What is the moment of inertia?

Moment of inertia is one of the basic geometric quantities describing the distribution of mass or surfaces around an axis of rotation. In structural mechanics, the moment of inertia of a surface (a.k.a. the second moment of area) describes the resistance of a cross-section to bending. The greater the moment of inertia, the greater the stiffness of a structural member.

Moment of inertia of a circle - formula

For a circle (solid disk) of radius , the moment of inertia of the surface with respect to the axis passing through the center and perpendicular to the surface of the circle is:

$I_{x},I_{y} = \frac{4} R^4$

If you know the diameter , you can use the transformed formula:

$I_{x},I_{y} = \frac{64} D^4$

Calculation example for a circle

Below is an example of calculating the moment of inertia of a circle in our online calculator. Just select the shape "circle", enter the radius, and the calculator will automatically calculate the value of the moment of inertia.

Moment of inertia of the semicircle - no symmetry

For a semicircle, the formula for the moment of inertia differs because the figure is not symmetrical about the horizontal axis. The moment of inertia of a semicircle with respect to the horizontal axis passing through the semicircle's center of gravity:

$I_{x} = 0.11 R^4$

Moment of inertia of the semicircle with respect to the vertical axis passing through the semicircle's center of gravity:

$I_{y} = \frac{8} R^4$

That is, as much as half the moment of inertia of a full circle.

Calculation example for a semicircle

Below is an example of calculating the moment of inertia of a semicircle in our online calculator. Just select the shape "semicircle", choose the orientation on the plane specify the radius, and the position of the center of the semicircle, and the calculator will automatically calculate the value of the moment of inertia.

Tubular profile - lightweight and strong

Tubular sections (i.e. rings) have a very favorable ratio of moment of inertia to mass. Thanks to their hollow interior, they maintain high stiffness with much less material consumption. The formula for the moment of inertia of a ring (tube) with an outer and inner radius:

$I_{x},I_{y} = \frac{ (R^4 - r^4)}{4}.$

Calculation example of a tubular profile

Below is an example of how to calculate the moment of inertia of a pipe section in our online calculator. Just select the "circle" shape, specify the radius, then select "circle" again for the hole specify the diameter and select the "cut" option to cut the hole. Then press the solve button and the calculator will automatically calculate the value of the moment of inertia.

As can be seen, the moment of inertia of the tubular section is reduced by the cutout. However, since the cutout is close to the center of mass, its effect on resistance against bending is minimal, making the use of material in this case inefficient. As a result, removing this part of the section leads to an improvement in the efficiency of the design, as it allows better distribution of material in areas that have a greater impact on the stiffness of the section.

Summary

Moment of inertia is important for both bending and deformation forces. This can be seen in their formulas, in which the moment of inertia is in the denominator:

Formula for stress from bending:

$\■sigma = ■frac{M y}{J}$

Formula for deflection of a cantilevered bale loaded with force at the end:

$f= \frac{P l^3}{3EJ}$

The moment of inertia in circular cross sections is characterized by the fact that it is the same in both axes. This makes sense because the cross-section is symmetrical about both X and Y axes. We will analyze this by comparing it to an I-beam, where the moment of inertia can vary from axis to axis. Using this symmetry is advantageous when the load does not always act along the stronger axis of the member, since we predict the strength of the member regardless of the direction of the load.

Moments of inertia for other basic figures can be found in this entry

Moment of inertia is a key quantity in structural strength analysis. You now know the formulas for a circle, semicircle and tubular profile, and our calculator allows you to quickly and accurately perform the necessary calculations.
Try our moment of inertia calculation tool to test the above calculations yourself.

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