Second polar moment of area

The second polar moment of area, also known as "polar moment of inertia" or even "moment of inertia", is a quantity used to describe resistance to torsional deformation (deflection), in objects (or segments of an object) with an invariant cross-section and no significant warping or out-of-plane deformation. It is a constituent of the second moment of area, linked through the perpendicular axis theorem. Where the planar second moment of area describes an object's resistance to deflection (bending) when subjected to a force applied to a plane parallel to the central axis, the polar second moment of area describes an object's resistance to deflection when subjected to a moment applied in a plane perpendicular to the object's central axis (i.e. parallel to the cross-section). Similar to planar second moment of area calculations (

I

x

{\displaystyle I_{x}}

,

I

y

{\displaystyle I_{y}}

, and

I

x

y

{\displaystyle I_{xy}}

), the polar second moment of area is often denoted as

I

z

{\displaystyle I_{z}}

. While several engineering textbooks and academic publications also denote it as

J

{\displaystyle J}

or

J

z

{\displaystyle J_{z}}

, this designation should be given careful attention so that it does not become confused with the torsion constant,

J

t

{\displaystyle J_{t}}

, used for non-cylindrical objects.

Simply put, the polar moment of area is a shaft or beam's resistance to being distorted by torsion, as a function of its shape. The rigidity comes from the object's cross-sectional area only, and does not depend on its material composition or shear modulus. The greater the magnitude of the second polar moment of area, the greater the torsional stiffness of the object.

Definition

The equation describing the polar moment of area is a multiple integral over the cross-sectional area,

A

{\displaystyle A}

, of the object.