How to calculate the second moment of inertia

Written by grant d. mckenzie
How to calculate the second moment of inertia
Second moment of inertia (Author images)

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.

How to calculate the second moment of inertia
Derivation of rectangular cross-section area moment of inertia

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.

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

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

How to calculate the second moment of inertia
Rudimentary I-beam

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) = (6_4^3)/12 = 32 Section: I(cc) = (4_6^3)/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.

How to calculate the second moment of inertia
Calculating combined area moment of inertia

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

Tip

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.

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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