Stress Concentration Factor Calculator

Stress concentration factor (Kt): A dimensionless ratio that compares the highest local elastic stress at a geometric change to the nominal stress. Kt = 1 means no increase above nominal stress [1].

Nominal stress: The reference stress away from the notch or based on the standard section used by the selected method.

Local maximum elastic stress: The estimated peak elastic stress at the notch root or fillet.

Primary geometry ratio 1: The main dimensionless check for the selected case, such as W/d or D/d.

Primary geometry ratio 2: A second dimensionless check when needed, often r/d or r divided by net width.

Method validity note: A message that tells you whether your geometry ratios are within the supported fitted-equation range, near its edge, or outside it.

MPa: Megapascal, a stress unit equal to 1 N/mm^2.

Mistake: Entering a hole diameter or notch depth in Feature size, d or h (mm) that is equal to or larger than Major width, W or D (mm) .
Fix: For a central hole, make sure W > d. For a U-notch, make sure W - 2h stays positive. For a shoulder fillet, enter the larger width as Major width, W or D (mm) and the smaller width as Feature size, d or h (mm) .

Mistake: Using the wrong stress in Nominal stress (MPa) , such as the peak notch stress instead of the reference-section stress.
Fix: Enter the average or reference-section stress in Nominal stress (MPa) , and use Peak or local maximum stress (MPa) only for the local notch-root stress in stress-ratio mode.

Mistake: Leaving Nominal stress (MPa) at 0 or entering a negative value in stress-ratio or local-stress mode.
Fix: Enter a positive nominal stress so the calculator can compute Stress concentration factor (Kt) or Local maximum elastic stress without division-by-zero or sign errors.

Mistake: Typing a value below 1 into Stress concentration factor, Kt for a notch problem.
Fix: Recheck the source of your Kt value. For standard stress-raising features, Stress concentration factor (Kt) is usually 1 or more.

Mistake: Ignoring Primary geometry ratio 1 , Primary geometry ratio 2 , or the Method validity note .
Fix: Compare the shown ratios with your intended geometry. If the validity note says edge-of-range or out-of-range, treat the result as less reliable or switch Out-of-range geometry policy to strict.

Mistake: Entering dimensions in inches while the labels show mm.
Fix: Convert all geometry values to millimeters before using Major width, W or D (mm) , Feature size, d or h (mm) , and Fillet or notch radius, r (mm) .

• The geometry mode gives a theoretical elastic Kt estimate for the selected case only. It is not a general finite element solver.
• Results depend on the chosen geometry family and loading case. A bar in tension and the same bar in bending can have different Kt values.
• The method assumes the dimensions are entered consistently in mm and the stress is entered in MPa.
• The central-hole calculation uses the Peterson polynomial cited below, with an explicit gross or net stress basis. Shoulder fillets use the published Noda approximation described below. U-notch fits remain unvalidated approximations; their ratio ranges do not establish accuracy.
• For U-notch cases, use a matching published chart or verified finite-element result and enter its Kt in Local stress mode. For shoulder fillets, use nominal stress at the smaller section; the bending mode is in-plane bending, not out-of-plane bending or a round shaft.
• The calculator uses theoretical elastic stress concentration factor, not fatigue notch factor. Real fatigue behavior can differ because of material notch sensitivity.
• Negative or zero Nominal stress (MPa) is not accepted in modes that divide by nominal stress or multiply by Kt.
• Impossible geometry is blocked, such as W less than or equal to d for a central hole, W - 2h less than or equal to 0 for a U-notch, or D less than or equal to d for a shoulder fillet.

Core equations

The calculator uses the standard definition of theoretical stress concentration factor [1].

Kt = sigma_max / sigma_nom

sigma_max = Kt * sigma_nom

Here, sigma_nom is the nominal stress and sigma_max is the local maximum elastic stress at the notch root.

Central circular hole in a thin plate

For uniaxial tension, let q = d/W. The net-section factor is Kt_net = 3 - 3.14q + 3.667q^2 - 1.527q^3. Gross-section Kt is Kt_net / (1 - q). Choose the basis matching your nominal stress. A 50 mm plate with a 10 mm hole gives Kt_net = 2.506464 and Kt_gross = 3.13308. A gross stress of 100 MPa or equivalent net stress of 125 MPa both give a peak of 313.308 MPa.

Ansys: Peterson central-hole correlation and net-section stress comparison. Plate length, thickness and loading must fit the thin-plate uniaxial model; the formula alone does not validate a real part.

Shoulder fillets in flat bars

The shoulder modes use the generalized Neuber approximation in Noda, Takase and Monda (1997), Equations 4, 5, 9, 10 and 11, with exponent 1.6 for tension and 1.4 for in-plane bending. They do not use the higher-accuracy Appendix II correction. Against 120 tabulated reference values, differences reached about 3.7% in tension and 6.5% in bending; these checks are not an error guarantee for every geometry. The range check uses 0.03 ≤ 2r/D ≤ 1 and 0.02 ≤ (D-d)/D ≤ 0.9. Outside this range, Warn permits an explicitly out-of-range estimate and Strict blocks it. Nominal stress is P/(d*t) in tension and 6M/(d*d*t) in in-plane bending, where t is thickness.

Noda et al.: shoulder fillets in round and flat bars (flat-bar cases d and e)

Geometry ratios used for each case

In Geometry to Kt mode, the calculator first converts your dimensions into dimensionless ratios. Those ratios are what the fitted equations use.

Central hole in plate: x = W / d

U-notch: d_net = W - 2*h

U-notch: x1 = W / d_net

U-notch: x2 = r / d_net

U-notch: x3 = h / r

Shoulder fillet: h = (D - d) / 2

Shoulder fillet: x1 = D / d

Shoulder fillet: x2 = r / d

Shoulder fillet: x3 = h / r

The result cards show the main ratios so you can verify your inputs before trusting the Kt value.

How outputs are produced

In stress-ratio mode, the calculator directly divides Peak or local maximum stress (MPa) by Nominal stress (MPa) to get Stress concentration factor (Kt).

In local-stress mode, it multiplies Stress concentration factor, Kt by Nominal stress (MPa) to get Local maximum elastic stress.

In geometry mode, it computes the controlling geometry ratios, applies the fitted equation for the selected case, then reports Kt. If Nominal stress (MPa) is also provided, it then calculates local maximum elastic stress from Kt.

Worked mini-example

If Peak or local maximum stress (MPa) is 150 and Nominal stress (MPa) is 100, then:

Kt = 150 / 100 = 1.5

If you already know Kt = 2.3 and Nominal stress (MPa) is 80, then:

sigma_max = 2.3 * 80 = 184 MPa

For a shoulder fillet with D = 60 mm, d = 40 mm, and r = 4 mm, the displayed check ratios include:

D / d = 60 / 40 = 1.5

r / d = 4 / 40 = 0.1

Those ratio outputs help confirm that the geometry was entered as intended.

Validation logic

The calculator blocks impossible geometry, divide-by-zero, and hidden-field misuse. For the unvalidated U-notch approximations, the existing ratio policy remains a numerical range check only. The visible note explicitly warns that this is not evidence of accuracy.

Assumptions behind the result

The Kt value is a theoretical elastic concentration estimated from the selected geometry relationship, not a plasticity correction, fatigue notch factor, or full 3D stress analysis. Real parts can differ because of material behavior, thickness effects, multiaxial loading, manufacturing details, and how nominal stress is defined for the reference section.