Polar moment of inertia

The polar second moment of area (also referred to as "polar moment of inertia") is a measure of an object's ability to resist torsion as a function of its shape. It is one aspect of the second moment of area linked through the perpendicular axis theorem where the planar second moment of area uses a beam's cross-sectional shape to describe its resistance to deformation (bending) when subjected to a force applied in a plane parallel to its neutral axis, the polar second moment of area uses a beam's cross-sectional shape to describe its resistance to deformation (torsion) when a moment (torque) is applied in a plane perpendicular to the beam's neutral axis. While the planar second moment of area is most often denoted by the letter, $I$, the polar second moment of area is most often denoted by either, $I_z$, or the letter, $J$, in engineering textbooks.

The calculated values for the polar second moment of area are most often used describe a solid or hollow cylindrical shaft's resistance to torsion, as in a vehicle's axle or drive shaft. When applied to non-cylindrical beams or shafts, the calculations for the polar second moment of area becomes erroneous due to warping of the shaft/beam. In these instances, a torsional constant should be used, where a correctional constant is added to the value's calculation.

A schematic showing how the polar second moment of area ("Polar Moment of Inertia") is calculated for an arbitrary shape of area, R, about an axis o, where ρ is the radial distance to the element dA.

The polar second moment of area carries the units of length to the fourth power (

$L^4$

); meters to the fourth power (

$m^4$

) in the metric unit system, and inches to the fourth power (

$i{n}^4$

) in the imperial unit system. The mathematical formula for direct calculation is given as a multiple integral over a shape's area,

$R$

, at a distance

$\rho$

from an arbitrary axis

$O$

.

$J_O=\underset{R}{\iint }{\rho }^{2}dA$.

In the most simple form, the polar second moment of area is a summation of the two planar second moments of area, $I_x$ and $I_y$. Using the Pythagorean theorem, the distance from axis $O$, $\rho$, can be broken into its $x$ and $y$ components, and the change in area, $dA$, broken into its $x$ and $y$ components, $dx$ and $dy$.

Given the two formulas for the planar second moments of area:

$I_x=\underset{R}{\iint }{x}^{2}dxdy$, and $I_y=\underset{R}{\iint }{y}^{2}dxdy$

The relation to the polar second moment of area can be shown as:

$J_O=\underset{R}{\iint }{\rho }^{2}dA$

$J_O=\underset{R}{\iint }(x^2+y^2)dxdy$

$J_O=\underset{R}{\iint }{x}^{2}dxdy+\underset{R}{\iint }{y}^{2}dxdy$

$\therefore J=I_x+I_y$

In essence, as the magnitude of the polar second moment of area increases (i.e. large object cross-sectional shape), more torque will be required to cause a torsional deflection of the object. However, it must be noted that this does not have any bearing on the torsional rigidity provided to an object by its constituent materials; the polar second moment of area is simply rigidity provided to an object by its shape alone. Torsional rigidity provided by material characteristics is known as the shear modulus, $G$. Linking these two components of rigidity, one can calculate the angle of twist of a beam, $\theta$, using:

$\theta =\frac{Tl}{JG}$

Where $T$ is the applied moment (torque) and $l$ is the length of the beam. As shown, higher torques and beam lengths lead to higher angular deflections, where higher values for the polar second moment of area, $J$, and material shear modulus, $G$, reduces the potential for angular deflections.

Related pages

• Moment (physics)
• Second moment of area
• List of second moments of area for standard shapes
• Shear modulus

Other websites

• Torsion of Shafts - engineeringtoolbox.com
• Elastic Properties and Young Modulus for some Materials - engineeringtoolbox.com
• Material Properties Database - matweb.com