Shafts, Keys and Couplings
Shaft torque capacity, keys and keyways, splines, shaft stiffness and critical speed, and rigid and flexible couplings.
What shafts do
A shaft carries gears, pulleys, sprockets and rotors and transmits the torque between them. It is loaded in torsion by the torque, in bending by the forces of the gears and belts, and sometimes axially (helical gears, worm drives). The load rotates with respect to the shaft’s material, so the bending stress at any point alternates fully each turn: the shaft is a classic fatigue problem, and stress concentrations at keyways, shoulders and grooves decide whether it lives or dies (see Fatigue and Stress Concentration). The stress and strength formulas come from Torsion, Shear and Combined Loads; this page applies them to real shafts.
Torque capacity
τ = 16 T ÷ (π d³) d = (16 T ÷ (π τall))1/3 hollow: τ = 16 T D ÷ (π (D⁴ − d⁴))
The shear stress in a solid round shaft of diameter d carrying a torque T is τ = 16 T ÷ (π d³): the capacity rises with the cube of the diameter, so a small increase in size gives a large increase in strength. The ASME shaft code gives an allowable shear stress of 30 percent of the yield or 18 percent of the ultimate strength, whichever is smaller, reduced by 25 percent if there is a keyway.
Worked example: the smallest shaft for 9.55 N·m
- A medium-carbon steel (AISI 1045, Sy = 530 MPa, Su = 630 MPa) gives τall = min(0.30 × 530, 0.18 × 630) = min(159, 113) = 113 MPa; with a keyway 0.75 × 113 = 85 MPa.
- For T = 9.55 N·m (the 1.5 kW, 1500 rpm motor of Gear Strength and Design Calculations): d = (16 × 9 550 ÷ (π × 85))1/3 = 8.3 mm for pure torsion.
- A hollow shaft is more efficient: a tube of outer diameter 40 mm and bore 28 mm has a polar moment of 76 percent of the solid shaft but only 51 percent of its mass, so the strength to weight is much better; hollow shafts are used in vehicles and aircraft.
Bending and torsion together
A real shaft carries both. The maximum stress at the surface combines the bending stress σ = 32 M ÷ (π d³) and the torsional shear τ = 16 T ÷ (π d³) with the von Mises (distortion energy) criterion: σ′ = (32 ÷ (π d³)) √(M² + ¾ T²). Setting σ′ equal to Sy ÷ n (n the safety factor) gives the diameter for static loading: d = [32 n √(M² + ¾ T²) ÷ (π Sy)]1/3.
Worked example: the pinion shaft
- The 50 mm pinion of the gear example transmits Ft = 382 N and has Fr = Ft tan 20° = 139 N. The resultant on the shaft is √(382² + 139²) = 407 N.
- With the gear midway between two bearings 100 mm apart the bending moment is M = F L ÷ 4 = 407 × 100 ÷ 4 = 10 163 N·mm (10.16 N·m), and the torque is 9 550 N·mm.
- √(M² + ¾T²) = √(10 163² + 0.75 × 9 550²) = 13 103 N·mm. For n = 2 and Sy = 530 MPa: d = [32 × 2 × 13 103 ÷ (π × 530)]1/3 = 8.0 mm (static).
- The shaft rotates, so the stress alternates and the fatigue limit governs: after allowing for the keyway (stress concentration about 2), the surface finish, the size and the reliability, the endurance limit of the shaft is only a fraction of the static strength, so the diameter that results is larger than the static value. A 15 mm shaft is a standard size and gives a von Mises stress of 40 MPa (nominal, before stress concentration): see the fatigue procedure in Fatigue and Stress Concentration.
Stiffness, deflection and critical speed
A shaft must also be stiff. Bending deflection misaligns gear teeth (a deflection above about 0.025 mm at the gear, or slope above 0.001 rad at a ball bearing, begins to hurt), and torsional twist spoils timing and positioning. A simply supported shaft with a central load has a deflection δ = F L³ ÷ (48 E I) with I = π d⁴ ÷ 64 (see Beam Bending and Deflection).
Worked example: deflection and critical speed of the 15 mm shaft
- I = π × 15⁴ ÷ 64 = 2 485 mm⁴. The deflection under the gear force of 407 N is δ = 407 × 100³ ÷ (48 × 200 000 × 2 485) = 0.017 mm: within the 0.025 mm guide.
- Critical (whirling) speed: a rotating shaft with a mass has a natural frequency; if it turns at that speed the deflection grows until it fails. A gear of 0.4 kg at mid-span deflects the shaft statically by 0.00016 mm, and the first critical speed is nc ≈ 946 ÷ √δ = 946 ÷ √0.00016 = 73 761 rpm (δ in mm). The 1 500 rpm operating speed is far below it; keep it at least 20 to 30 percent away from any critical speed.
- Long, slender shafts (a conveyor roller, a lead screw) have low critical speeds: shorten the span, increase the diameter, or add a bearing.
Keys and keyways
A parallel key fits in a keyway in the shaft and in the hub and transmits the torque by its side faces. Its size is standard for each shaft diameter (ISO 773 / DIN 6885): width × height by diameter range:
| Shaft diameter (mm) | Key width × height (mm) |
|---|---|
| 12 to 17 | 5 × 5 |
| 17 to 22 | 6 × 6 |
| 22 to 30 | 8 × 7 |
| 30 to 38 | 10 × 8 |
| 38 to 44 | 12 × 8 |
| 44 to 50 | 14 × 9 |
| 50 to 58 | 16 × 10 |
Worked example: the length of a key
- A hub on a 30 mm shaft transmits 120 N·m with a standard 8 × 7 mm key. The force on the key is F = 2T ÷ d = 2 × 120 000 ÷ 30 = 8 000 N.
- Shear: τ = F ÷ (w L) ≤ 100 MPa gives L ≥ 8 000 ÷ (8 × 100) = 10.0 mm.
- Bearing on the side: σ = F ÷ ((h ÷ 2) L) ≤ 150 MPa gives L ≥ 8 000 ÷ (3.5 × 150) = 15.2 mm: this governs. A standard 20 mm key (or longer: the hub length of about 1 to 1.5 times the shaft diameter allows 30 to 45 mm) gives a large margin.
- Practice: use the key as the designed weak point only if you want protection; otherwise make it strong. For reversing or shock loads use a key one size up, a tighter fit, or a spline or taper fit instead.
- Keyway stress concentration: a keyway has a stress concentration factor of about 2 in bending (end-milled keyway) and reduces the shaft strength; sled-runner (profile) keyways have lower values. Give the keyway ends generous radii.
- Fit: a normal key fits tight on the sides and clear on the top; the shaft–hub fit is H7/k6 or similar (see Fits and Tolerances).
- Retain the hub axially with a circlip, a shoulder and a nut, or an end plate; the key does not hold the hub.
- Woodruff keys (half-moon) are easy to fit on small tapered shafts but cut deep into the shaft.
Splines, press fits, taper bushings and set screws
| Connection | Features | Use |
|---|---|---|
| Spline | Many keys cut into the shaft: high torque, hubs can slide; involute and straight-sided types | Gearboxes, vehicle drivelines, sliding gears |
| Press (interference) fit | Friction from an interference; no keyway, no stress concentration; see Press, Shrink and Expansion Fits | Permanent hubs, bearings; high speed |
| Taper-lock, locking assemblies | A tapered bushing or a locking ring clamps the hub with friction; no keyway, adjustable, removable | Pulleys, sprockets, couplings |
| Set screw | A screw onto the shaft: small torque only | Light drives, positioning; not for repeated high torque |
| Pin (taper pin, roll pin) | A cross pin in shear | Light, low-cost connections; a shear pin as a fuse |
| Clamp hub | A split hub tightened with a screw | Servo shafts, precision; no backlash |
Couplings
A coupling joins two shafts to transmit torque while allowing for some misalignment (angular, parallel and axial) and sometimes absorbing shock and vibration. A rigid coupling needs perfect alignment; a flexible coupling accepts a small error, which is always present.
| Coupling | Misalignment allowed | Backlash | Properties and use |
|---|---|---|---|
| Rigid (sleeve, clamp) | None | None | Very stiff; needs perfect alignment; line shafting |
| Jaw (spider) | Small to moderate | Small (elastomer) | Cheap, damps vibration; steppers, general drives |
| Bellows | Small angular and axial | None | Torsionally stiff; servo axes |
| Disc (diaphragm) | Angular, axial | None | High torque and stiffness, high speed |
| Oldham | Large parallel offset | Low | Light drives with offset shafts |
| Gear coupling | Moderate angular | Small | Heavy, high torque, needs lubrication |
| Grid, chain, tire | Moderate | Moderate | Heavy-duty, shock absorption |
| Universal joint (cardan) | Large angle | Some | Variable angles; speed fluctuation in single joints |
| Magnetic, hydraulic (fluid) | N/A | N/A | Sealed drives; torque limiting, soft start |
Worked example: choosing a coupling for a servo motor
The nominal torque is 9.55 N·m with a service factor of 1.5 for frequent reversing: the coupling must be rated for at least 1.5 × 9.55 = 14.3 N·m and, because a servo peak can be 3 times the nominal, for the peak torque as well. For a positioning axis choose a backlash-free, torsionally stiff bellows or disc coupling with a clamping hub (see Mechanical Transmission), check the bore sizes, the maximum speed and the allowed misalignment against the real alignment you can achieve.
A shaft design checklist
- Loads: torque, the force and direction of each gear, pulley or chain, the weight, and the axial thrust.
- Layout: bearings as close as practical to the loads, steps in diameter for the shoulders (one fixed bearing, one free to expand), the length of each hub.
- Reactions and moments: bending-moment and torque diagrams (see Beam Bending and Deflection) in two planes, and the resultant.
- Strength: static check by the von Mises equation, then fatigue at the keyways, shoulders and grooves (see Fatigue and Stress Concentration).
- Stiffness: deflection at the gears and the slope at the bearings; torsional twist if it matters.
- Critical speed: at least 20 to 30 percent away from the operating speed.
- Details: fillet radii at shoulders (at least 5 to 10 percent of the diameter), keyway ends, surface finish, tolerances (see Fits and Tolerances) and the bearing seat sizes (Bearings and Bearing Selection).
- Material and heat treatment: 1045 for general use, 4140 or 4340 hardened for high strength, stainless for corrosion; ground and polished surfaces where fatigue matters.
Troubleshooting
| Symptom | Likely cause | Check |
|---|---|---|
| Shaft breaks at a shoulder or a keyway | Fatigue from stress concentration, a sharp corner | Fillet radius, keyway ends, surface finish; redesign with a larger diameter or radius |
| Key sheared or rounded | Overload, a short key, a loose hub | Key length and fit; spline or taper-lock instead |
| Hub loose on the shaft, fretting | A loose fit, an overload, no axial retention | Fit tolerance; a locking assembly; retain axially |
| Vibration at a certain speed | Critical speed, unbalance, misalignment | Natural frequency; balance; alignment |
| Coupling wear or noise | Misalignment, overload, a worn spider or elements | Align with a dial indicator or laser; check the ratings |
| Bearings overheat | Shaft bending, misalignment, tight fit, preload | Deflection, alignment, fits (see Bearings and Bearing Selection) |
Related
- Gears and Power Transmission: the overview of this guide.
- Gear Strength and Design Calculations, Belts and Pulleys and Chains and Sprockets: the loads on the shaft.
- Bearings and Bearing Selection: supporting the shaft.
- Torsion, Shear and Combined Loads, Beam Bending and Deflection and Fatigue and Stress Concentration: the underlying stress and strength theory.
- Fits and Tolerances and GD&T: tolerancing shafts and bores.
FAQ
How do I choose the diameter of a shaft?
Compute the torque and the bending moment, find the minimum diameter with the von Mises equation for static loading, then check the fatigue life with stress concentrations and the stiffness (deflection and critical speed). Round up to a standard size.
Why do shafts usually fail at keyways and shoulders?
Those features concentrate the stress (by a factor of 2 or more), and the rotating shaft makes the stress alternate every turn, so cracks start there and grow by fatigue.
Do I need a flexible coupling?
Almost always: perfect alignment is impossible. Choose the stiffness from the application: torsionally stiff (bellows, disc) for positioning, soft (jaw, elastomer) for damping and shock.
Can I use a set screw to hold a gear?
Only for light loads: the point of the screw digs into the shaft and loosens with vibration. Use a key, a clamp hub or a taper bushing for real torque.
References
- Shigley, J. E. and Mischke, C. R., Mechanical Engineering Design, McGraw-Hill: shafts and shaft components (the ASME elliptic and DE-Goodman methods).
- ISO 773 and DIN 6885 (parallel keys), DIN 5480 and ISO 4156 (splines); ANSI/ASME B106.1M (design of transmission shafting).
- Manufacturer catalogs for couplings (Ruland, KTR, Rexnord) for ratings and misalignment limits.
Last reviewed: October 2026. Strengths and allowable stresses are typical; a final shaft design needs a full fatigue analysis.