What is the formula for minimum shaft diameter under torsion?

For a solid circular shaft: d = ∛(16T/(πτ_all)), where T is the applied torque and τ_all is the allowable shear stress. This comes from the torsion formula τ = Tc/J = 16T/(πd³) for a solid shaft.

How do I choose the allowable shear stress?

The allowable shear stress is typically 50-60% of the tensile yield strength: τ_all = 0.577 × σ_y (von Mises criterion). For steel AISI 1045: τ_all ≈ 180 MPa. Apply a safety factor of 1.5-3.0 depending on application criticality and loading type.

How does a hollow shaft compare to a solid shaft?

A hollow shaft with the same outer diameter has slightly less torsional strength but much less weight. The strength ratio is (1-k⁴) where k = d_i/d_o. For k = 0.5 (ID = half OD), you retain 93.75% of strength at about 75% of the weight.

Should I account for keyways and stress concentrations?

Yes. Keyways reduce shaft strength by 25-50% depending on type. Apply a stress concentration factor Kt (typically 1.6-3.0 for keyways). The effective stress becomes τ_eff = Kt × τ. Size the shaft for the concentrated stress or reduce the allowable stress accordingly.

What about combined bending and torsion?

When both bending moment M and torque T act, use the equivalent torque T_eq = √(M² + T²) (maximum shear stress theory) and size the shaft using T_eq in place of T. For the von Mises criterion, use the equivalent bending moment M_eq = √(M² + 0.75T²) with the bending stress formula σ = 32·M_eq/(πd³) checked against the allowable normal stress.

Free Engineering Tool #071

Shaft Diameter for Torsion Calculator

Calculate the minimum shaft diameter required to transmit a given torque. Enter torque, allowable shear stress, and material to get the required diameter with safety factor.

Shafts & RotorsSolid & HollowAuto-Calc

Torque T (N·m)

Allowable Shear Stress τ_all (MPa)

Safety Factor

Shaft Type

Quick presets

Results

Minimum Shaft Diameter

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Actual Shear Stress

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Safety Factor Applied

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Polar Moment J

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Angle of Twist (per m)

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Minimum Diameter (Solid Shaft)

Hollow Shaft

Angle of Twist

Material	σ_y (MPa)	τ_all (MPa)	G (GPa)
AISI 1020 (mild)	210	40	79
AISI 1045 (medium)	310	80	79
AISI 4140 (alloy)	415	110	80
AISI 4340 (high-str)	470	140	80

⚠️ Note: This calculator sizes for pure torsion only. For combined bending + torsion, use the equivalent torque method. Always account for keyways and stress concentrations.