Stress Concentration

The flow of force, and what gets in its way

It helps to picture stress as the flow of force through a solid, like streamlines of water through a channel. In a plain bar pulled in tension, those lines are straight and evenly spaced — the stress σ = P / A is uniform across every cross-section. Put a hole in the bar and the force lines can no longer pass through where the material is missing; they have to swerve around it. Like water speeding up through a constriction, the lines crowd together at the sides of the hole, and where they crowd, the stress rises. That crowding is stress concentration.

The amount of amplification is captured by the elastic stress concentration factor, Kt:

Kt = σ_max / σ_nom

where:
  σ_max = peak local stress at the discontinuity (Pa)
  σ_nom = nominal stress, computed on a reference area (Pa)
  Kt    = dimensionless, geometry-only factor (≥ 1)

The crucial word is geometry. Kt is a purely geometric multiplier — it does not depend on the material or on how hard you pull. A steel plate and an aluminium plate with the same hole have the same Kt = 3. The factor tells you nothing about whether the part survives; it tells you how the load it does see gets concentrated. One subtlety: σ_nom must be defined consistently. It can be referenced to the gross section (full width) or the net section (width minus the hole). Handbook charts always state which — mixing them is a common error worth a factor of two.

Kt = 3: the canonical hole

The most famous result in the field is that a small circular hole in a wide plate under uniaxial tension has Kt = 3. This is not a rule of thumb — it is exact, from the Kirsch solution (Ernst Kirsch, 1898) of the elasticity equations. Along the edge of the hole perpendicular to the load, the stress is exactly three times the remote stress, regardless of hole diameter, as long as the hole is small relative to the plate width.

The full Kirsch field for a hole of radius a in a plate under remote tension σ gives the tangential (hoop) stress around the hole edge as:

σ_θθ (at hole edge, r = a) = σ · (1 − 2·cos 2θ)

  θ = 90° (sides, perpendicular to load):  σ_θθ = 3σ   ← peak, Kt = 3
  θ = 0°  (top/bottom, along the load):    σ_θθ = −σ   ← compression!

Two surprises hide in that one line. First, the peak is three times the applied stress. Second, at the poles of the hole — top and bottom, along the loading direction — the stress is negative, i.e. compression, even though the plate is being pulled apart. This is why a hole edge has both a tension hot-spot and a compression cold-spot 90° apart, and why fatigue cracks always launch from the sides of a loaded hole, never the top.

The effect is also local. The hoop stress decays roughly as 1/r² away from the hole; by about three radii out you are back to nominal. Stress concentration is intense but short-ranged — which is exactly why relieving it (with a fillet or a relief groove) is so effective: you only have to fix a small region.

The elliptical notch and the road to infinity

Inglis (1913) generalised the hole to an ellipse, and his result is the one that haunts every fracture engineer. For an elliptical hole with semi-axis a perpendicular to the load and tip radius of curvature ρ:

Kt = 1 + 2·√(a / ρ)

  a = half the notch depth / crack length (m)
  ρ = radius of curvature at the notch tip (m)

Set a = ρ (a circular hole, a = ρ = radius) and you recover Kt = 1 + 2 = 3. But now shrink the tip radius ρ toward zero — sharpen the notch into a crack — and Kt blows up without bound. A perfectly sharp crack has infinite elastic stress at its tip. That is the bridge between stress concentration (a geometry problem) and fracture mechanics (an energy problem): once Kt is meaningless because ρ → 0, you switch to the stress intensity factor K and fracture toughness K_IC instead. Stress concentration governs blunt features; fracture mechanics governs sharp ones.

Common features and their Kt

Real parts are full of stress raisers. The table below gives representative elastic Kt values for the classic cases (uniaxial tension or bending, referenced to the net section, typical proportions). Exact values come from Peterson's charts or finite-element analysis, but these capture the magnitudes a designer should carry in their head.

Feature	Typical Kt	Driving ratio	How to reduce it	Where it bites
Circular hole, wide plate (uniaxial)	3.0	hole small vs. width	biaxial load (→2), reinforced edge	Bolt/rivet holes, ports
Circular hole, biaxial tension	2.0	—	inherent to load case	Pressure-vessel openings
Elliptical hole, a/b = 3	~7	a/ρ (sharpness)	round the ends	Slots, windows
Shoulder fillet (stepped shaft), r/d = 0.05	~2.0–2.5	r/d and D/d	increase fillet radius r	Shaft shoulders, axles
Shoulder fillet, r/d = 0.30	~1.4	r/d and D/d	already good	Well-designed shafts
Sharp re-entrant corner (r ≈ 0)	→ ∞ (singular)	radius → 0	add ANY fillet	Square pockets, keyway ends
U-groove / circumferential notch	~2–4	r/d, t/r	larger groove radius	Snap-ring grooves, threads
Transverse hole in round bar (tension)	~2.5–3	d_hole / D_bar	smaller relative hole	Cotter-pin holes
Sharp crack tip	∞ (use K_IC)	ρ → 0	fracture mechanics regime	Fatigue / brittle fracture

Worked example: the fillet that triples fatigue life

A stepped steel shaft transmits a bending moment that produces a nominal stress σ_nom = 80 MPa in the smaller diameter d = 40 mm, stepping up to D = 50 mm (so D/d = 1.25). Compare a near-sharp shoulder, r = 1 mm, against a generous fillet, r = 8 mm.

Case A — tight fillet:  r = 1 mm  →  r/d = 0.025  →  Kt ≈ 2.5
  σ_max = Kt · σ_nom = 2.5 × 80 = 200 MPa

Case B — generous fillet: r = 8 mm  →  r/d = 0.20  →  Kt ≈ 1.5
  σ_max = Kt · σ_nom = 1.5 × 80 = 120 MPa

Peak stress drop: 200 → 120 MPa, a 40% reduction.

That 40% cut in peak stress is enormous for fatigue. The S–N curve of steel in the high-cycle regime follows roughly σ ∝ N^(−0.1), i.e. life N scales as roughly σ^(−10) over the relevant range. Dropping the local stress by a factor 200/120 = 1.67 multiplies fatigue life by about 1.67^10 ≈ 160×. Even after correcting for notch sensitivity (using Kf instead of Kt) and finite-life corrections, the generous fillet routinely buys an order of magnitude in life. The fillet costs nothing but a slightly larger tool radius on the lathe — it is the highest-leverage decision on the drawing.

From Kt to Kf: notch sensitivity

The elastic Kt overstates the real fatigue penalty, because metal is not perfectly elastic and the highly-stressed volume at a small notch is tiny. The reduction actually seen in fatigue strength is the fatigue notch factor Kf, linked to Kt by the notch sensitivity q:

Kf = 1 + q · (Kt − 1)      with   0 ≤ q ≤ 1

  q = 0  → notch has no fatigue effect (fully insensitive)
  q = 1  → full theoretical effect (Kf = Kt)

Peterson's empirical form:  q = 1 / (1 + a/r)
  a = material constant (~0.025 mm for hard steel, larger for soft)
  r = notch root radius

The lesson buried in q = 1/(1 + a/r): small notches (small r) have low q and are forgiving, while large, gentle notches (large r, low Kt anyway) approach full sensitivity. High-strength steels have a small material constant a, so they are more notch-sensitive — exactly the materials you reach for when you need strength are the ones most punished by a careless corner. This is a recurring trap: upgrading to a stronger alloy without improving the geometry can lower the part's fatigue life.

Static loading: why ductile parts forgive notches

Under a single static load on a ductile metal, stress concentration is often a non-issue. When σ_max reaches the yield strength, the material at the notch root yields and flows plastically, redistributing load to its neighbours. The part keeps carrying load until the entire net section yields, so the static failure load is governed by net-section area, not by Kt. This is why a tension member with a bolt hole is designed on net-section yield, and Kt is ignored for static strength of ductile steel.

Brittle materials are the opposite. Cast iron, ceramics, glass, fully-hardened steel and concrete cannot yield to relieve the peak. There, the local σ_max is real: a brittle part fails when Kt·σ_nom reaches the material's fracture stress, far below the net-section strength a ductile part would survive. The single most important question when assessing a stress raiser is therefore: is the load static or cyclic, and is the material ductile or brittle?

Situation	Does Kt matter?	Governing check
Static load, ductile metal	Largely no	Net-section yield (σ_nom ≤ σ_y)
Static load, brittle material	Strongly yes	Kt·σ_nom ≤ fracture stress
Cyclic load, any metal	Yes — always	Kf·σ_a ≤ endurance limit
Pre-existing sharp crack	Use K, not Kt	K_I ≤ K_IC (fracture toughness)

Failure modes and design trade-offs

• Fatigue crack initiation. The dominant real-world consequence. Cyclic loading at a notch accumulates local plastic strain until a microcrack nucleates at the notch root, then propagates. Design for the local stress range Kf·Δσ, not the nominal stress. The de Havilland Comet airliner's 1954 crashes traced to fatigue cracks launched from near-square cabin window corners — the fix was simple rounding.
• Brittle fracture. In brittle materials or cold ferritic steel below its ductile-to-brittle transition, a notch peak can trigger sudden, complete fracture with no warning. The Liberty ships of WWII split in two at square hatch corners and weld defects in cold seas.
• Threads, keyways and splines. Bolt threads concentrate stress at the root of the first engaged thread (Kt ≈ 3–4); most bolt fatigue failures occur there. Rolled threads (which leave compressive residual stress) outlast cut threads several-fold for this reason.
• The reinforcement trade-off. You cannot always avoid a hole — you need the bolt, the oil port, the lightening hole. Options are to add material around it (a boss or doubler), to put the hole where the nominal stress is low, or to accept Kt and qualify the part by test. Adding a second, smaller relief hole on either side can even lower the peak by smoothing the force flow.
• Multiple raisers interacting. Two close holes do not simply add their fields; they can shield or amplify depending on spacing relative to the load direction. Closely spaced holes in a row (a perforated plate) raise the net-section stress and must be analysed as a ligament.
• Surface finish as a micro-notch. A rough machined surface is a field of tiny notches. Polishing, shot-peening (which adds compressive residual stress) and avoiding sharp tool marks raise fatigue strength substantially — surface finish factors of 0.7–0.9 are applied to the endurance limit for exactly this reason.