Torsion Shaft Diameter
Determine minimum shaft diameter under torsion for solid or hollow shafts from allowable shear stress.
Inputs
Formula
Solid d = ∛(16T/(π[τ])); Hollow dₒ = ∛(16T/(π(1−k⁴)[τ]))
Fundamentals
A shaft under torsion carries shear stress — central to strength and stiffness of drive shafts.
Max surface shear $\tau=Tr/J=T/W_t$, polar moment $J=\pi d^4/32$, torsional section modulus $W_t=\pi d^3/16$. Twist per length $\varphi=T/(GJ_p)$.
History
Torsional shear stress and polar moment theory for circular shafts was laid by Coulomb (1784) and Saint-Venant (1855).
Engineering applications
Used for drive shafts, drill strings and torque wrenches that carry pure torsion in circular sections.
Glossary
| Torque $T$ | Couple that rotates the shaft about its axis. |
| Shear $\tau$ | In-plane internal stress, max at the surface. |
| Polar moment $J$ | Geometric resistance to twist. |
How to use
- Fill in Torque T, Allow. shear [τ], Hollow ratio k (0=solid) in the Inputs section (watch the unit on each field).
- Click Calculate; the tool evaluates the formula shown above.
- Read Solid dia., Hollow outer (k>0) in the results area.
Formula notesTorsional shear τ = T·r / J; solid J = π d⁴/32, d = ∛(16T/(π·τ)); hollow J = π d⁴(1−k⁴)/32.
Formula · Worked Example · Knowledge
Formula
Solid shaft under torsion: $d=\sqrt[3]{16T/(\pi[\tau])}$; hollow $d_o=\sqrt[3]{16T/(\pi(1-k^4)[\tau])}$ (T in N·mm).
Worked Example
T=400 N·m, [τ]=45 MPa → d≈27.2 mm (solid).
Key Points
- Hollow shafts are lighter at equal strength — good for high speed.
- Use lower allowable stress for reversing torque.
- With bending, use combined method (see related tools).
Parameters
Inputs
| Parameter | Symbol | Unit | Default |
|---|---|---|---|
| Torque T (N·m) | T | N·m | 400 |
| Allow. shear [τ] (MPa) | tau | MPa | 45 |
| Hollow ratio k (0=solid) | k | 0 |
Outputs
| Result | Symbol | Unit |
|---|---|---|
| Solid dia. | d_solid | mm |
| Hollow outer (k>0) | d_hollow | mm |
Applications
- Common engineering use cases
FAQ
What formula does this tool use?
This tool computes per ISO / AGMA / ASME standard formulas: Solid d = ∛(16T/(π[τ])); Hollow dₒ = ∛(16T/(π(1−k⁴)[τ]))
How accurate are the results?
Results match input precision, based on SI units and common engineering approximations; for critical duty re-check with a safety factor.
Where is it used?
Common engineering use cases