The number Kt giving how many times the average stress the maximum stress at a hole or notch root is. For an elliptical hole it is 1 + 2√(a/ρ), and for a circular hole it is 3.
Japanese name: 応力集中係数
Stress concentration factor appears in 1 practice question across 1 topic, including Fracture mechanics and fatigue.
The stress concentration factor Kt is the maximum stress σmax at the root of a notch or hole divided by a reference nominal stress σn: Kt = σmax/σn. The nominal stress may be calculated on the gross section without the hole or on the net section reduced by the hole, and the value of Kt depends on which is used as the reference. When using handbook values, therefore, always check how the reference was taken. Kt is determined by elastic calculation and depends only on geometry and the way the load is applied, not on the kind of material. For the same geometry, it has the same value for steel and for aluminum alloys.
When uniaxial tension acts on a circular hole in an infinitely wide plate, the stress is greatest at the two points on the edge of the hole lying across the load direction, and Kt = 3. Away from the edge the stress falls quickly; with hole radius a and distance r from the center, the stress on the section beside the hole is σ(2 + a²/r² + 3a⁴/r⁴)/2. At r = 2a, that is, only one radius from the edge, it has already fallen to about 1.22 times. The value also changes with the type of loading: Kt = 2 for equal biaxial tension and 4 for pure shear. What matters is that the high stress is confined very close to the hole.
The drop in fatigue strength is often not as large as Kt would predict. The fatigue notch factor Kf, the fatigue limit of an unnotched specimen divided by that of a notched specimen, is therefore used, and the relation between the two is expressed by the notch sensitivity q = (Kf − 1)/(Kt − 1). q takes values from 0 to 1 and tends to be smaller for smaller notch root radii and for softer, lower-strength materials. High-strength steels are notch sensitive, with q close to 1, and feel almost the full effect of Kt. The sharper the notch, the larger the gap between Kf and Kt, and designing on Kt alone can be overly conservative.
When a static load acts on a ductile material, the stress redistributes to the surroundings even if the notch root yields, so the effect of Kt is small and strength can often be evaluated from the average stress on the net section. For brittle materials, for fatigue, and for materials that become brittle at low temperature, however, Kt translates directly into a loss of strength. In design, measures include giving large fillets to the corners of stepped shafts, avoiding abrupt changes of section and adding relief grooves nearby to smooth the flow of stress. The emergency measure of drilling a small hole at a crack tip to blunt it and stop its growth is based on the same idea.
Answer: A. About 1100 MPa
a/ρ = 1.0/0.040 = 25, so √(a/ρ) = 5, the stress concentration factor is Kt = 1 + 2×5 = 11, and the maximum stress is σmax = 100×11 = about 1100 MPa. The longer the hole and the sharper its end (the smaller ρ), the more the stress concentrates; when ρ shrinks to about an interatomic distance, as at a crack, the stress at the end reaches the theoretical strength even under a low average stress. About 5100 MPa skips the square root and uses 1 + 2a/ρ, and about 140 MPa swaps a and ρ to use √(ρ/a). For a circular hole (a = ρ), Kt = 3, and the formula also shows why rounding the root of a notch eases the stress concentration.
Not in the elastic range. Kt is obtained from elasticity theory and depends only on geometry and loading. How much that concentration affects strength, however, differs between materials: little for the static strength of ductile materials and a lot for brittle materials and fatigue.
Because stress that cannot pass through the hole detours around its sides and concentrates near the edge. In the formula for an elliptical hole, Kt = 1 + 2√(a/ρ), a = ρ for a circle, so Kt = 1 + 2 = 3. It does not depend on hole size: in a wide plate, a small hole and a large hole both give 3.
For simple shapes it is taken from elasticity solutions or handbook charts, and for complex shapes it is calculated by the finite element method. Experimentally, the strain at the notch root is measured with strain gauges, photoelasticity or digital image correlation. In finite element analysis, the maximum stress is underestimated unless the elements at the notch root are fine enough.
a/ρ = 1.0/0.040 = 25, so √(a/ρ) = 5, the stress concentration factor is Kt = 1 + 2×5 = 11, and the maximum stress is σmax = 100×11 = about 1100 MPa.
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