Shaft Diameter Calculator — Torsional Stress / Twist

Diameter equations for stress and twist

For a concentric hollow circular shaft with fixed diameter ratio k = d/D:

The calculator selects:

Use k = 0 for a solid shaft. Both equations come from the same linear-elastic circular-shaft relationships used by the Shaft Torsional Stress / Angle of Twist Calculator.

Worked solid-shaft example

Use the default inputs: T = 500 N·m, τ_allow = 60 MPa, L = 1,000 mm, an illustrative entered G = 79.3 GPa, θ_allow = 1°, and k = 0.

1. Stress sizing gives D_stress = 34.881591 mm.
2. Twist sizing gives D_twist = 43.798061 mm.
3. The angle-of-twist limit governs, so D_required = 43.798061 mm.
4. A forward calculation at that continuous diameter gives τ = 30.309268 MPa and θ = 1°.
5. Stress utilization is 50.52%; twist utilization is 100% at the unrounded theoretical diameter.

The example does not recommend 60 MPa, 1°, or 79.3 GPa for a particular shaft. Replace all three with verified project values.

Select allowable values before using the result

This calculator does not derive an allowable stress from yield strength, ultimate strength, hardness, or a material name. The entered τ_allow must already reflect the governing design approach, load factors, safety factors, stress concentration treatment, fatigue basis, reliability requirement, environment, and company or code rules.

Likewise, θ_allow is a functional requirement. Couplings, gears, encoders, seals, controls, alignment, backlash, and torsional dynamics may impose a much tighter limit than material strength. Enter the total allowable twist over the same length L used by the calculator.

Hollow-shaft ratio behavior

For a hollow shaft, the calculator holds k = d/D constant. Both diameter equations include 1 − k⁴, but the stress diameter uses a cube root while the twist diameter uses a fourth root. The result is exact for the ideal concentric circular tube model; it is not a thin-wall approximation.

The calculation does not check wall-thickness tolerances, ovality, local instability, machining access, joining, surface defects, or whether the solved inner and outer diameters form a practical tube or bored shaft.

Round upward and verify the actual shaft

D_required is not a preferred stock diameter, finished size, minimum material condition, or drawing dimension. Select a larger candidate consistent with procurement and manufacturing constraints. For a hollow shaft, define an actual D and d rather than assuming the exact ratio will be preserved after rounding.

The related link transfers the continuous result to Calculator #020. Replace the transferred diameters with the actual candidate before relying on its nominal stress and twist outputs.

Engineering scope and limitations

This calculator covers a straight, uniform, solid or concentric hollow circular shaft in linear-elastic Saint-Venant torsion under constant torque magnitude. It excludes:

• derivation or approval of allowable stress, allowable twist, material properties, load factors, or safety factors;
• keyways, splines, cross-holes, shoulders, grooves, threads, fillets, welds, fits, notches, and other stress concentrations;
• fatigue, variable amplitude, mean stress, shock, reversal, impact, yielding, plastic torsion, fracture, wear, and surface condition;
• combined bending, axial force, transverse load, thermal load, gear and belt reactions, bearing span, or deflection;
• critical speed, lateral whirl, torsional vibration, resonance, damping, transient response, coupling stiffness, and system compliance;
• local tube stability, tolerances, minimum wall, corrosion allowance, machining, heat treatment, inspection, stock availability, cost, certification, and final engineering approval.

Use the governing shaft-design standard or company method, verified project data, and qualified engineering review before releasing a drawing or selecting a safety-critical shaft.