# How to calculate the second moment of inertia

Written by grant d. mckenzie
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The second moment of inertia, or mass moment of inertia, is a mathematical representation of a beam's resistance to bending. The value relies on the cross-sectional area and the location of the centroid. In general, it conveys the way in which the cross-sectional area is dispersed around a reference axis. You can calculate the second moment of inertia for some shapes with a simple summation, but shapes that are more complex require integration using the formulas in the graphic.

Skill level:
Moderate

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## Instructions

1. 1

Derive the equation for the second moment of inertia for the cross-section you are analysing. In this example, the cross section is a vertical rectangle.

2. 2

Substitute actual lengths for the variables in the derived equation. h = 6 b = 4 I(xx) = (4*6^3)/12

3. 3

Evaluate the equation to obtain the second moment of inertia for the cross section. I(xx) = (4*216)/12 = 72

4. 4

Break the problem up to calculate the moments of inertia for each part if you are calculating the second moment of inertia for a complex section. Always use the centroidal axis, "cc" in this example, as your reference. Using the same values for "b" and "h" as before: Section: I(cc) = (64^3)/12 = 32 Section: I(cc) = (46^3)/12 = 72 Section: I(cc) = (6*4^3)/12 = 32 Notice the top and bottom segments, which are laying on their sides, are more prone to bending than the centre segment based on their second moments of inertia.

5. 5

Calculate the second moment of inertia for the entire section using the formula in the graphic. I(xx) = 632 + 72 + 632 = 1336

#### Tips and warnings

• The second moment of inertia has the units of length raised to the fourth power.
• A good engineering reference will have many of the most common cross-section formulas already derived, so you can skip the integration step if you have access to one.

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