets the stress a part actually feels — and that holes, fillets, and notches can multiply local stress several times over. Remember the anchors: a circular hole in a wide plate gives K_t = 3.0, sharper features give more, and the peak stress is what you compare against yield and fatigue limits. Design the fillet generously, carry K_t through every check, and the discontinuities in your part stop being the place it quietly fails.