The polar moment of inertia, also known as second polar moment of area, is a quantity used to describe resistance to torsional deformation, in cylindrical objects 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 planarsecond moment of area describes an object's resistance to deflection 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. Similar to planar second moment of area calculations, the polar second moment of area is often denoted as. While several engineering textbooks and academic publications also denote it as or, this designation should be given careful attention so that it does not become confused with the torsion constant,, 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,, of the object. where is the distance to the element. Substituting the and components, using the Pythagorean theorem: Given the planarsecond moments of area equations, where: It is shown that the polar moment of inertia can be described as the summation of the and planarmoments of inertia, and 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 :
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,, and the torsional constant,, no longer using to describe the polar moment of inertia. In objects with significant cross-sectional variation, 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 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,. Combining these two features with the length of the shaft,, one is able to calculate a shaft's angular deflection,, due to the applied torque, : As shown, the larger the material's shear modulus and polar moment of 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 is the torsional shear stress, is the applied torque, is the distance from the central axis, and 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:
Power carried by the shaft is 1000MW; this is typical for a large nuclear power plant.
Yield stress of the steel used to make the shaft is: 250×106N/m².
Electricity has a frequency of 50Hz; this is the typical frequency in Europe. In North America the frequency is 60Hz. This is assuming that there is a 1:1 correlation between rotational velocity of turbine and the frequency of mains power.
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 0.200m. If one adds a factor of safety of 5 and re-calculates the radius with the maximal stress equal to the yield stress/5 the result is a radius of 0.343m, or a diameter of 69cm, 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:
