View source for Second polar moment of area

{{Short description|Quantity of resistance to torsional deformation}}
{{distinguish|text=[[moment of inertia]], which characterizes an object's [[angular acceleration]] due to a [[torque]]}}

{{Multiple issues|
{{refimprove|date=August 2019}}
{{Cleanup|reason=excessive bold and italics|date=October 2023}}
}}
The '''second polar moment of area''', also known (incorrectly, colloquially) as "polar moment of inertia" or even "moment of inertia", is a quantity used to describe resistance to [[Torsion (mechanics)|torsional]] deformation ([[Deflection (engineering)|deflection]]), in objects (or segments of an object) with an invariant [[Cross section (geometry)|cross-section]] and no significant warping or out-of-plane deformation.<ref name=ugural>Ugural AC, Fenster SK. Advanced Strength and Applied Elasticity. 3rd Ed. Prentice-Hall Inc. Englewood Cliffs, NJ. 1995. {{ISBN|0-13-137589-X}}.</ref> 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 (geometry)|plane]] parallel to the central axis, the '''polar''' second moment of area describes an object's resistance to deflection when subjected to a [[moment (physics)|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 (<math>I_x</math>,<math>I_y</math>, and <math>I_{xy}</math>), the '''polar''' second moment of area is often denoted as <math>I_z</math>. While several engineering textbooks and academic publications also denote it as <math>J</math> or <math>J_z</math>, this designation should be given careful attention so that it does not become confused with the '''[[torsion constant]]''', <math>J_t</math>, used for non-cylindrical objects.

Simply put, the '''polar moment of area''' 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 second polar moment of area, the greater the torsional stiffness of the object.

==Definition==
[[File:Polar Moment of Inertia.svg|frame|right|A schematic showing how the '''second polar moment of area''' is calculated for an arbitrary shape about an axis <math>O</math>.  Where <math>\rho</math> is the radial distance to the element <math>dA</math>.]]

{{hatnote|'''Note:''' While it has become common to find the term '''moments of inertia''' used to describe the '''polar''' and '''planar''' second moments of area, this is primarily a construct of [[engineering]] fields. The term ''moment of inertia'', within physics and mathematics fields, is strictly the [[moment of inertia|second moment of mass]], used to describe a massive object's resistance to rotational motion, not its resistance to torsional deformation. While the '''polar''' and '''planar''' second moments of area are integrated over all the infinitesimal elements of a given ''area'' in some two-dimensional cross-section, the '''polar''' and '''planar''' second moments of '''mass''' (those regarding inertia) are integrated over all the infinitesimal elements of mass in a three-dimensional space occupied by an object. Simply put, the '''polar''' and '''planar''' second moments of area are an indication of rigidity, and the moment of inertia (second moment of '''mass''') is the rotational motion resistance of a massive object.}}

The equation describing the polar moment of area is a [[multiple integral]] over the cross-sectional area, <math>A</math>, of the object.

<math display="block"> J = \iint_A r^2 \, dA </math>
where <math>r</math> is the distance to the element <math>dA</math>.

Substituting the <math>x</math> and <math>y</math> components, using the [[Pythagorean theorem]]:
<math display="block"> J = \iint_A \left(x^2+y^2\right) dx \, dy </math>
<math display="block"> J = \iint_A x^2 \, dx\, dy + \iint_A y^2 \, dx \, dy </math>

Given the '''planar''' second moments of area equations, where:
<math display="block"> I_x = \iint_A y^2 dx \, dy </math>
<math display="block"> I_y = \iint_A x^2 dx \, dy </math>

It is shown that the polar moment of area can be described as the summation of the <math>x</math> and <math>y</math> '''planar''' moments of area, <math>I_x</math> and <math>I_y</math>
<math display="block">\therefore J = I_z = I_x + I_y</math>

This is also shown in the [[perpendicular axis theorem]].<ref>{{Cite web|url=http://www.efunda.com/math/areas/MomentOfInertia.cfm|title=Moment Of Inertia; Definition with examples|website=www.efunda.com}}</ref> For objects that have [[rotational symmetry]],<ref>{{cite book |first= Joaquin |last= Obregon |title= Mechanical Simmetry |isbn = 978-1-4772-3372-6 |year= 2012 |publisher= Author House |url = https://www.researchgate.net/publication/273061569 }}</ref> such as a cylinder or hollow tube, the equation can be simplified to:
<math display="block">J = 2I_x</math> or <math display="block">J = 2I_y</math>

For a circular section with radius <math>R</math>:
<math display="block">I_z = \int_0^{2\pi} \int_0^R r^2 (r \, dr \, d\phi) = \frac{\pi R^4}{2}</math>

==Unit==
The [[SI]] unit for '''polar second moment of area''', like the ''planar second moment of area'', is meters<!--was UK English originally -- see WP:MOSS --> to the fourth power (''m''<sup>4</sup>), and inches to the fourth power (''in''<sup>4</sup>) in [[United States customary units|U.S. Customary units]] and [[imperial units]].

==Limitations==
The polar second moment of area can be 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 second moment of area''', <math>I_z</math>, and the '''torsional constant''', <math>J_t</math>, no longer using <math>J</math> to describe the polar second moment of area.<ref>{{cite web| author=galtor| title=What is the difference between the Polar Second Moment of Area ("Polar Moment of Inertia"), IPIP and the torsional constant, JTJT of a cross section?| url=https://engineering.stackexchange.com/questions/8064/what-is-the-difference-between-the-polar-moment-of-inertia-i-p-and-the-tors}}</ref>

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 second moment of area is most often used to calculate the [[angular displacement]] of an object subjected to a moment ([[torque]]) applied parallel to the cross-section, 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]], <math>G</math>. Combining these two features with the length of the shaft, <math>L</math>, one is able to calculate a shaft's angular deflection, <math>\theta</math>, due to the applied torque, <math>T</math>:
<math display="block">\theta = \frac{TL}{JG}</math>

As shown, the larger the material's shear modulus and polar second moment of area (i.e. larger cross-sectional area), the greater resistance to torsional deflection.

The polar second moment of area appears in the formulae that describe torsional [[shear stress|stress]] and angular displacement.

Torsional stresses:
<math display="block">\tau = \frac{T \, r}{J_z}</math>
where <math>\tau</math> is the torsional shear stress, <math>T</math> is the applied torque, <math>r</math> is the distance from the central axis, and <math>J_z</math> is the polar second moment of area.

'''Note:''' In a circular shaft, the [[Stress (physics)|shear stress]] is maximal at the surface of the shaft.

==Sample calculation==
[[Image:Dampfturbine Laeufer01.jpg|thumb|250px|The rotor of a modern [[steam turbine]].]]

Calculation of the [[steam turbine]] shaft radius for a turboset:

Assumptions:
*Power carried by the shaft is 1000{{nbsp}}[[megawatts|MW]]; this is typical for a large [[nuclear power]] plant. 
*[[Yield stress]] of the steel used to make the shaft (''τ''<sub>yield</sub>) is: 250{{nbsp}}×{{nbsp}}10<sup>6</sup>{{nbsp}}N/m<sup>2</sup>.
*Electricity has a frequency of 50{{nbsp}}[[hertz|Hz]]; this is the typical frequency in Europe.  In North America the frequency is 60{{nbsp}}Hz. 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:
<math display="block">\omega = 2 \pi f</math>

The torque carried by the shaft is related to the [[power (physics)|power]] by the following equation:
<math display="block">P = T \omega</math>

The angular frequency is therefore 314.16{{nbsp}}[[radian|rad]]/[[second|s]] and the torque 3.1831{{nbsp}}×{{nbsp}}10<sup>6</sup>{{nbsp}}[[newton metre|N·m]].

The maximal torque is:
<math display="block">T_\max = \frac{ \tau_\max J_z }{r}</math>

After substitution of the ''polar second moment of area'' the following expression is obtained:
<math display="block">r = \sqrt[3]{\frac{2 T_\max}{\pi \tau_\max}}</math>

The [[radius]] is ''r''=0.200{{nbsp}}m = 200{{nbsp}}mm, or a [[diameter]] of 400{{nbsp}}mm.  If one adds a [[factor of safety]] of 5 and re-calculates the radius with the admissible stress equal to the ''τ''<sub>adm</sub>=''τ''<sub>yield</sub>/5 the result is a radius of 0.343{{nbsp}}m, or a diameter of 690{{nbsp}}mm, the approximate size of a turboset shaft in a nuclear power plant.

==Comparing polar second moments of area and moments of inertia (second moments of mass)==
=== Hollow Cylinder ===

Polar second moment of area:
<math display="block">I_z = \frac{\pi \left(D^4 - d^4\right)}{32} </math>

Moment of inertia:
<math display="block">I_c = I_z \rho l = \frac{\pi \rho l\left(D^4 - d^4\right)}{32}  </math>

=== Solid cylinder ===
Polar second moment of area
<math display="block">I_z = \frac{\pi D^4}{32} </math>

Moment of inertia
<math display="block">I_c = I_z \rho l = \frac{\pi \rho l D^4}{32} </math>
where:
*<math>d</math> is the [[inner diameter]] in meters (m)
*<math>D</math> is the [[List of gear nomenclature#outside diameter|outer diameter]] in meters (m)
*<math>I_c</math> is the moment of inertia in kg·m{{sup|2}}
*<math>I_z</math> is the polar second moment of area in meters to the fourth power (m<sup>4</sup>)
*<math>l</math> is the length of cylinder in meters (m)
*<math>\rho</math> is the specific mass in kg/m{{sup|3}}

==See also==
* [[Torsion constant]]
* [[Torsion spring]]
* [[Torsion bar suspension]]
* [[Second moment of area]]
* [[Shear modulus]]
* [[List of second moments of area]]

==References==
<references/>

==External links==
*[http://www.engineeringtoolbox.com/torsion-shafts-d_947.html Torsion of Shafts] - engineeringtoolbox.com
*[http://www.engineeringtoolbox.com/young-modulus-d_417.html Elastic Properties and Young Modulus for some Materials] - engineeringtoolbox.com
*[http://www.matweb.com Material Properties Database] - matweb.com

[[Category:Continuum mechanics]]
[[Category:Mechanical quantities]]
[[Category:Moment (physics)]]