Polar moment of inertia

The polar moment of inertia, also known as second polar moment of area, is a quantity used to describe resistance to torsional deformation (deflection), in cylindrical objects (or segments of cylindrical 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 (formula_1,formula_2, and formula_3), the polar second moment of area is often denoted as formula_4. While several engineering textbooks and academic publications also denote it as formula_5 or formula_6, this designation should be given careful attention so that it does not become confused with the torsion constant, formula_7, used for non-cylindrical objects.
Simply put, the polar moment of inertia 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 polar moment of inertia, the greater the torsional resistance of the object.
Definition.
The equation describing the polar moment of inertia is a multiple integral over the cross-sectional area, formula_8, of the object.
where formula_10 is the distance to the element formula_11. 
Substituting the formula_12 and formula_13 components, using the Pythagorean theorem:
Given the planar second moments of area equations, where:
It is shown that the polar moment of inertia can be described as the summation of the formula_12 and formula_13 planar moments of inertia, formula_1 and formula_2 
This is also shown in the perpendicular axis theorem. For objects that have rotational symmetry
, such as a cylinder or hollow tube, the equation can be simplified to:
For a circular section with radius formula_25:
Unit.
The SI unit for polar moment of inertia, like the "area moment of inertia", is meters to the fourth power ("m"4), and inches to the fourth power ("in"4) in U.S. Customary units and imperial units.
Limitations.
The polar moment of inertia is insufficient for use to analyze beams and shafts with non-circular cross-sections, due their tendency to warp when twisted, causing out-of-plane deformations. In such cases, a torsion constant should be substituted, where an appropriate deformation constant is included to compensate for the warping effect. Within this, there are articles that differentiate between the polar moment of inertia, formula_4, and the torsional constant, formula_7, no longer using formula_5 to describe the polar moment of inertia. 
In objects with significant cross-sectional variation (along the axis of the applied torque), which cannot be analyzed in segments, a more complex approach may have to be used. See 3-D elasticity.
Application.
Though the polar moment of inertia is most often used to calculate the angular displacement of an object subjected to a moment (torque) applied parallel to the cross-section, it must be noted that the provided value of rigidity does not have any bearing on the torsional resistance provided to an object as a function of its constituent materials. The rigidity provided by an object's material is a characteristic of its shear modulus, formula_30. Combining these two features with the length of the shaft, formula_31, one is able to calculate a shaft's angular deflection, formula_32, due to the applied torque, formula_33:
As shown, the larger the material's shear modulus and polar moment of area (i.e. larger cross-sectional area), the greater resistance to torsional deflection.
The polar moment of area appears in the formulae that describe torsional stress and angular displacement.
Torsional stresses:
Where formula_36 is the torsional shear stress, formula_33 is the applied torque, formula_10 is the distance from the central axis, and formula_6 is the polar moment of area.
Note: In a circular shaft, the shear stress is maximal at the surface of the shaft.
Sample calculation.
Calculation of the steam turbine shaft radius for a turboset:
Assumptions:
The angular frequency can be calculated with the following formula:
The torque carried by the shaft is related to the power by the following equation:
The angular frequency is therefore 314.16rad/s and the torque 3.1831×106N·m.
The maximal torque is:
After substitution of the "polar moment of inertia" the following expression is obtained:
The radius is "r=0.200"m = 200mm, or a diameter of 400mm. If one adds a factor of safety of 5 and re-calculates the radius with the admissible stress equal to the "τ"adm="τ"yield"/5" the result is a radius of 0.343m, or a diameter of 690mm, the approximate size of a turboset shaft in a nuclear power plant.
Comparing polar and mass moments of inertia.
Hollow Cylinder
Polar moment of inertia:
Mass moment of inertia:
Solid cylinder
Polar moment of inertia
Mass moment of inertia
where: