How to Find the Center of Mass of an Asymmetric Beam

T-Beam, C-Beam, L-Beam

I-beams and rectangular beams are symmetric along both axes, thus their centers of gravity (centroids) are at the intuitive geometric center. However, many kinds of beams have cross-sections that are symmetric along one axis and asymmetric along the other, while some are asymmetric along both axes. When one end of the beam has more mass than the other end, the centroid is closer to the heavier end.

Finding the exact location of the center of mass can be tricky for complex shapes. Luckily, if you can partition the cross-sectional shape of the beam into rectangles, you can easily find the x- and y-coordinates of its center of mass.

The Rectangular Partition Method

(1) Draw the cross-section to scale on a piece of paper and designate one corner to be the origin (0,0). Draw the x-axis and y-axis and note the figure's position in relation to the axes. Label the lengths and widths of edges.

(2) Divide the figure into several smaller rectangles. There may be more than one way to do this; you will arrive at the same final answer no matter which partition you work with. Label the widths and lengths of each rectangle.
(3) Label the coordinates of the center of mass for each rectangle and also record the area of each rectangle. Remember, the center of mass for each rectange will coincide with its geometric center.

(4) Multiply the x-coordinate of each rectangular centroid by the area of its corresponding rectangle. Add the numbers and divide by the total area of the figure. This new number is the x-coordinate of the figure's center of mass.

(5) Multiply the y-coordinate of each rectangular centroid by the area of its corresponding rectangle. Add the numbers and divide by the total area of the shape. This number is the y-coordinate of the figure's center of gravity.
The x- and y-coordinates of the figure's centroid are weighted averages of the centroids of the smaller rectangles.

Example 1: T-Beam

A T-beam is symmetric about the vertical axis, so the x-coordinate of the centroid lies along the vertical axis. Finding the y-coordinate requires the rectangle method. See image below.

Example 2: C-Beam

A C-beam is symmetric about the horizontal axis, so the y-coordinate of its center of gravity lies along the horizontal axis. To find the x-coordinate of the centroid, you can use the rectangular partion method. See image below.

Example 3: L-Beam

An L-beam is not symmetric in either direction, so you must use the rectangle technique to find both the x- and y-coordinates of its centroid. See image below.