Second Moment of Area

In this entry you will find formulas for Second Moment of Area (area moment of inertia) for basic plane figures and how to use these formulas when calculating the Second Moment of Area for figures composed of several simple figures.

1. Second Moment of Area (area moment of inertia)

2. Product of inertia for shapes

3. List of second moments of area

4. Example calculation of second moments of area

Second Moment of Area (area moment of inertia)

Second Moment of Area of a figure is the sum of the products of the elementary fields dA and the squares of their distances from this axis.

Fig.1 Second Moment of Area - formulas

Product of inertia for shapes

Product of inertia of the figure (product moment of area) relative to the axis is the sum of the products of the elementary fields dA and their distances from the axis.

Fig.2 Product of inertia - formula

If a figure has at least one axis of symmetry, the product of inertia of such a figure is zero.

We can use the above formulas to determine the Second Moment of Area of any figures by definition using integrals. In this entry, however, I would like to discuss how to calculate the Second Moment of Area of figures using formulas for simple figures without using definitions and integrals. You will encounter this type of tasks very often.

To use this method we will use Steiner's Theorem. You can find more information about this method in this post .

List of second moments of area

In the figure below you will find formulas for the  second moments of area and product of inertia for some shapes. The formulas included in this table are sufficient to solve problems involving complex figures.

Note that for a triangle and a quarter circle, the sign of the product of inertia depends on the orientation of the figure relative to the coordinate system

Fig.3 List of second moments of area

Example calculation of second moments of area

The figure below shows a figure composed of a square, a triangle and a cut-out circle. For this figure we will calculate the second moment of area and product of inertia.

The diagram of the figure, all calculations and the graphs of the figure with centroidal axis are generated in my moments of inertia calculator. You can create any figure consisting of simple figures and determine its center of gravity and second moment of area.

Fig.4 A figure composed of several simple figures

Below are instructions on how to perform this type of tasks:

1. Division of a figure into simple figures (rectangles, triangles, circles...)

2. Calculation of surface areas and centers of gravity for these simple figures.

3. Calculation of the center of gravity of the entire figure .

4. Calculation of the second moment of inertia and product of inertia for all simple figures (rectangles, triangles, circles...) from Fig.3

5. Calculation of the central moments of inertia and the product of inertia for the entire figure using Steiner's Theorem .

According to the above instructions, for our example, we divide the figure into simple figures:

• A1 - square, A2 - triangle, A3 - circle.

We calculate the areas of the figures and their centers of gravity:

• x1,x2,x3 and y1,y2,y3

Then we determine the center of gravity of the entire figure, which is described in this entry .

Fig.5 Determining the center of gravity of a figure

After calculating the center of gravity of the figure, we proceed to calculate the central moment of inertia. We use Fig. 3 and Steiner's Theorem .

Fig.6 Determination of second moment of area to centroidals

Note that solid figures (square and triangle in our example) are summed in the second moment of area calculation and cut-out shapes (circle in our example) are subtracted.

Finally, we create a drawing of our figure with the centroidals marked.

Fig.7 Drawing of the figure with the central axes marked

This concludes the entry Second Moment of Area. Thank you 😊