Second polar moment of area

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.  

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 (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** 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 = \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 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