A stress concentration factor calculator deals with the thing that average stress hides. Divide a load by an area and you get a number that is correct on average and wrong everywhere that matters. Put a hole in a plate and the stress at the edge of that hole is three times the average. Cut a sharp notch and it can be five or ten times. Parts do not fail where the average stress is highest; they fail at the root of the sharpest feature, and Kt is how that local peak is expressed.
Arb Digital publishes free engineering calculators that each own one job, and this page owns the local peak. The live stress and strain calculator deliberately computes the average stress across whatever cross-section you give it and states on its own page that it models no concentration effects. This page starts from that nominal stress and applies the geometric amplification. Use that page for the nominal number; use this one for what happens at the hole.
What This Stress Concentration Factor Calculator Does
It computes Kt for three published geometries, and it accepts a Kt you have read from your own chart for anything else. From that factor and the load, thickness and section you enter, it returns the nominal stress on the net section, the peak elastic stress at the root, and the fatigue notch factor Kf if you supply a notch sensitivity.
What the page does not do is supply a material. No allowable stress, no fatigue limit, no notch sensitivity table. Those are properties of the alloy, the heat treatment, the surface finish and the environment in front of you, and they belong in your own data, not in a web page's defaults.
How to Use It
- Pick the geometry that matches your feature. If none of them does, get Kt from a published chart for your actual shape and switch to manual entry.
- Enter the two dimensions in the order the labels give. For the circular hole it is diameter then plate width; for the ellipse it is the semi-axis across the load then the tip radius; for the notch it is depth then root radius.
- Enter the load and thickness so the page can compute the nominal stress on the net section. If you already have a nominal stress, set the load and thickness to give it and check the reported value.
- Add your notch sensitivity if you are working on fatigue. Setting q to 1 makes Kf equal to Kt, which is the conservative assumption; setting it to 0 removes the notch effect entirely, which is not.
- Read the validity note. The calculator says when a ratio has gone outside the range the published relation was fitted over, rather than silently extrapolating.
The Formulas, And Where They Come From
For a circular hole in a plate of finite width W, loaded in tension, the standard published approximation with Kt referenced to the net section is the Howland cubic in λ = d/W:
Kt = 3.00 − 3.13λ + 3.66λ2 − 1.53λ3
As the hole becomes small relative to the plate, λ tends to zero and Kt tends to exactly 3, which is the Kirsch solution for a circular hole in an infinite plate. That value of 3 is the single most quoted number in the whole subject.
For an elliptical hole of semi-axis a perpendicular to the load, with the tip radius ρ = b2/a, the Inglis solution gives
Kt = 1 + 2a/b = 1 + 2√(a/ρ)
which again returns 3 for a circular hole, where a = b. For a deep U-notch of depth t and root radius r on both edges of a bar in tension, Neuber's relation gives Kt = 1 + 2√(t/r), and this too collapses to 3 for a semicircular notch where the depth equals the radius. Three different geometries converging on the same number is not a coincidence: it is the same circular boundary seen three ways.
The canonical published source for Kt charts across hundreds of geometries is Peterson's Stress Concentration Factors, now maintained by Pilkey. Anything not covered by the three relations above should be read from Peterson or an equivalent standard reference, which is what the manual mode is for. MIT's 3.11 Mechanics of Materials course covers the underlying continuum stress analysis, and OpenStax sets out the elastic behaviour these relations assume in section 12.4, Elasticity and Plasticity.
Work the default by hand. A 20 mm hole in a 60 mm wide, 10 mm thick plate gives λ = 0.3333, so Kt = 3.00 − 1.0433 + 0.4067 − 0.0567 = 2.307. The net section is (60 − 20) × 10 = 400 mm2, so a 20 kN load gives a nominal stress of 50 MPa and a peak of 2.307 × 50 = 115.3 MPa. With a notch sensitivity of 0.8, Kf = 1 + 0.8(2.307 − 1) = 2.045.
Kt Is Elastic, And That Is A Real Limitation
Every relation on this page comes from linear elastic theory. It assumes the material obeys Hooke's law everywhere, including at the root of the notch, and that nothing yields. In a ductile metal under a static load, that assumption fails almost immediately at any serious concentration: the peak reaches yield, the material at the root deforms plastically, and the load redistributes into the surrounding elastic material.
The practical consequence is that a static, monotonic load on a ductile part is often barely affected by the notch at all. Local yielding blunts the peak, and the part carries close to the net-section load a smooth part would. This is why Kt is usually applied at full strength to brittle materials and to fatigue, and applied with judgement to ductile static cases.
Notch Sensitivity: Why Kf Is Smaller Than Kt
Fatigue behaves differently from static loading, and this is where concentration factors earn their reputation. A notch does reduce fatigue strength, but not by the full factor Kt. The measured reduction is the fatigue notch factor Kf, and the two are linked through the notch sensitivity q:
Kf = 1 + q(Kt − 1)
with q running from zero, meaning the notch has no fatigue effect, to one, meaning it has its full theoretical effect. Real values depend on the material and on the root radius. Coarse-grained and low-strength materials are less notch-sensitive; high-strength steels approach q = 1. Very small radii are less sensitive than the elastic factor suggests, because the highly stressed volume becomes so small that it contains too few grains for a crack to initiate easily.
Take q from your own fatigue reference for your material and radius. Using q = 1 is conservative and is a defensible starting point when you have no data. It is worth remembering that fatigue is where most notch failures actually occur, so the von Mises stress calculator and a static margin alone will not catch them.
Net Section Versus Gross Section, And Why Charts Disagree
Two published charts for the same hole can give Kt values that differ by fifty per cent, and both can be right. The difference is which nominal stress the factor is meant to multiply. A net-section factor multiplies the load divided by the reduced area through the hole. A gross-section factor multiplies the load divided by the full plate area as if the hole were not there.
The two are related exactly: Kt,gross = Kt,net ÷ (1 − d/W). For the default case, the net-section factor of 2.307 becomes a gross-section factor of 3.46. Both produce the same peak stress when paired with their own nominal stress, and either produces nonsense when paired with the other. This calculator uses the net section throughout and states so in the results, and if you enter a Kt manually you must know which basis your chart used.
Where Concentrations Actually Bite In Practice
The features that cause trouble are rarely the ones drawn on the design sheet. A generous fillet at a shaft shoulder is planned for; a tool mark left in that fillet by a worn insert is not, and its radius may be a fraction of a millimetre. Corrosion pits, weld toes, stamped part numbers, machining chatter and sharp corners on keyways all act as notches, and none of them appears in the nominal stress calculation.
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Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Mixing net and gross nominal stress — a chart factor is tied to one basis or the other, and pairing it with the wrong nominal stress changes the peak by the full area ratio.
- Applying Kt to a ductile static case at full value — local yielding redistributes the load and blunts the elastic peak. The elastic factor is an upper bound there, not a prediction.
- Treating Kf as equal to Kt without saying so — it is a conservative assumption and often a reasonable one, but it should be a stated choice rather than an accident.
- Extrapolating a fitted curve past its range — the hole approximation is fitted over a limited span of diameter-to-width ratio, and pushing it towards a hole that nearly severs the plate produces a number with no basis behind it.
- Ignoring the features that were never drawn — tool marks, corrosion pits, weld toes and stamped characters are all notches, and they are where cracks usually start.
Related Free Tools From Arb Digital
Get the nominal stress first with the stress and strain calculator, then combine a multiaxial state with the von Mises stress calculator. For section properties and stability use the section modulus calculator, the moment of inertia calculator, the beam deflection calculator and the column buckling calculator. Pressure and shear cases are covered by the hoop stress calculator and the shear stress calculator, and direction-dependent materials by the stiffness matrix calculator. Everything Arb Digital publishes is on the free online tools hub.
Frequently Asked Questions
It is the ratio of the peak local stress at a geometric feature to the nominal stress in the surrounding section. A factor of three at a hole means the material right at the hole edge sees three times the average stress in that cross-section.
It comes from the Kirsch elastic solution for a circular hole in an infinite plate under uniaxial tension. The peak occurs at the two points on the hole edge perpendicular to the load, and the analysis gives exactly three times the far-field stress there.
Kt is the theoretical elastic factor from geometry alone. Kf is the measured reduction in fatigue strength caused by the same notch, and it is usually smaller. They are linked by the notch sensitivity q, where Kf equals one plus q times Kt minus one.
Much less than the elastic factor suggests. The material at the root yields, the load redistributes into the surrounding elastic region, and the part carries close to its net-section capacity. Concentration factors matter most for brittle materials and for fatigue.
Whichever your chart is referenced to, paired with the matching nominal stress. They differ by the area ratio and give the same peak stress when used consistently. This calculator uses the net section and says so with every result.
The standard reference is Peterson's Stress Concentration Factors, now edited by Pilkey, which collects elastic solutions and finite element results for a very large set of geometries. The manual entry mode exists so you can use a value read from it or an equivalent source.
Increase the root radius, spread the change of section over a longer transition, or move the feature to a lower-stress region. Sharpness is what drives the factor, so the largest radius the design allows is almost always the cheapest improvement.
No. It gives an elastic factor and a peak stress from published relations. Qualifying a load-bearing component needs material allowables, a failure criterion, a fatigue assessment and review by a qualified engineer.
This tool is provided for educational and preliminary study use only. It applies published elastic stress concentration relations to dimensions you supply and takes every material property, allowable stress and notch sensitivity as your own input; it publishes no material data and applies no safety factor. Any load-bearing component must be analysed and signed off by a qualified mechanical or structural engineer working from qualified material allowables and a fatigue assessment.
