Polar moment of inertia

Note: The polar moment of inertia must not be confused with the moment of inertia, which characterizes an object's angular acceleration due to an applied torque.

The 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 area moment of inertia, or 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 moment of inertia 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 moment of inertia 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.

File:PolarMomentOfInertia.jpg
A schematic showing how the 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 moment of inertia 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 ($in^4$) 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 $O$.

$J_{O} = \iint\limits_R \rho^2 dA$.

In the most simple form, the polar moment of inertia 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:

$I_x = \iint\limits_R x^2 dxdy$, and $I_y = \iint\limits_R y^2 dxdy$

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

$J_{O} = \iint\limits_R \rho^2 dA$

$J_{O} = \iint\limits_R (x^2+y^2) dx dy$

$J_{O} = \iint\limits_R x^2 dxdy + \iint\limits_R y^2 dxdy$

$\therefore J = I_x + I_y$

In essence, as the magnitude of the polar moment of inertia 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 moment of inertia 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 moment of inertia, $J$, and material shear modulus, $G$, reduces the potential for angular deflections.