Shaft Diameter Calculator
Minimum shaft diameter for combined bending and torsion by the DE-Goodman fatigue criterion.
Find the minimum diameter of a rotating or stationary shaft carrying bending, torque and axial force. The calculator follows the distortion energy (DE) Goodman method from Shigley, with Marin surface, size and reliability factors and Neuber notch sensitivity for shoulders and keyways.
- Live results
- Full step-by-step working
- PASS / FAIL checks
- Free PDF report
Results
Step-by-step calculation (12 steps)
- 1Unmodified endurance limitFormulaSe' = 0.5·Sut (Sut ≤ 1400 MPa)SolutionSe' = 0.5 × 700ResultSe' = 350 MPa
- 2Surface factor (Marin)Formulaka = a·Sut^bSolutionka = 4.51 × 700^(-0.265) [Machined / cold drawn]Resultka = 0.795
- 3Size factor (iterated with d)Formulakb = 1.24·d^(−0.107) (2.79 ≤ d ≤ 51 mm); 1.51·d^(−0.157) (51 < d ≤ 254 mm)Solutiond = 36.28 mmResultkb = 0.844
- 4Reliability factorFormulake at 90% reliabilitySolutionkc = 1 (combined loading, von Mises), kd = 1 (room temperature)Resultke = 0.897
- 5Corrected endurance limitFormulaSe = ka·kb·kc·kd·ke·Se'SolutionSe = 0.795 × 0.844 × 1 × 1 × 0.897 × 350ResultSe = 210.7 MPa
- 6Fatigue stress concentration (Neuber)Formulaq = 1 / (1 + √a / √r), Kf = 1 + q·(Kt − 1)Solution√a = 0.307 √mm (bending), 0.234 √mm (torsion); q = 0.765, qs = 0.811ResultKf = 1.54, Kfs = 1.41
- 7Load componentsFormulaMa, Mm, Ta, Tm from shaft condition and torque typeSolutionRotating shaft: bending fully reversed. Torque steady.ResultMa = 250, Mm = 0, Ta = 0, Tm = 300 N·m
- 8Minimum diameter (DE-Goodman)Formulad = { (16n/π)·[ √(4(Kf·Ma)² + 3(Kfs·Ta)²) / Se + √(4(Kf·Mm)² + 3(Kfs·Tm)²) / Sut ] }^(1/3)Solutiond = { (16 × 2 / π) × [ 767698 / 210.7 + 730199 / 700 ] }^(1/3) [moments in N·mm]Resultdmin = 36.28 mm
- 9Standard diameter (ISO 15 bearing bore series)Formulad ≥ dmin, next standard sizeSolution36.28 mm → next sizeResultd = 40 mm
- 10Von Mises stresses at selected diameterFormulaσa′ = √(σa² + 3τa²), σm′ = √(σm² + 3τm²)Solutionσa = 61.09, τa = 0, σm = 0, τm = 33.55 MPaResultσa′ = 61.09 MPa, σm′ = 58.11 MPa
- 11Fatigue safety factor (Goodman)Formulanf = 1 / (σa′/Se + σm′/Sut)Solutionnf = 1 / (61.09 / 208.5 + 58.11 / 700) [kb = 0.836 at d = 40]Resultnf = 2.66 ≥ 2PASS
- 12First-cycle yield checkFormulany = Sy / √((σa + σm)² + 3(τa + τm)²)Solutionny = 490 / 84.31Resultny = 5.81 ≥ 2PASS
Engineering takeaways
- Shaft size is driven by the alternating bending stress at the notch. Reducing Kt (larger fillet radius, relief groove) is often cheaper than a bigger diameter.
- Check shaft deflection and slope at bearings and gears separately. Stiffness often governs over strength.
- Check every critical section (shoulders, keyways, grooves). The worst section is not always at maximum moment.
Method: Shigley, Mechanical Engineering Design, DE-Goodman criterion with Marin factors and Neuber notch sensitivity.
The PDF report contains every input, formula, substitution and check shown above, in LTM calculation-sheet format.
Worked Example
The calculator selects the next standard diameter from the ISO 15 bearing bore series, then verifies fatigue and first-cycle yield safety factors at that size. Materials include E250, EN8, C45, EN19 / 42CrMo4 and EN24.
The calculator above opens with this example. Change any input to run your own case.
Inputs
| Bending moment M (at critical section) | 250 N·m |
|---|---|
| Torque T | 300 N·m |
| Axial force Fa (0 if none) | 0 N |
| Shaft condition | Rotating (bending fully reversed) |
| Torque type | Steady |
| Material | C45 / CK45 Q&T (EN 10083-2) (Sut 700, Sy 490 MPa) |
| Surface finish | Machined / cold drawn |
| Reliability | 90% (ke = 0.897) |
| Stress concentration Kt (bending) | 1.7 |
| Stress concentration Kts (torsion) | 1.5 |
| Notch radius r | 1 mm |
| Design factor of safety n | 2 |
Results
| Minimum diameter | 36.28 mm |
|---|---|
| Selected standard diameter | 40 mm |
| Fatigue safety nf | 2.66 PASS |
| Yield safety ny | 5.81 PASS |
Shaft Diameter Formulas Used
Corrected endurance limit
- kasurface factor a·Sut^b
- kbsize factor 1.24·d^-0.107
- kereliability factor
Fatigue stress concentration
- Ktgeometric stress concentration
- rnotch radius
DE-Goodman diameter
- Ma, Mmalternating and mean bending moment
- Ta, Tmalternating and mean torque
- ndesign factor
Yield check
Basis: Shigley DE-Goodman criterion; Marin factors; Neuber notch sensitivity. Results are for preliminary design and should be confirmed against supplier data and the governing standard.
Shaft Diameter Calculator FAQ
Why is bending treated as fully reversed on a rotating shaft?
A point on the surface of a rotating shaft passes from the tension side to the compression side every revolution, so a steady bending load produces a fully reversed stress. Steady torque, on the other hand, produces a mean shear stress.
What stress concentration factor should I use?
A shoulder fillet with r/d ≈ 0.1 gives Kt ≈ 1.7 in bending and 1.5 in torsion. Sharp shoulders (r/d ≈ 0.02) reach 2.7. End-milled keyways are about 2.1 bending and 3.0 torsion. Retaining ring grooves can reach 5.
What design factor is typical for shafts?
Use 1.5 to 2 for well-known loads and materials, and 2 to 3 where loads are uncertain or failure is costly. The calculator reports both fatigue and yield safety at the chosen standard diameter.
Is strength the only check?
No. Shafts often fail to function before they break: check deflection and slope at bearings and gear meshes, critical speed and key or spline strength. Stiffness frequently governs over fatigue.
What standard sizes are suggested?
The calculator rounds up to the ISO 15 bearing bore series (17, 20, 25, 30 ... mm) because the bearing seat is usually the controlling diameter.
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