Shaft Calculation per DIN 743
Load capacity verification for steel shafts and axles right in your browser: notch effect, component fatigue strength and safety factors against fatigue fracture and permanent deformation – computed live with every intermediate value on display.
Strength verification at the notch cross section
σB = 1100 N/mm², σS = 900 N/mm² (at dB = 16 mm)
Geometry & conditions
Loading
Maximum value = |mean| + amplitude. Include any application factor KA in the mean and amplitude beforehand – this split strongly affects the result.
Result
Show intermediate values
- K1 (σB)
- 0.871
- K1 (σS)
- 0.797
- σB(d) [N/mm²]
- 958.5
- σS(d) [N/mm²]
- 717.4
- σmv [N/mm²]
- 110.3
- τmv [N/mm²]
- 63.7
- Cross section A [mm²]
- 1,257
- Section modulus Wb [mm³]
- 6,283
- Section modulus Wt [mm³]
- 12,566
| Quantity | Tension/compr. | Bending | Torsion |
|---|---|---|---|
| σm or τm in N/mm² | 0 | 0 | 63.7 |
| σa or τa in N/mm² | 0 | 79.6 | 15.9 |
| σmax or τmax in N/mm² | 0 | 79.6 | 79.6 |
| Concentration factor α | 1.968 | 1.801 | 1.405 |
| Stress gradient G′ in 1/mm | 0.874 | 0.874 | 0.383 |
| Support number n | 1.043 | 1.043 | 1.028 |
| Fatigue notch factor β | 1.887 | 1.727 | 1.366 |
| Size factor K2 | 1 | 0.888 | 0.888 |
| Roughness factor KF | 0.895 | 0.895 | 0.94 |
| Total influence K | 2.004 | 2.061 | 1.602 |
| Fatigue strength WK in N/mm² | 191.3 | 232.5 | 179.5 |
| Mean stress sensitivity ψ | 0.111 | 0.138 | 0.103 |
| Static support K2F | 1 | 1.2 | 1.2 |
| Increase factor γF | 1.05 | 1.05 | 1 |
| Yield limit FK in N/mm² | 753.3 | 903.9 | 497 |
| Tolerable amplitude ADK in N/mm² | 191.3 | 195.2 | 99.4 * |
* limited by the yield limit (kink in the Smith diagram)
Formulas and fundamentals
DIN 743 requires two separate verifications for every critical cross section of a shaft or axle: the safety against fatigue fracture S_D from the stress amplitudes and the fatigue strength of the notched component, and the safety against permanent deformation S_F from the maximum stresses and the component yield limit. The minimum safety factor of the standard is S_min = 1.2; it only covers the uncertainty of the method itself, so values of 1.5 to 2.0 are commonly agreed in practice depending on load knowledge and failure consequences.
The starting point are the nominal stresses in the notch cross section, always computed with the notch root diameter d: σ_zd = F/A, σ_b = M_b/W_b and τ_t = M_t/W_t, each split into mean value and amplitude. For a cross hole, net section moduli apply; for a keyway the full circular cross section is used because the weakening is fully contained in the fatigue notch factor.
For shoulders, circumferential grooves and cross holes the fatigue notch factor β follows from the stress concentration factor α and the support number n, which captures the material's micro-support effect via the relative stress gradient G′: β = α/n. For keyways the standard provides experimentally determined β values, converted from the 40 mm specimen diameter to the component diameter with the geometric size factor K3. Together with the size factor K2, the surface roughness factor K_F (input as Rz, not Ra!) and the surface treatment factor K_V, the total influence factor K is formed, giving the component fatigue strength σ_WK = K1·σ_W/K.
The technological size factor K1 converts the material properties from the 16 mm reference diameter to the effective diameter d_eff – with different factors for tensile strength and yield strength, a frequent source of errors. Via the equivalent mean stress σ_mv and the mean stress sensitivity ψ the tolerable amplitude σ_ADK follows, either for loading case 1 (constant mean stress) or case 2 (constant σ_m/σ_a ratio, usually more conservative). At high mean stresses the yield limit caps the amplitude – the calculator detects this kink in the Smith diagram automatically. The safety factors follow from S_D = 1/√((σ_zda/σ_zdADK + σ_ba/σ_bADK)² + (τ_ta/τ_tADK)²) and analogously S_F with the maximum stresses and component yield limits.
Worked example
Reference example (University of Bayreuth, ZN743): shaft shoulder with D = 50 mm, d = 40 mm, r = 3 mm made of quenched and tempered 36CrNiMo4 (σ_B = 1100 N/mm², σ_S = 900 N/mm² at 16 mm), Rz = 5 µm, d_eff = 50 mm. Loading as nominal stresses: tension/compression 200 ± 50, bending 300 ± 60, torsion 100 ± 40 N/mm², loading case 1.
The calculation yields stress concentration factors α = 1.97 / 1.80 / 1.40 (tension, bending, torsion), support numbers around 1.03 and thus fatigue notch factors β = 1.89 / 1.73 / 1.37. With K1(σ_B) = 0.87, K2 = 0.89 and K_F ≈ 0.90 the component fatigue strengths come to about 191 / 232 / 179 N/mm² and the tolerable amplitudes to 133 / 159 / 148 N/mm² at an equivalent mean stress of 529 N/mm².
Result: S_D = 1.25 and S_F = 1.28 – both above the minimum safety factor of 1.2, the verification is passed. The source uses the simplifying convention of applying K1(σ_B) to the yield strength as well and obtains S_F = 1.40; this calculator uses the separate K1 factors as intended by the standard and is therefore on the safe side.
Frequently asked questions
What minimum safety factor does DIN 743 require?
The standard states S_min = 1.2 for both verifications. This value only covers the uncertainty of the calculation method. In practice, higher values of 1.5 to 2.0 are agreed depending on how well the loads are known and the consequences of a failure – the minimum safety factor is therefore adjustable in the calculator.
Why are two verifications needed – fatigue and yielding?
Both verifications are mandatory and address different failure modes: the dynamic verification compares the stress amplitudes with the component fatigue strength (fatigue fracture after many load cycles), the static one compares the maximum stresses with the component yield limit (permanent deformation at peak load). At high mean loads the static verification often governs.
What do loading case 1 and case 2 mean?
The cases describe how the mean stress behaves when the load increases. Case 1: the mean stress stays constant, only the amplitude grows. Case 2: the ratio of mean stress to amplitude stays constant, both grow together. Case 2 is usually more critical and therefore the conservative choice when in doubt – the calculator defaults to case 2.
Why are the material properties converted with K1?
Tabulated material properties apply to the 16 mm reference diameter. Larger cross sections harden less thoroughly during quenching and tempering, so the strength drops. The technological size factor K1 captures this via the effective diameter d_eff – with different factors for tensile and yield strength. Forgetting K1 is the most common mistake in DIN 743 calculations.
Do I enter Rz or Ra?
The roughness factor K_F requires the mean roughness depth Rz in µm, not the arithmetic mean roughness Ra. Entering Ra would give far too favourable results. Typical values: polished 1, ground 4, fine-turned 10, turned 25, mill scale 200 µm. For keyways the calculator sets K_F = 1 because the reference roughness is already contained in the experimental fatigue notch factor.
Why does the keyway need no notch radius?
There is no closed-form stress concentration formula for keyways. DIN 743 instead uses experimentally determined fatigue notch factors that depend only on the tensile strength of the component and are converted from the 40 mm specimen diameter to the component diameter with the size factor K3. The nominal stresses are formed with the full circular cross section. With two keyways in the same cross section, the fatigue notch factor increases by a factor of 1.15.
Which components does the verification apply to?
DIN 743 applies to non-welded steel shafts and axles at −40 to +150 °C in a non-corrosive environment, under tension/compression, bending and torsion (assumed in phase). The endurance limit refers to 10⁷ load cycles. Transverse shear, buckling and residual stresses are not covered explicitly by the method.