Torsion Shaft Diameter

Calculate shaft diameter under torsional moment for solid and hollow shafts.

Inputs

T

Torque

N·mm

\tau

Shear Stress

MPa

Formula Interpretation

Solid Shaft

d = \sqrt[3]{\dfrac{5T}{\tau}} \quad \text{(mm)}

Apply when the shaft is a solid cylinder. $T$ is the maximum torque (N·mm), $\tau$ is the allowable torsional shear stress (MPa).

Hollow Shaft

d_2 = \sqrt[3]{\dfrac{5T}{\tau(1-k^4)}} \quad \text{(mm)}

Apply when the shaft is hollow. $k$ is the ratio of inner to outer diameter ( $k$ = $d_1$ / $d_2$ ). A hollow shaft is lighter but has higher section modulus.

Key Concepts

Purpose of Shafts

Shafts transmit rotational power. They are primarily subjected to torsion, bending, and combined loads depending on the application.

Allowable Stress Condition

The maximum stress acting on the shaft must not exceed the allowable stress. This condition determines the minimum required diameter.

Hollow vs. Solid Shafts

Hollow shafts have a higher polar section modulus per unit weight than solid shafts, making them lighter while maintaining torsional and bending strength.

Worked Example

Find the solid shaft diameter under a torque of 2 × 10⁵ N·mm. Also, if the shaft is hollow with an outer diameter of 30 mm, find the inner diameter. The allowable torsional stress is 50 MPa.

Step 1 — Solid shaft diameter (Formula ①)

Substituting T = 2 × 10⁵ N·mm and τ = 50 MPa:

d = \sqrt[3]{\dfrac{5T}{\tau}} = \sqrt[3]{\dfrac{5 \times 2 \times 10^5}{50}} \approx 27.1 \text{ (mm)}

Therefore, select a shaft diameter of 28 mm.

Step 2 — Hollow shaft inner diameter (outer diameter d₂ = 30 mm)

First compute the diameter ratio k:

k = \sqrt[4]{1 - \dfrac{5T}{d_2^3 \tau}} = \sqrt[4]{1 - \dfrac{5 \times 2 \times 10^5}{30^3 \times 50}} \approx 0.713

Since $k$ = $d_1$ / $d_2$ :

d_1 = k \cdot d_2 = 0.713 \times 30 \approx 21.4 \text{ (mm)}

Therefore, select an inner diameter of 20 mm.

Solid shaft: d = 28 mm. Hollow shaft (d₂ = 30 mm): d₁ = 20 mm.

Extended Knowledge

•Shaft dimensions are standardized by JIS (Japanese Industrial Standards). Always select the next larger standard diameter from the calculated value.
•The maximum shear stress in a shaft under torsion occurs at the outer surface. For a hollow shaft, the material near the centre contributes little to torsional resistance, which is why removing it (making the shaft hollow) does not significantly reduce strength.
•When a shaft is subjected to both bending and torsion simultaneously, the equivalent stress must be computed using the combined stress formula before sizing the diameter.