OpenSpecEngineering

Finds the smallest solid shaft diameter that carries a bending moment and a torque together, by the ASME code for transmission shafting [2]. It then suggests a key for that diameter. Pick a steel to take the code’s allowable shear stress, or type your own. Machine designers use it for a first shaft size before a full fatigue and deflection check.

The sizing equation

d³ = (16/(π·τ_allow))·√((K_f·M)² + (K_fs·T)²)

d is the minimum shaft diameter, M the largest bending moment at the section and T the torque there. K_f and K_fs are factors of 1 or more that raise the bending and torsion for stress concentration, shock or fatigue; enter 1 for a steady load on a plain shaft. τ_allow is the allowable shear stress [2]:

• Commercial steel shafting: 8000 psi (55.2 MPa), or 6000 psi (41.4 MPa) with a keyway.
• A listed steel: the smaller of 0.30 S_y and 0.18 S_ut, less 25 % with a keyway. S_y and S_ut are the yield and tensile strengths [1].

The key is either a square key about d/4 on a side, a DIN 6885-1 parallel key [3] or an ASME B17.1 square key [4], taken from the band that holds d.

The key suggested is the one listed for a shaft of the minimum diameter. A larger stock shaft may take the next key size up.

Limits and common mistakes

• The code method is a simple static check. A shaft with reversing bending also needs a fatigue check, and deflection and critical speed are not checked here.
• Use the moment and torque at the same section. The largest moment and the largest torque are often at different places.
• The result is a minimum. Round up to a stock size, then check that the key still fits that size.
• Pick “With keyway” whenever the section has one; the keyway weakens the shaft.
• The page gives the key’s cross-section only. Check its length in shear and bearing separately.

Worked example

A cold-drawn AISI 1045 shaft carries a gear on a key. At the gear the bending moment is 300 N·m and the torque 500 N·m. Find the minimum diameter and a metric key.

1. Enter Bending Moment 300 N·m and Torque 500 N·m.
2. Choose Allowable Shear Stress Basis AISI 1045 cold-drawn (Sut 630, Sy 530 MPa) and Keyway in the Shaft With keyway (ASME: 25 % lower allowable).
3. Leave Fatigue Factor, Bending and Fatigue Factor, Torsion at 1, and choose Key Size Parallel key, DIN 6885-1 (metric).
4. Press Solve. The Solution shows: Minimum Shaft Diameter (d): 32.6847 mm
   Allowable Shear Stress (τ_allow): 85.05 MPa
   Allowable Stress Basis: ASME code: 0.18 Sut, less 25 % for the keyway
   Parallel Key for this Diameter (DIN 6885-1, b × h): 10 × 8 mm (shaft over 30 up to 38 mm)

Check by hand. 0.18 × 630 = 113.4 MPa is below 0.30 × 530 = 159 MPa, and 0.75 × 113.4 = 85.05 MPa. √(300² + 500²) = 583.1 N·m, so d³ = 16 × 583,100 / (π × 85.05) = 34,920 mm³ and d = 32.7 mm. A 35 mm shaft is still in the 30 to 38 mm band, so it takes the same 10 × 8 key.