DIN 743 Shaft Calculator for Fatigue and Static Strength

What is the DIN 743 strength verification for?

DIN 743 is the German standard for the load-capacity calculation of steel shafts and axles. For every critical section it answers two questions: does the shaft survive the recurring stress amplitudes from rotating bending, torsion and axial load without fatigue fracture (safety against fatigue, DIN 743-1 clause 4)? And does it stay free of permanent deformation under the maximum load that occurs (safety against exceeding the yield limit, clause 5)? Both checks work with nominal stresses at the notched section and convert the strength of the smooth test bar to the real component through notch effect, size, surface finish and mean stress. The standard requires at least Smin = 1.2 for both checks; higher values are to be agreed where the load assumption is uncertain or consequential damage is high.

Pre-sizing or complete verification?

A shaft pre-sizing derives the minimum diameter of a smooth solid shaft from bending moment, torque, yield strength and safety; the hollow-shaft pre-sizing does the same for an annulus. Both work statically with the equivalent stress and know neither notches nor fatigue. They are sufficient to fix an order of magnitude early on or to compare variants.

The complete DIN 743 verification becomes necessary as soon as the real geometry is known: every shoulder, keyway, relief groove and circumferential groove raises the local stress, and every rotating shaft under a transverse force sees alternating bending stress. Only the DIN 743 check rates these notches with βσ and βτ, accounts for component size, roughness and surface strengthening, and separates fatigue safety from static safety. Pre-sizing does not replace it – it supplies the starting value that is verified here with shoulders, grooves and transitions.

Which loads the calculator accounts for

DIN 743-1 covers axial load, bending and torsion individually and combined, each split into mean value and amplitude. The calculator therefore takes Fzdm and Fzda, Mbm and Mba, Mtm and Mta per section and forms the nominal stresses σzdm, σzda, σbm, σba, τtm and τta according to Table 1 of the standard; the section properties A, Wb and Wt apply to solid and hollow shafts. Dominant transverse shear is explicitly not covered.

The static check needs the maximum load. Without your own entry the calculator uses Mbmax = Mbm + Mba (likewise for torque and axial force); if a start-up shock or tilting moment is known, enter the maximum values directly. Mean stress and amplitude must not be added and treated as one load: the amplitude decides on fatigue fracture, the mean stress only shifts the bearable amplitude, and only the maximum stress decides on permanent deformation.

Formula and variables

S_D = 1 / √[(σzda/σzdADK)² + (σba/σbADK)² + (τta/τtADK)²] ≥ Smin

• σba = Mba/Wb · 10³, τta = Mta/Wt · 10³, σzda = Fzda/A (DIN 743-1, Table 1)
• σbWK = σbW(dB) · K1(deff) / Kσ (DIN 743-1, eq. 6)
• Kσ = (βσ/K2(d) + 1/KFσ − 1) / KV (DIN 743-1, eq. 8)
• σmv = √((σzdm + σbm)² + 3·τtm²), τmv = σmv/√3 (eq. 23, 24)
• σbADK = σbWK − ψbσK · σmv (case 1, eq. 11) or σbADK = σbWK / (1 + ψbσK·σmv/σba) (case 2, eq. 16)
• σbFK = K1(deff) · K2F · γF · σS(dB), τtFK = K1(deff) · K2F · σS(dB)/√3 (eq. 31, 32)
• S_F = 1 / √[((σzdmax/σzdFK + σbmax/σbFK)² + (τtmax/τtFK)²] (eq. 25)

Choose the inputs correctly

Per section: diameter d (plus dᵢ for a hollow shaft), notch case with large diameter D and radius r (shoulder, relief groove, circumferential groove) or keyway, roughness Rz, surface strengthening, and mean and amplitude of bending moment, torque and axial force – optionally your own maximum values. For the shaft: tensile strength σB and yield strength σS at the reference diameter dB, material group, load case 1 or 2 and the required minimum safety factors. Mean and alternating moments describe different damage components. Surface, size and notch factors change fatigue strength, not the external moment; they must not be treated as extra load.

How to use the calculator

Start with the shaft data: σB and σS at the reference diameter, material group, load case (case 1 if only the amplitude changes with load, case 2 if mean and alternating stress grow proportionally) and the required minimum safeties. Then create one entry per critical section with notch case, d, D, r, Rz, strengthening and the load components – typically the smallest shoulder, the keyway under the hub (capacity of the joint itself: parallel key calculator) and a relief groove at the bearing seat. The calculator reports nominal stresses, factors, component strengths, both safeties and the utilisation per section and flags the governing section.

Worked example

Shoulder d = 42 mm to D = 50 mm with r = 5 mm, Rz = 5 µm, quenched and tempered steel σB = 1,000 MPa, σS = 800 MPa (dB = 16 mm), Mbm = 3,636 N·m, Mba = 364 N·m, Mtm = 1,455 N·m, Mta = 436 N·m: the calculator returns K1 = 0.871/0.832, K2 = 0.885, βσ = 1.50, βτ = 1.25, Kσ = 1.80, Kτ = 1.47, σbADK = 156.6 MPa, τtADK = 142.8 MPa, fatigue safety 2.62 and static safety 1.40 – identical to the worked example in DIN 743 Supplement 1:2024-07, clause 5.2.

How to read fatigue safety and static safety

The calculator reports two safety factors per section. The fatigue safety S_D compares the existing stress amplitudes σzda, σba and τta with the bearable amplitudes σzdADK, σbADK and τtADK of the notched component at the existing mean stress; it must reach the required minimum. The static safety S_F compares the maximum nominal stresses with the component yield limits σFK and τtFK. Which of the two governs depends on the load case: in the supplement example with high mean stress and a small amplitude the static safety (1.40) determines the diameter although the fatigue safety is 2.62. Utilisation relates both safeties to the required minimum; the section with the highest utilisation is flagged as governing.

Diameters and radii in millimetres, moments in N·m, forces in N, roughness Rz in µm, strengths and stresses in MPa (= N/mm²), factors and safeties dimensionless. Moments are converted to N·mm internally so that σ = M/W comes out directly in MPa.

Nominal stress, notch effect, component strength and safety

The nominal stress is the stress obtained from load and section with the elementary formulas of strength of materials – without the stress peak at the notch root. DIN 743 deliberately keeps this nominal-stress method and moves the local influence entirely to the strength side: the notch factor βσ (or βτ) is, per DIN 743-2 clause 4.1, the ratio of the fatigue strength of the smooth specimen to that of the notched component, both expressed as nominal stress. For shoulders, relief grooves and circumferential grooves the calculator derives it from the shape factor ασ,τ (DIN 743-2 eq. 9) and the support factor n (eq. 5) via the relative stress gradient; for keyways the material-dependent tabulated values are used. A single notch case can be assessed beforehand with the notch factor calculator.

The fatigue strength of the smooth test bar becomes the component strength: σbWK = σbW(dB)·K1(deff)/Kσ is the fatigue strength of the notched component at zero mean stress (DIN 743-1 eq. 6). The mean-stress sensitivity ψσK (eq. 20–22) and the equivalent mean stress σmv (eq. 23) shift it to the bearable amplitude σADK – in load case 1 with constant mean stress per eq. 10–12, in case 2 with a constant ratio of mean to alternating stress per eq. 15–17. Once the mean stress exceeds the boundary stated in the standard, the amplitude is limited by the component yield limit (eq. 13/14 or 18/19); the calculator switches to that branch automatically.

The fatigue safety S_D follows from eq. 2 as the quadratic superposition of the three amplitude ratios. The check against permanent deformation uses, per clause 5, the component yield limits σFK and τtFK (eq. 31, 32) with static support effect K2F (Table 3) and increase factor γF (Table 2) and relates them to the maximum nominal stresses according to eq. 25. Both safeties are reported separately per section and compared with the required minimum.

Which influences enter Kσ, Kτ and σFK

Component size: K1(deff) reduces tensile and yield strength of large sections because the hardness attainable by quenching and tempering or case hardening falls with diameter (DIN 743-2 clause 6.2, eq. 10–14, separately for structural, quenched-and-tempered and case-hardening steel). K2(d) describes per eq. 16 how bending and torsional fatigue strength converge on the axial fatigue strength with growing diameter; for annular sections the outside diameter counts. Surface: KFσ follows from roughness Rz and the tensile strength at the component per eq. 18, KFτ from it per eq. 19; for keyways KF = 1 because the experimental notch factors already include the surface influence (Table 1). Surface strengthening: KV per DIN 743-2 clause 8 and Table 4 captures residual stress and hardness of a nitrided, case-hardened, roller-burnished or shot-peened layer; the calculator picks a conservative value per process and diameter, custom KV values remain possible.

Notch effect: βσ and βτ enter the component fatigue strength through Kσ = (βσ/K2 + 1/KFσ − 1)/KV and Kτ (DIN 743-1 eq. 8, 9). Hollow section: dᵢ changes A, Wb and Wt per Table 1 and reduces the static support effect K2F (bending 1.1 instead of 1.2, torsion 1.0 instead of 1.2). Material data: σB and σS at the reference diameter give the fatigue strengths σbW ≈ 0.5·σB, σzdW ≈ 0.4·σB and τtW ≈ 0.3·σB per DIN 743-3 eq. 1–3; the component yield limit σFK = K1(deff)·K2F·γF·σS(dB) uses K1 for the yield strength and γF from Table 2 (1.00 for ασ ≤ 1.5 up to 1.15 for ασ > 3, always 1 for torsion).

What DIN 743-1, -2 and -3 each contribute

DIN 743-1:2012-12 (fundamentals) defines the procedure: nominal stresses (Table 1), safety equations against fatigue (eq. 2) and permanent deformation (eq. 25), component fatigue strength and overall influence factors (eq. 5–9), mean-stress influence with case 1 and case 2 (eq. 10–24) and the component yield limits with K2F and γF (eq. 31, 32, Tables 2 and 3). Exactly this procedure is implemented; the crack-initiation check for hard surface layers per 5.1.2 and the compressive regime with negative mean stress are not included.

DIN 743-2:2012-12 (shape factors and notch factors) supplies the factors: keyway notch factors (Table 1), shape factors for shoulder, circumferential groove and shoulder with relief groove (eq. 9 with Table 2), support factor from the relative stress gradient (eq. 5), size factors K1 and K2 (eq. 10–16), roughness factors KF (eq. 18, 19) and strengthening factor KV (Table 4). Cross holes, splines and V-notches can be covered through the custom notch case with your own βσ/βτ.

DIN 743-3:2024-07 (material strength values) relates tensile strength to the fatigue strengths (clause 5.2, eq. 1–3) and tabulates the common shaft steels at the reference diameter. The calculator uses these ratios; fatigue strengths measured on the component can be entered directly. DIN 743 Supplement 1:2024-07 contains fully worked examples; the shoulder example in clause 5.2 is stored as an automated reference test and is reproduced by the calculator to rounding accuracy.

Typical applications

Complete strength verification of gearbox, drive and machine shafts with shoulders, relief grooves, circumferential grooves and keyways; re-checking a pre-sized shaft with its real geometry; comparing several critical sections in one run; assessing the effect of a material change, surface strengthening or a larger notch radius on fatigue and static safety.

Assumptions, limits and common mistakes

DIN 743-1:2012-12 (clause 4: eq. 2, 5–9, 10–19, 20–24; clause 5: eq. 25, 31, 32, Tables 1–3), DIN 743-2:2012-12 (eq. 5, 9–14, 16, 18, 19; Table 1 keyway), DIN 743-3:2024-07 (clause 5.2, eq. 1–3) and DIN 743 Supplement 1:2024-07 (worked example 5.2 as reference test). Limits: steels at 20 °C, constant or damage-equivalent amplitudes, nominal-stress method for solid and hollow circular sections; no transverse shear, no load spectra, no crack-initiation check for hard surface layers, no compressive regime with σzdm + σbm < 0; keyway notch factors are the tabulated values for dBK = 40 mm without K3 conversion.

Common mistake: Do not set notch, size and surface factors to 1 by default – a real shoulder or keyway can halve the bearable amplitude compared with the smooth shaft and is the very reason for the verification. For rotating shafts enter the bending moment from a stationary transverse force as amplitude, not as mean. Do not confuse deff with d: K1 uses the diameter governing the heat treatment, usually the larger one. And do not overlook the static safety – at high mean stress it often governs even though the fatigue safety looks comfortable. Stiffness and vibration are not part of this verification: check angle of twist and critical speed separately.