Design of shafts and keys

Why it matters

Every gearbox, pump, machine-tool spindle and conveyor has shafts that carry torque while gears, pulleys and bearings load them in bending. Keys lock the hubs to the shafts. Sizing both correctly, for strength and for stiffness, is one of the most common design tasks and a regular GATE numerical.

Key ideas

Shafts, axles and spindles. A shaft rotates and transmits torque (and usually carries bending loads). An axle supports rotating parts but transmits no torque (it may be stationary or rotate). A spindle is a short shaft such as a machine-tool spindle. Shafts are usually medium-carbon or alloy steel, chosen for strength, fatigue resistance, machinability and, where needed, hardenability or corrosion resistance.

Loads on a shaft.

• Torque from power and speed: T = 60P/(2πN).
• Bending from gear tooth forces, belt tensions and weights, found from the shaft's free-body diagram in two planes and combined: M = √(M_H² + M_V²).
• Axial loads from helical gears are usually small and often ignored in simple design.

Strength design (static, ductile). At the section of largest M and T the bending stress σ = 32M/(πd³) and torsional shear τ = 16T/(πd³) act together. Combine them with a failure theory:

• Maximum shear stress (Tresca/Guest): uses the equivalent torque T_e = √(M² + T²), with τ_allow = 16T_e/(πd³). Standard choice for ductile shafts.
• Maximum principal stress (Rankine): uses the equivalent bending moment M_e = ½[M + √(M² + T²)], with σ_allow = 32M_e/(πd³). For brittle materials.
• For shock and fatigue, the ASME code multiplies M and T by combined shock and fatigue factors K_m and K_t (take values from your data book): T_e = √((K_m·M)² + (K_t·T)²). A full fatigue design uses Goodman or Soderberg with the corrected endurance limit (see the fatigue topic).

Rigidity design. Machine-tool and gear shafts are often sized for stiffness: a limit on angle of twist (often about 0.25° to 1° per metre, from your data book) or on lateral deflection and slope at gears and bearings.

• Twist: θ = T·L/(G·J) (radians).

Hollow shafts. For the same strength a hollow shaft is lighter and stiffer per kilogram, because the material near the centre carries little stress. With k = d_i/d_o, J = π·d_o⁴(1 − k⁴)/32.

Keys. A key fits in a keyway in the shaft and hub and transmits torque by shear on its width and crushing on its sides.

• Sunk keys: rectangular (width ≈ d/4, thickness ≈ 2b/3), square (b = h ≈ d/4), parallel or taper (1 in 100). Gib-head taper keys can be driven out.
• Woodruff key: semicircular, sits in a deep recess, aligns itself, used on tapered shafts and light duty; the deep slot weakens the shaft.
• Saddle keys (friction only, light duty), tangent keys (heavy, reversing torque), round keys, and splines (many integral keys: larger torque, uniform load, and axial sliding of the hub, as in gearboxes).
• A keyway lowers shaft strength (stress concentration); Moore's empirical factor or a data-book K_f accounts for it.

Formulas

T = 60·P/(2π·N) — T in N·m, P in W, N in rev/min.

τ = 16T/(πd³), σ_b = 32M/(πd³) — solid shaft; Pa with N·m and m.

Hollow: τ = 16T/(π·d_o³·(1 − k⁴)), σ_b = 32M/(π·d_o³·(1 − k⁴)), k = d_i/d_o.

T_e = √(M² + T²) (Tresca), M_e = ½[M + √(M² + T²)] (Rankine).

θ = T·L/(G·J) — θ in rad, L length (m), G shear modulus (Pa), J = πd⁴/32 (m⁴).

Key (width b, height h, length l, shaft diameter d): shear T = l·b·τ·(d/2); crushing T = l·(h/2)·σ_c·(d/2).

Worked examples

Example 1 (standard). A solid steel shaft transmits 20 kW at 300 rev/min. Allowable shear stress is 50 MPa. Find the diameter, then the length of a square key 12 mm × 12 mm for τ_key = 60 MPa and σ_c = 120 MPa.

1. T = 60 × 20 000/(2π × 300) = 636.6 N·m.
2. d = (16T/(π·τ))^(1/3) = (16 × 636.6/(π × 50 × 10⁶))^(1/3) = 0.0402 m → use 45 mm (the next standard size, leaving margin for the keyway).
3. Key in shear: l = 2T/(d·b·τ) = 2 × 636.6/(0.045 × 0.012 × 60 × 10⁶) = 0.0393 m.
4. Key in crushing: l = 4T/(d·h·σ_c) = 4 × 636.6/(0.045 × 0.012 × 120 × 10⁶) = 0.0393 m.
5. The two are equal because for a square key with σ_c = 2τ the key is equally strong in shear and crushing.
6. d = 45 mm, key length ≥ 39.3 mm (in practice the key runs the full hub length, often about 1.5d).

Example 2 (GATE level). At its critical section a steel shaft carries M = 1.2 kN·m and T = 0.9 kN·m. Find the diameter by (a) the maximum shear stress theory with τ_allow = 60 MPa and (b) the maximum principal stress theory with σ_allow = 100 MPa. (c) What hollow shaft with k = 0.6 meets (a), and how much lighter is it?

1. T_e = √(1200² + 900²) = 1500 N·m; M_e = ½(1200 + 1500) = 1350 N·m.
2. (a) d = (16 × 1500/(π × 60 × 10⁶))^(1/3) = 0.0503 m = 50.3 mm.
3. (b) d = (32 × 1350/(π × 100 × 10⁶))^(1/3) = 0.0516 m = 51.6 mm.
4. (c) d_o = (16T_e/(π·τ·(1 − k⁴)))^(1/3) = (16 × 1500/(π × 60 × 10⁶ × 0.8704))^(1/3) = 52.7 mm; d_i = 0.6 × 52.7 = 31.6 mm.
5. Mass ratio = area ratio = 52.7² × (1 − 0.36)/50.3² = 0.70.
6. (a) 50.3 mm, (b) 51.6 mm (round up to a standard size); (c) d_o ≈ 52.7 mm, d_i ≈ 31.6 mm, about 30 % lighter.

Common mistakes

• Calling √(M² + T²) the equivalent bending moment; it is the equivalent torque. M_e = ½[M + √(M² + T²)].
• Forgetting to combine horizontal and vertical bending moments vectorially.
• Using 32 instead of 16 in the torsion formula (or vice versa for bending).
• Leaving the power in kW with torque in N·m.
• Designing for strength only when the shaft carries gears that need stiffness.
• Taking the key crushing area as l·h instead of l·h/2.

For GATE PI

Expect diameter of a shaft under torque alone or combined bending and torsion with a named theory, torque from power and speed, key length from shear and crushing, comparison of hollow and solid shafts, and angle of twist. Practise the free-body diagram of a shaft with a gear and a pulley, and the equivalent torque and moment formulas.

Quick check

1. A shaft transmits 50 kW at 1500 rev/min with τ = 40 MPa. What diameter is needed?
2. M = 500 N·m and T = 300 N·m. Find T_e and M_e.
3. Why is a spline used in a gearbox sliding gear?
4. What is the difference between an axle and a shaft?

Answers: 1. 34.3 mm (use 35 mm); 2. 583 N·m and 541.5 N·m; 3. it transmits torque while letting the gear slide axially; 4. an axle carries no torque.