Shafts

Shafts transmit torque and carry gears, pulleys and bearings. Learn loads on shafts, combined bending and torsion, equivalent stress theories, fatigue-based design (DE-Goodman), keyways and shoulders, deflection and critical speed.

A shaft looks simple, but it is loaded in torsion by the power it transmits, in bending by gear and belt forces, and it rotates — so every point sees fully reversed bending stress each revolution. Most shaft failures are fatigue failures at shoulders and keyways.

Design proceeds from layout (where bearings and components sit) to loads, to diameters from strength and fatigue, then checks of deflection, slope and critical speed.

An analogy

A shaft is the spine of a machine: it carries everything mounted on it and passes the 'motion signal' (torque) from motor to load.

Loads on a shaft

Torque from the driver; transverse forces from gear tangential and radial forces, belt tensions and component weights; axial forces from helical gears. Draw the shaft as a beam on bearing supports and find bending moments in two planes, then combine: M = √(M_xy² + M_xz²).

Static strength

For a first size, the maximum-shear-stress theory gives d from √(M² + T²). Ductile shafts are then checked with von Mises. Standard sizes and bearing bores (multiples of 5 mm) are chosen above the calculated minimum.

Stress raisers

Shoulders (for locating bearings), keyways, retaining-ring grooves and press fits create stress concentrations. Use generous fillet radii (r/d ≥ 0.05), relief grooves, sled-runner keyways and place raisers where bending is low.

Fatigue, deflection and critical speed

Rotating bending is fully reversed; steady torque is a mean stress. DE-Goodman or DE-ASME elliptic criteria combine them with Marin-corrected endurance limits. Slopes at bearings (≈ 0.001 rad for ball bearings) and gear mesh deflections (≈ 0.1 mm) limit stiffness; critical speed must be well away (typically ≥ 2× operating speed) from running speed to avoid whirl.

Formulas

• Max-shear-stress (static): d³ = (16 / π τ_allow) · √(M² + T²)
• Von Mises equivalent: σ′ = √(σ² + 3τ²), σ = 32M/πd³, τ = 16T/πd³
• DE-Goodman (rotating, steady T): 1/n = (16/πd³) [ 2K_fM_a/S_e + √3 K_fsT_m/S_ut ]
• Critical speed (Rayleigh, single mass): ω_c = √(g/δ_st)

Worked example: Static shaft diameter

Given: M = 300 N·m, T = 400 N·m; τ_allow = 50 MPa

Find: Minimum diameter.

• √(M² + T²) = 500 N·m
• d³ = 16 × 500 000/(π × 50) = 50 930
• d = 37.1 mm → choose 40 mm

Answer: ≈ 37 mm → 40 mm

Worked example: Critical speed

Given: Static deflection under the rotor mass = 0.1 mm

Find: First critical speed.

• ω = √(9.81/0.0001) = 313 rad/s
• n_c = 313 × 60/2π = 2990 rpm

Answer: ≈ 2990 rpm — keep running speed below ≈ 1500 rpm or stiffen

Common mistakes

• Wrong: Sizing only for static torque. — Right: Rotating shafts see reversed bending — design for fatigue with K_f at shoulders and keyways.
• Wrong: Forgetting to combine bending planes. — Right: M = √(M_y² + M_z²) from both planes.

Applications

• Gearbox shafts: Input, intermediate and output shafts carrying helical gears.
• Machine spindles: Stiffness-driven design for accuracy.
• Pump & fan shafts: Critical-speed and seal-deflection limits.

Key takeaways

• Draw the shaft as a beam; combine bending planes.
• Static: d³ = 16√(M² + T²)/(πτ).
• Design for fatigue at shoulders and keyways.
• Check deflection, slope and critical speed.

Going further

• DE-ASME elliptic: A less conservative fatigue criterion for shafts, widely used in industry and standards.
• Rotordynamics: Campbell diagrams, bearing stiffness/damping and gyroscopic effects govern high-speed turbomachinery shafts.
• Press fits: Interference fits transmit torque by friction and create fretting-fatigue stress raisers at hub edges.