What it does
Two linked checks for a rotating shaft and the bearings that carry it.
Bearing life. A rolling bearing eventually fails by fatigue of its races. Its catalogue gives a dynamic load rating C, the load at which 90 % of bearings last a million revolutions. The life at another load P follows L10 = (C/P)^p, with p = 3 for ball bearings and 10/3 for roller bearings. The tool works out:
- the equivalent load P from radial and axial loads, with the catalogue’s e, X and Y and a load factor for shocks,
- the L10 life in millions of revolutions and in hours at your speed,
- the adjusted life for a higher reliability (from a Weibull spread of lives) and a life factor aISO for lubrication and cleanliness,
- or, switched round, the C you need for a target life in hours.
Shaft diameter. A shaft under a steady load turns its bending stress into a fully reversed cycle every revolution, while the torque stays steady. The DE-Goodman criterion combines the alternating and mean von Mises stresses with your endurance limit Se and tensile strength Sut to give the diameter for a safety factor, or the safety factor of a diameter. It also checks first-cycle yield and gives the plain static diameters by the von Mises and maximum shear stress theories.
How to use it
- Bearing. Choose ball or roller, then type C, the loads, and e, X and Y from the maker’s catalogue (the page’s values are made up). Give the speed, the reliability you want and, if you know it, aISO.
- Shaft. Enter the alternating and mean bending moments and torques at the section you are checking (often a shoulder or keyway near a bearing or gear), the fatigue stress concentration factors Kf and Kfs, and your material’s corrected endurance limit Se, tensile strength Sut and yield strength Sy.
Worked example
All values are made up. A ball bearing with C = 30 kN carries Fr = 3.5 kN and Fa = 1.2 kN at 1200 rpm, with e = 0.3, X = 0.5, Y = 1.6 and a load factor of 1.2. Fa/Fr = 0.343 is above e, so P = 1.2 × (0.5 × 3.5 + 1.6 × 1.2) = 4.404 kN. The basic life is (30 / 4.404)³ = 316.1 million revolutions, 4390 hours. At 95 % reliability with a Weibull slope of 1.5, a1 = 0.6189 and the adjusted life is 2717 hours. For 20 000 hours at 95 % you would need C of at least 58.36 kN.
The shaft carries a fully reversed bending moment of 150 N·m and a steady torque of 120 N·m at a shoulder with Kf = 1.7 and Kfs = 1.5; Se = 210 MPa, Sut = 700 MPa and Sy = 560 MPa. For a safety factor of 1.5 the DE-Goodman diameter is 28 mm, where σa′ = 118.3 MPa and σm′ = 72.32 MPa. First-cycle yield is no problem (factor 4.04). A 30 mm shaft would have a fatigue safety factor of 1.84.
Tips
- Bearing life is very sensitive to load: 10 % more load cuts a ball bearing’s life by about a quarter.
- The weakest shaft section is usually not where the moment is largest but where a shoulder, groove or keyway raises the stress. Check each one.
- Se must already include the surface, size, loading, temperature and reliability factors for your part; using the plain test-bar value overestimates the life.
FAQ
Why are there no bearing tables? C, e, X and Y belong to each bearing and maker, so you type them from the catalogue. The page never guesses them.
Is DE-Goodman conservative? It is a little conservative for most steels, which is why it is a popular design criterion. Gerber or ASME elliptic criteria give somewhat smaller shafts.
Can I rely on these numbers for a machine? They are for study and planning, not a substitute for professional judgement. Shaft deflection, critical speeds, fits and the bearing maker’s own life method need checking too.
Where the numbers come from
- Basic rating life L10 = (C/P)^p with p = 3 for ball and 10/3 for roller bearings, the equivalent load P = X Fr + Y Fa, and the adjusted life a1 · aISO · L10, as defined in ISO 281 and machine design texts (equations only; no catalogue or factor tables are reproduced).
- Reliability factor from a two-parameter Weibull distribution of bearing life through the 90 % point, a standard result (e.g. Shigley's Mechanical Engineering Design, chapter on rolling-contact bearings).
- Shaft sizing by the distortion-energy Goodman (DE-Goodman) criterion and the static maximum shear stress and distortion energy criteria: Shigley's Mechanical Engineering Design (Budynas and Nisbett), chapter on shafts (equations only).
- All bearing data (C, e, X, Y), load factors, Weibull slope, strengths and fatigue factors on the page are made-up example values chosen by fibub and labelled as such.
- Checked against an independent NumPy/SciPy calculation: the Goodman diameter by root-finding on σ = Mc/I and τ = Tc/J, static diameters by root-finding, and the reliability factor from scipy.stats.weibull_min.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.