Polar Moment of Inertia Calculator

Polar Moment of Inertia for Circular Shafts

The polar moment of inertia, denoted J, describes how a circular cross-section resists torsion. It appears in shear stress and twist equations for drive shafts, drill rods, and other torque-loaded members.

Formulas

Solid circular shaft:

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

Hollow circular shaft:

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

$D$ is the outer diameter and $d$ is the inner diameter. The result has units of length to the fourth power, such as mm⁴ or in⁴.

Engineering Use

Shear stress in a circular shaft is $\tau = T\rho / J$, where $T$ is applied torque and $\rho$ is radial distance from the axis. Angle of twist is $\phi = TL / (JG)$, where $L$ is shaft length and $G$ is shear modulus. These relations apply to circular sections where the cross-section remains plane under torque.

Frequently Asked Questions

What is the polar moment of inertia of a 5 cm diameter solid shaft?

Using J = πD⁴/32, a 5 cm diameter solid shaft has J ≈ 61.36 cm⁴.

How is polar moment different from area moment of inertia?

Polar moment governs torsion resistance. Area moment of inertia governs bending deflection. For a circle, J equals the sum of the two principal area moments.

Can I use these formulas for rectangular bars?

No. The torsion equations that use J are derived for circular cross-sections. Noncircular shapes need a torsion constant instead.

What units should I use?

Use consistent length units throughout. Common choices are mm⁴ in metric drawings and in⁴ in US customary practice.

Does a hollow shaft always have lower J than a solid shaft?

For the same outer diameter, yes, because material is removed from the center. Hollow shafts can still be efficient when weight savings matter.

Report

Polar Moment of Inertia Calculator

Polar Moment of Inertia for Circular Shafts

Formulas

Engineering Use

Frequently Asked Questions

Related Tools

Recent Visits