Polar Moment of Inertia, J ( I_p )

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:

• The greater the Polar Moment of Inertia, the more torque is require to turn the shaft by a certain angle.
• The greater the Polar Moment of Inertia, the smaller the Shear Stress required to produce a given torque.
• For the same area, the further away the material of a cross-section is away from the axis about which it twists, the greater the Polar Moment of Inertia.

Cross-section	Polar Moment of Inertia, J	Cross-section
Solid Circular Shaft	Jsolid = pR4 2
Hollow (Thick-walled) Shaft	Jthick = p(Ro4 Ri4) 2
Thin-Walled Shaft, t<<R	Jthin = 2pt(Rave)3,   Rave = Ro + Ri 2

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