Skip to main content Skip to page footer
TechMechAcademy logo TechMechAcademy logo

Mechanics of Materials

Topic: Area Moment of Inertia

Here, you can learn what an area moment of inertia (also known as second moment of area, or second-order area moment) is, where it finds application in engineering mechanics, and how to calculate it.

Let's explore these essential concepts together.

On this Page
Practice Exercises
Table of Contents

2. Calculation Methods for Arbitrary Shapes

In the following, we will explain the methods for calculating the moments of inertia \(I_y\), \(I_z\), and \(I_{yz}\) for areas of any shape using Cartesian coordinates and polar coordinates. There are a total of six different approaches available, of which the following four will be presented in more detail, each illustrated with an example. The choice of which approach to use depends on how we can construct the infinitesimal area element \(dA\), as it is contained in equations (6.2) - (6.4), and whether Cartesian coordinates or polar coordinates are employed.

2.1 Use of Cartesian Coordinates

2.1.1 Infinitesimal Area Element

We are working with a tiny area element \(dA\) with side lengths \(dy\) and \(dz\), as shown in Figure 6.2.1.

A rectangular surface element dA with side lengths dy and dz is exemplarily depicted in an arbitrary cross-sectional area. Its position is dimensioned based on an arbitrarily arranged yellow coordinate system with variables x, y, and z, where y and z are used for measurement.
Fig. 6.2.1: Calculation Method Infinitesimal Area Element in Cartesian Coordinates

Since this area element is infinitesimal in both coordinate directions, we need a double integral to perform the calculation:

$$ \begin{align} I_y&= \int\limits_{(A)} z^2 \ \mathrm{d}A = \int\limits_{(y)} \int\limits_{(z(y))} z^2 \ \mathrm{d}y \mathrm{d}z\\[7pt] I_z&= \int\limits_{(A)} y^2 \ \mathrm{d}A = \int\limits_{(y)} \int\limits_{(z(y))} y^2 \ \mathrm{d}y \mathrm{d}z\\[7pt] I_{yz} &= -\int\limits_{(A)} y \cdot z \ \mathrm{d}A = -\int\limits_{(y)} \int\limits_{(z(y))} y \cdot z \ \mathrm{d}y \mathrm{d}z \end{align} $$

(6.5)

To apply this formula, it is necessary to determine the functional relationship \(z(y)\). If this is not possible, an alternative option is to use the integration limits \(z\) and \(y(z)\).

Example 6.1: Determining Area Moment of Inertia using the Calculation Method for Arbitrary Shapes Infinitesimal Area Element in Cartesian Coordinates

For the quarter-circle area depicted, the following area moments of inertia with respect to the illustrated \(y\), \(z\)-coordinate system are to be determined using the calculation method infinitesimal area element in Cartesian coordinates:

A rectangular area element dA with side lengths dy and dz is exemplified within a quarter-circle region. The coordinate system is depicted in yellow, with y being positive to the left and z being positive downward. The position of the area element is dimensioned from the origin in terms of y and z.
Fig. B6.1.1: Quarter-Circle Area
  1. Axial Area Moment of Inertia \(I_y\)
  2. Axial Area Moment of Inertia \(I_z\)
  3. Biaxial Area Moment of Inertia \(I_{yz}\)
Solution

To apply the required calculation method, it is necessary to determine the functional relationship \(z(y)\).

This can be found by examining the relationship between any value \(y\), its corresponding value \(z(y)\), and the radius of the (quarter) circle:

Figure
Fig. B6.1.2: Circle Equation

We recognize a right-angled triangle, and by using the Pythagorean theorem, we obtain the equation of the circle.

Continue with TechMechAcademy+

Everything. Always. Everywhere.

With TechMechAcademy+ full access to all content.

For Beginners: TechMechAcademy+. Full access to all content for one day.

Overview of the benefits:

  • Most cost-effective offer.
  • Ideal if you need access to already identified content for a short period.
  • Unlimited access to all existing and newly created content throughout the entire premium membership.
  • Guaranteed premium membership for 24 hours. Access automatically ends at 0:00 CET (Central European Time) on the following day. No cancellation necessary.

€3.99

TechMechAcademy+. Full access to all content for one week.

Overview of the benefits:

  • Cost-effective offer.
  • Ideal for short-term exam preparation with the content of TechMechAcademy.
  • Unrestricted entry to every piece of content, both existing and newly generated, across the entirety of the premium membership.
  • Guaranteed premium membership for one week. Access starts on the day of activation and ends automatically at 0:00 CET (Central European Time) on the following week's day. No cancellation necessary.

€9.99

Bestseller: TechMechAcademy+. Full access to all content for one month.

Overview of the benefits:

  • Benefit from the bestseller.
  • Ideal for effective exam preparation with the content of TechMechAcademy.
  • Complete access to all content, both existing and newly produced, under the umbrella of the premium membership.
  • Guaranteed premium membership for one month. Access starts on the day of activation and ends automatically at 0:00 CET (Central European Time) on the following month's day. No cancellation necessary.

€14.99

Become a TechMechAcademy+ Premium Member!

Are you already a TechMechAcademy+ premium member? Then please log in here to enjoy full access to all content.