Torsional Constant Calculator

Torsional Constant of Circular Shafts

The torsional constant $J$ (polar moment of inertia) measures a shaft's resistance to twisting. For a solid circular cross-section, $J$ depends only on the outer diameter. Hollow shafts subtract the inner bore contribution.

Formulas

Solid circular shaft:

$$J = \frac{\pi D^4}{32}$$

Hollow circular shaft:

$$J = \frac{\pi (D^4 - d^4)}{32}$$

Here $D$ is the outer diameter and $d$ is the inner diameter. Both must be in the same units. The result is in length to the fourth power (m⁴ or mm⁴).

Relation to Shear Stress

$$tau = \frac{T \cdot r}{J}$$

Maximum shear stress $\tau$ at the outer radius $r = D/2$ equals torque $T$ times radius divided by $J$. A larger $J$ reduces stress for the same applied torque.

Related tools: Torsional Spring Calculator and Torque Calculator.

Frequently Asked Questions

What units should I use for diameter?

Enter diameters in millimeters. The calculator converts to meters internally and reports $J$ in both m⁴ and mm⁴.

When is a hollow shaft preferred?

Hollow shafts provide similar torsional stiffness with less weight and material. Drive shafts and propeller shafts often use thin-walled tubes.

What happens if inner diameter equals outer diameter?

The wall thickness becomes zero and $J$ is undefined. The inner diameter must be strictly less than the outer diameter for a valid hollow shaft.

Is J the same as the polar moment of inertia?

For circular sections in torsion, yes. The symbol $J$ specifically denotes the polar second moment of area used in torsion formulas.

How does J affect angle of twist?

Angle of twist $\theta = TL / (GJ)$, where $G$ is shear modulus and $L$ is shaft length. A larger $J$ reduces twist for a given torque.

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Torsional Constant

Torsional Constant of Circular Shafts

Formulas

Relation to Shear Stress

Frequently Asked Questions

Related Tools

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