Check a solid circular shaft under combined bending, torsion and axial load. Returns von Mises equivalent stress with safety factor against yield, a fatigue margin, the first critical speed, and shear and bearing stress in the key.
Bending and torsion act at the same point on the shaft surface, so they must be combined rather than checked separately. The von Mises criterion reduces the biaxial state to one equivalent stress comparable with the uniaxial yield strength.
M bending moment (N·m) ·
T torque (N·m) ·
Fa axial force (N) ·
d diameter ·
L bearing span ·
Lk key length ·
Sy yield ·
Su tensile strength
A 50 mm C45 steel shaft on a 600 mm bearing span carrying 500 N·m bending and 800 N·m torque, with a 40 mm long key.
The shaft is comfortable, but notice the margins are uneven: static SF is 5.60 while the key bearing stress sits at 63 % of yield. Shorten the key to 20 mm and bearing stress doubles to 492 MPa — the key would fail while the shaft still shows SF 5.60. The key is the weak link long before the shaft is.
| Quantity | Symbol | Unit | Accepted range |
|---|---|---|---|
| Shaft diameter | d | mm or in | > 0 |
| Bearing span | L | mm | > 0 |
| Bending moment | M | N·m | ≥ 0 |
| Torque | T | N·m | ≥ 0 |
| Axial force | Fa | N | any |
| Yield strength | Sy | MPa | > 0 |
| Tensile strength | Su | MPa | > 0 |
| Required safety factor | SF | — | ≥ 1 |
| Key length | Lk | mm | > 0 |
Diameters in inches are converted at 25.4 mm. Key width is derived as d/4, rounded to the nearest millimetre with a 4 mm floor, and the key is assumed square.
| Material | Sy (MPa) | Su (MPa) | Se = 0.504·Su | Typical use |
|---|---|---|---|---|
| C45 steel | 390 | 620 | 312 | General machine shafts |
| 4140 alloy | 655 | 1 020 | 514 | High-duty, heat-treated |
| Stainless | 207 | 517 | 261 | Corrosive and hygienic service |
Bending produces a normal stress of 32M/πd³ and torsion produces a shear stress of 16T/πd³. Because both act at the same point on the surface, they must be combined rather than checked separately. The von Mises equivalent stress is √(σ² + 3τ²), and that combined figure is compared against yield.
Von Mises is the distortion energy criterion, which predicts yielding of ductile metals more accurately than maximum shear or maximum normal stress. A shaft carrying both bending and torsion has a biaxial stress state at its surface, and von Mises reduces that to a single equivalent number that can be compared directly against the uniaxial yield strength from a tensile test.
Critical speed is the rotational speed at which the shaft's natural bending frequency is excited once per revolution, causing resonance and large whirl amplitudes. It is estimated from the static deflection under self weight using the Rayleigh relation. Operating speed is normally kept below about 70 % of the first critical speed, or well above it if the shaft is designed to run supercritically.
Key width is conventionally about one quarter of the shaft diameter, with height equal to width for a square key. The key must be checked twice: for shear across its width, where the force is 2T/d spread over width × length, and for bearing on the half of its height that engages the hub. Bearing usually governs, and the fix is to lengthen the key rather than widen it.
For static loading a safety factor of 1.5 to 2 against yield is common in general machinery, rising to 3 or more where loads are uncertain or the consequences of failure are severe. Rotating shafts under bending experience fully reversed stress every revolution, so fatigue rather than yield usually governs, and the fatigue margin should be checked separately against the endurance limit.