Polar Moment of Inertia, J ( Ip )

The Polar Moment of Inertia is a geometric property of a cross section. Physically, it is a measure of how difficult it is to turn a cross-section about an axis perpendicular to it (the inherent rotational stiffness of the cross-section). This means:

Cross-section Polar Moment of Inertia, J Cross-section
Solid Circular Shaft
Jsolid =

Hollow (Thick-walled) Shaft
Jthick =
p(Ro4 Ri4)

Thin-Walled Shaft, t<<R
Jthin = 2pt(Rave)3,   Rave =
Ro + Ri


The units of J are generally expressed in: mm4 or in4.