Stress Concentration Factor K_t — A Practical Mechanical Design Guide

Every mechanical component has at least one place where the geometry changes — a drilled hole, a keyway, a shaft step at a bearing seat. At each of those places, stress doesn’t just go up a little. It spikes. That spike is captured by the stress concentration factor K_t, a dimensionless multiplier that says how much higher the peak stress is compared with what a simple load-over-area calculation would predict. Get it wrong in a static design and you’ve added an invisible safety margin tax. Get it wrong in a fatigue design and you’ve got a crack initiation site that will shorten component life by an order of magnitude.

This guide covers everything you need to apply K_t confidently: the physical meaning, the key geometric cases, the worked example with numbers you can check yourself, and the connection to the fatigue stress concentration factor K_f. Use the Stress Concentration Simulator alongside it to explore the curves interactively.

What K_t Actually Means — Force Flow Lines

The clearest way to understand K_t is through the idea of force flow lines — imaginary lines that represent how stress travels through a loaded body. In a smooth, uniform bar under tension, those lines are evenly spaced and parallel. They’re like water flowing at constant speed through a straight pipe.

Now punch a hole in the bar. The force flow lines can’t go through the hole, so they crowd around it — just like water speeding up as it squeezes through a narrow channel. The more abrupt the geometric change, the tighter the crowding, and the higher the local stress. A sharp 90° corner is the limiting case: the lines have nowhere to go but infinitely tight, which is why a true zero-radius corner gives a theoretically infinite K_t. In the real world, every machined edge has some small radius, and that finite radius is what keeps the stress finite — but it can still be very high.

The fundamental relationship is simple:

\[\sigma_{\max} = K_t \times \sigma_{\text{nom}}\]

where \(\sigma_{\text{nom}}\) is the nominal stress calculated from the net cross-section using basic formulas (force divided by net area, or bending moment divided by section modulus at the net section). K_t is purely a function of geometry — shape ratios, not material properties. A steel component and an aluminium component of identical geometry have the same K_t.

Plate with a Central Hole — The Benchmark Case

The circular hole in a wide plate under uniaxial tension is the foundation of stress concentration theory. Kirsch’s exact elasticity solution (1898) shows that as the hole diameter d becomes vanishingly small relative to the plate width D (i.e., d/D → 0), the tangential stress at the hole edge is exactly three times the applied stress:

\[K_t \to 3.0 \quad \text{as } \frac{d}{D} \to 0\]

This is one of the most important numbers in stress analysis. It means that even a tiny hole — one you might dismiss as negligible — triples the local stress at its edge. For the default simulator geometry (D=100 mm, d=20 mm, d/D=0.2), K_t ≈ 2.5, because as the hole grows the net section shrinks and the nominal stress calculation already captures some of the effect.

The full relationship follows Peterson’s curve fit. As d/D increases from 0 toward 1, K_t decreases from near 3.0 toward lower values, but the actual peak stress σ_max doesn’t necessarily fall — because σ_nom rises as the net area shrinks. Designers often misread the chart: a lower K_t at larger d/D doesn’t mean a safer component.

Worked Example — Plate with Hole, Default Geometry

Use the simulator defaults to trace through the full calculation.

Given: Plate width D = 100 mm, hole diameter d = 20 mm, applied nominal stress σ_nom = 100 MPa.

Step 1: Find d/D.

\[\frac{d}{D} = \frac{20}{100} = 0.20\]

Step 2: Read K_t from Peterson’s chart (or the simulator’s curve fit). At d/D = 0.20 for a plate with a central hole under uniaxial tension, the Peterson curve gives:

\[K_t \approx 2.50\]

Step 3: Compute peak stress.

\[\sigma_{\max} = K_t \times \sigma_{\text{nom}} = 2.50 \times 100\,\text{MPa} = 250\,\text{MPa}\]

So while the average stress across the net section is 100 MPa, the plate actually reaches 250 MPa right at the hole edge. If the material’s yield strength is 280 MPa, that’s a margin of only 12% — something a nominal-stress-only analysis would completely miss.

Shaft with Shoulder Fillet — The Most Common Design Case

Shaft steps at bearing seats, gear seats, and snap-ring grooves are everywhere in rotating machinery. Each step is a stress concentrator. The simulator’s shaft-with-fillet geometry (default: D=80 mm, d=50 mm, r=5 mm) covers both tension and torsion loading.

For a shaft under torsion, the relevant factor is often called K_ts (the “s” for shear), but it works the same way: \(\tau_{\max} = K_{ts} \times \tau_{\text{nom}}\). Combine this with the torsion shear stress formula to find the true peak shear stress at the fillet root, not just the nominal value from \(Tc/J\).

The dominant design variable at a shaft step is the fillet radius r. Consider a shaft with D=80 mm, d=50 mm (D/d = 1.6). At r=0.5 mm (r/d ≈ 0.01), K_t is in the range of 4–6 depending on the loading type. At r=5 mm (r/d = 0.10), K_t drops to roughly 1.5–1.8. That’s a factor-of-three reduction in peak stress for a fillet radius that grew by 10 mm — one of the highest-return design improvements available.

The improvement flattens out as r/d exceeds about 0.15–0.20. Beyond that, enlarging the fillet further yields diminishing returns and may interfere with bearing seating requirements. This trade-off is exactly what the K_t vs r/d chart communicates: spend your geometry budget where the curve is steep, not where it’s flat.

Other Geometry Types in the Simulator

Beyond the two most common cases, the simulator covers four additional geometry types, each matching a Peterson chart:

• Plate edge notch: D=100 mm, notch depth t=15 mm, notch radius r=5 mm. Common in structural members with cut-outs or reliefs.
• Plate shoulder fillet in tension and bending: D=100 mm, d=60 mm, r=5 mm. The bending case generally gives higher K_t than tension at the same geometry because the stress gradient is steeper.
• Shaft with fillet in tension: D=80 mm, d=50 mm, r=5 mm. Relevant when the shaft carries axial load from thrust bearings or press fits.
• Shaft with transverse hole: D=80 mm, hole d=15 mm. A cross-drilled hole for an oil passage or pin seat creates a stress raiser in both the hoop and axial directions simultaneously.

Each geometry type uses Peterson’s handbook curve fits rather than hand-read charts, so the K_t value updates continuously as you drag the sliders.

K_t in Fatigue — Introducing Notch Sensitivity

Static design uses K_t directly in the yield or fracture check. Fatigue design is more nuanced, because real materials aren’t fully sensitive to the theoretical elastic stress peak. Tiny cracks and inclusions already present in the material blunt the notch effect at the microscale. The degree to which a material “feels” the stress concentration is captured by the notch sensitivity q, which runs from 0 (completely insensitive) to 1 (fully sensitive).

The fatigue stress concentration factor K_f combines q and K_t:

\[K_f = 1 + q\,(K_t - 1)\]

For q = 0, K_f = 1: the notch has no effect on fatigue life, and you can ignore the stress concentration entirely. For q = 1, K_f = K_t: the full theoretical amplification applies. High-strength steels (S_u > 700 MPa) tend toward q ≈ 1 because their fine grain structure means the material’s natural “crack blunting” ability is limited. Cast iron and some softer aluminium alloys have q much closer to 0 — they already contain enough internal discontinuities that an external notch adds relatively little.

In a fatigue analysis using the S-N curve (see the principal stress approach via Mohr’s circle for combined loading), the peak stress fed into the endurance limit calculation is K_f × σ_nom, not K_t × σ_nom. Using K_t everywhere in fatigue would be conservative for low-strength materials but appropriate for high-strength steels where q ≈ 1.

Design Scenario — Shaft Step at a Bearing Seat

Here’s a concrete situation that comes up in almost every rotating machinery design. A motor shaft transitions from 80 mm down to 50 mm at the inboard bearing seat. The shoulder must carry the bearing axial load (tension) while the shaft transmits torque from a coupling. The current fillet radius is r=1 mm, machined to keep costs down.

Entering these values into the simulator (shaft fillet, tension: D=80, d=50, r=1, r/d=0.02) returns K_t ≈ 4.1 for tension and K_ts ≈ 3.0 for torsion. The nominal torsional shear stress from the coupling torque is 45 MPa. That means the actual peak shear stress at the fillet root is about 135 MPa — well into the fatigue damage range for a steel with an endurance limit of 200 MPa, considering the stress cycles from torque fluctuations during start-stop.

Increasing the fillet radius to r=5 mm (r/d=0.10) changes the picture dramatically. K_ts drops to around 1.65, giving a peak shear stress of 74 MPa — a 45% reduction. The endurance limit margin is now much more comfortable, and the change costs almost nothing: it’s just a toolpath adjustment on the CNC lathe. This is why fillet radius is always the first variable to look at when a fatigue failure occurs at a shaft step.

Remedies for High Stress Concentration

When the geometry of a component produces an unacceptably high K_t, there are several proven approaches to bring it down:

• Increase fillet radius — the first and most effective lever at shaft shoulders and plate steps. Even a 2× increase in r from a small baseline can halve K_t.
• Add relief grooves (undercuts) — cutting a groove adjacent to the step moves the force flow lines away from the critical corner, effectively splitting one severe concentration into two milder ones.
• Use symmetric geometry — a symmetric double-edge notch concentrates stress less than a single-sided notch for the same net section, because the load path is shared evenly.
• Reduce the step ratio D/d — a gradual taper rather than a sharp step spreads the force flow lines over a longer length, reducing their local density.
• Shot peening — introduces compressive residual stresses at the surface that directly oppose the tensile stress peak from K_t. Especially effective for fatigue in notched regions.
• Cold-working holes — expanding a hole with an oversized mandrel or interference-fit bushing leaves compressive hoop stresses at the bore, reducing the effective K_f for cyclic loading.

None of these remedies changes the underlying K_t number from Peterson’s chart (except the first two, which alter the geometry). Shot peening and cold-working work by offsetting the stress peak with a residual stress of the opposite sign — they don’t reduce K_t, they reduce its consequence.