Moment of Inertia Formula and Equations

Fundamentals of Moment of Inertia

The moment of inertia can be derived as getting the moment of inertia of the parts and applying the transfer formula: I = I₀ + Ad². We have a comprehensive article explaining the approach to solving the moment of inertia.

Fundamentally, the moment of inertia is the second moment of area, which can be expressed as the following:

[math] I_{x}=int int y^{2}dA [math] [math] I_{y}=int int x^{2}dA [math]

To observe the derivation of the formulas below, we try to find the moment of inertia of an object such as a rectangle about its major axis using just the formula above. To get the moment of inertia, the limits have to be determined such that they are taken from the axis of rotation to its extreme fiber. This would be the limits of the outer integral. The inner integral has a limit from 0 to b. That said, we can also express dA as xdy, which will become bdy. As the axis of rotation is at the neutral axis, the moment of inertia can be integrated with an upper limit of h/2 and a lower limit of 0 and multiplied twice due to the symmetry of the rectangle. This leaves us with the integral below.

[math] I_{x}=2int_{0}^{frac{h}{2}} y^{2}bdy [math] Integrating, [math] I_{x}=2b left [ frac{y^{3}}{3} right ]_{0}^{frac{h}{2}} [math] [math] I_{x}=2b left [ frac{h^{3}}{24}-0 right ] [math] [math] I_{x}=frac{bh^{3}}{12} [math]

Moment of Inertia Formula for Beam Sections

SkyCiv has compiled a summary of moment of inertia equations for beam sections (second moment of area). Moment of inertia equations is extremely useful for fast and accurate calculations. The formulas have been summarised into their simplest forms for your convenience.

SkyCiv also offers a Free Moment of Inertia Calculator for quick calculations or to check you have applied the formula correctly. The moment of inertia formula for rectangle, circle, hollow and triangle beam sections have been given. Some important things to remember regarding a beam’s area moment of inertia are:

• Area moment of inertia is different from the mass moment of inertia
• It is also known as the second moment of area
• It is a significant factor of deflection (the greater the I_x, the lower deflection will be)
• The units are in length to the power of 4
• The below equations give the moment of inertia with respect to the centroid of the section

REFERENCE	Ixx	Iyy
Moment of Inertia Equation for Rectangle or Rectangular Sections
	[math] dfrac{bh^3}{12} [math]	[math] dfrac{b^{3}h}{12} [math]
Moment of Inertia Equation for a Hollow Rectangular Section
	[math] dfrac{bh^3}{12} – dfrac{b_1h_1^3}{12} [math]	[math] dfrac{b^3h}{12} – dfrac{b_1^3h_1}{12} [math]
Moment of Inertia Equation for a Circle or Circular Section
	[math] dfrac{pi}{64}D^4 [math]	[math] dfrac{pi}{64}D^4 [math]
Moment of Inertia Equation for a Hollow Circular Section
	[math] dfrac{pi}{64}D^4 – dfrac{pi}{64}d^4 [math]	[math] dfrac{pi}{64}D^4 – dfrac{pi}{64}d^4 [math]
Moment of Inertia Equation for a Isosceles Triangle
	[math] dfrac{bh^3}{36} [math]	[math] dfrac{3b^3h}{144} [math]
Moment of Inertia Equation of an I-Section
	[math] frac{TFwtimes TFt^{3}}{12} +frac{Wttimes Wh^{3}}{12} + frac{BFwtimes BFt^{3}}{12} +[math] [math] TFwtimes TFttimes left ( BFt + Wh +frac{TFt}{2} -bar{y}_{bot} right )^{2} +[math] [math] Wttimes Whtimes left ( BFt +frac{Wh}{2} -bar{y}_{bot} right )^{2} +[math] [math] BFwtimes BFttimes left ( frac{Wh}{2} -bar{y}_{bot} right )^{2} [math]	[math] frac{TFttimes TFw^{3}}{12} + frac{Whtimes Wt^{3}}{12} + frac{BFttimes BFw^{3}}{12} [math]
Moment of Inertia Equation of a T-Section
	[math] frac{TFwtimes TFt ^{3}}{12} + frac{Wttimes Wh ^{3}}{12} +[math] [math] TFwtimes TFtleft ( Wh + frac{TFt}{2} -bar{y}_{bot} right )^{2} +[math] [math] Wttimes Wh times left ( frac{Wh}{2} – bar{y}_{bot} right )^{2} [math]	[math] frac{TFttimes TFw ^{3}}{12} + frac{Whtimes Wt ^{3}}{12} [math]
Moment of Inertia Equation of a Channel Section
	[math] frac{TFwtimes TFt^{3}}{12} + frac{BFwtimes BFt^{3}}{12} + frac{Wt times h^{3}}{12} +[math] [math] TFw times TFt times left ( h – frac{TFt}{2} – bar{y}_{bot} right )^{2} +[math] [math] BFw times BFt times left ( frac{BFt}{2} – bar{y}_{bot} right )^{2} +[math] [math] Wt times h times left ( frac{h}{2} – bar{y}_{bot} right )^{2} [math]	[math] frac{TFttimes TFw^{3}}{12} + frac{BFttimes BFw^{3}}{12} + frac{h times Wt^{3}}{12} +[math] [math] TFt times TFw times left ( Wt + frac{TFw}{2} – bar{x}_{left} right )^{2} +[math] [math] BFt times BFw times left ( Wt + frac{BFw}{2} – bar{x}_{left} right )^{2} +[math] [math] h times Wt times left ( frac{Wt}{2} – bar{x}_{left} right )^{2} [math]
Moment of Inertia Equation of Angles
	[math] frac{BFwtimes BFt^{3}}{12} + frac{ LFt times LFh^{3}}{12} +[math] [math] BFwtimes BFttimes left ( frac{BFt}{2}-bar{y}_{bot} right )^{2} +[math] [math] LFt times LFhtimes left ( BFt + frac{LFh}{2}-bar{y}_{bot} right )^{2} [math]	[math] frac{BFttimes BFw^{3}}{12} + frac{ LFh times LFt^{3}}{12} +[math] [math] BFttimes BFwtimes left ( frac{BFw}{2}-bar{x}_{left} right )^{2} +[math] [math] LFh times LFttimes left ( frac{LFt}{2}-bar{x}_{left} right )^{2} [math]

Use SkyCiv Section Builder for Hand Calculations

In this article, we have walked you through the moment of inertia formula. For more information on this topic, visit our tutorial on Moment of Inertia of Beam Section.

Did you know SkyCiv Section Builder also shows the full-hand calculations for the following shapes?

• Rectangular, Hollow Rectangular
• Circular, Hollow Circular
• I-Beam, T-Beam
• Angle (L-Beam), Channel
• Triangular sections

We hope you find the above table valuable for how to calculate a moment of inertia of a circle, triangle, and rectangle moment of inertia among other shapes. We also have a useful Moment of Inertia Calculator, a simplified of SkyCiv Section Builder, that handles these calculations for you or sign up today to get started with SkyCiv software!