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Strength of Materials: Area Moment of Inertia



Area Moment of Inertia

Knowing the area moment of inertia is a critical part of being able to calculate stress on a beam. It is determined from the cross-sectional area of the beam and the central axis for the direction of interest. For basic shapes there are tables that contain area moment of inertia equations which can be viewed below. However, there are certain cases where the area moment of inertia will have to be calculated either through calculus or by manipulating the equations found in this table.

Calculate area moment of inertia for square triangle circle and half circle

Area Moment of Inertia Calculator (Javascript must be enabled to use)

Width (b): Height (h): Radius (r):
Rectangle Triangle Semicircle Circle
Answer =

To calculate the area moment of inertia through calculus equation 1 would be used for a general form.

Area Moment of Inertia general equation using Calculus (1)

Area Moment of Inertia for Multiple Sectioned Beams

As mentioned earlier in some cases, such as an I-beam, the equations above would have to be manipulated to calculate the area moment of inertia for that shape. To calculate the area moment of inertia equations 2 and 3 would be used. Refer to the example below for a better understanding.

Equation for area moment of inertia in the x direction (2)

Equation for area moment of inertia in the y direction (3)

Example of calculating the area moment of inertia on an I-Beam

Finding the Centroid of a Beam

On the figures above you may have noticed the letter C next to a dot. This is the centroid of the part. The centroid is important in determining the area moment of inertia because, as seen in the previous example, sections relate of the centroid. Basically what the centroid does is it splits the area of the cross-section evenly across an x and y axis. To determine the centroid, equations 4 and 5 would be used, or you could take symmetry into consideration if it applies. Refer to the example below.

Equation used to calculate the centroid in the x direction for a cross-sectional area (4)

Equation used to calculate the centroid in the y direction for a cross-sectional area (5)

Example of calculating a centroid on a cross-section



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