Power Transmission Shaft Diameter

Calculate transmission shaft diameter (solid or hollow) satisfying both torsional rigidity and allowable shear stress conditions.

Inputs

P

Power

N

Rotational Speed

rpm

G

Shear Modulus

MPa

Steel ≈ 79,000–83,000 MPa

\tau_a

Allowable Shear Stress

MPa

Formula Interpretation

① Torsional Rigidity Method

d = 2173\,\sqrt[4]{\dfrac{P}{N \cdot G}} \quad \text{(mm)}

$P$ is power (kW); $N$ is rotational speed (rpm); $G$ is shear modulus (MPa); twist limit is 1/4° per metre.

② Solid Shaft — Shear Strength

d = 365\,\sqrt[3]{\dfrac{P}{\tau N}} \quad \text{(mm)}

$\tau_a$ is the allowable shear stress (MPa); $P$ and $N$ are as in Formula ①.

③ Hollow Shaft — Shear Strength (outer diameter)

d_2 = 365\,\sqrt[3]{\dfrac{P}{(1-k^4)\,\tau N}} \quad \text{(mm)}

$k$ is the inner-to-outer diameter ratio $k = d_1/d_2$ ; $\tau_a$ is the allowable shear stress (MPa).

Knowledge: Transmission Shaft Design

When designing a transmission shaft, calculate the diameter for both the torsional rigidity condition (Formula ①) and the shear strength condition (Formulas ②③), and take the larger value. Design should also account for stress concentrations from keyways, diameter steps and grooves. Gears, pulleys, sprockets and couplings should be located as close to bearings as possible.

Worked Example

A shaft transmits 30 kW at 300 rpm. Find the required shaft diameter. Allowable shear stress is $30\,\text{MPa}$ , shear modulus is $80{,}000\,\text{MPa}$ , and maximum twist is 1/4° per metre.

Step 1 — Torsional Rigidity Method (Formula ①)

d = 2173\,\sqrt[4]{\dfrac{30}{300 \times 80{,}000}} = 2173\,\sqrt[4]{1.25 \times 10^{-6}} \approx 71.4 \text{ (mm)}

Step 2 — Shear Strength Method (Formula ②)

d = 365\,\sqrt[3]{\dfrac{30}{30 \times 300}} = 365\,\sqrt[3]{3.33 \times 10^{-3}} \approx 54.5 \text{ (mm)}

The rigidity method gives the larger diameter ( $71.4\,\text{mm}$ > $54.5\,\text{mm}$ ), so the recommended diameter is $75\,\text{mm}$ (select nearest standard value).

Extended Knowledge

•When gears, pulleys, sprockets or couplings are mounted on a transmission shaft, the bearing span is a primary cause of bending deflection and vibration.
•Transmission shafts are primarily loaded by torque. Even when shear stress is below the allowable value, the torsional angle must be checked if the torque is large.
•A shaft under torsion must have sufficient rigidity. The standard criterion is to limit the twist angle to no more than 1/4° per metre of shaft length.