Polar moment of inertia
Polar moment of inertia is a quantity used to predict an object's ability to resist torsion, in objects (or segments of objects) with an invariant circular cross section and no significant warping or out-of-plane deformation. It is used to calculate the angular displacement of an object subjected to a torque. It is analogous to the area moment of inertia, which characterizes an object's ability to resist bending and is required to calculate displacement.
The larger the polar moment of area, the less the beam will twist, when subjected to a given torque.
Polar moment of area should not be confused with moment of inertia, which characterizes an object's angular acceleration due to a torque. See moment (physics).
The polar moment of area cannot be used to analyze shafts with non-circular cross-sections. In such cases, the torsion constant can be substituted instead.
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.
However the polar moment of area can be used to calculate the moment of inertia of an object with arbitrary cross-section.
For a circular section with radius r:
The SI unit for polar moment of area, like the area moment of inertia, is metre to the fourth power (m4).
By the perpendicular axis theorem, the following equation relates Iz to the area moments of inertia about the other two mutually perpendicular axes:
For non circular cross section analytical formulas are needed which can be found here
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