Lesson 6

The Centroid of an Area

🟦  6.1  A conceptual introduction

Two books are stacked on a table, but not in a symmetric pile. 

We can use a moment equilibrium equation to determine the (x, y) location of the force of the table on the lower surface of the bottom book. 

This procedure is similar to the way we will calculate centroids (center of area), and therefore serves as a useful introduction to the topic.

Work through the flipbook, paying close attention to how each moment calculation is set up.

❏ Flipbook: an asymmetric stack of books

🟦  6.2  Moment of area

Up until now, we have the used the word moment for a tendency to rotate (a force multiplied by a perpendicular distance to an axis). We might explain this idea as:

The moment of a force = force times perpendicular distance.

To calculate the location of the centroid of an area, we will use the moment of an area. This idea is extremely abstract, but can be expressed like this:

The moment of an area = area times perpendicular distance.

In other words, we can use the word "moment" to refer to a mathematical operation: to multiply something (a force, an area, a mass, etc.) by a perpendicular distance.

Force times distance

Area times distance

🟦  6.3  Two analogies for centroids

Let's say you cut out a random shape out of cardboard, such as the blue body below. You punch a hole in it anywhere (not necessarily at the centroid), and tie a string through the hole. When you let go, the body swings and oscillates until it reaches static equilibrium. The line of the cable defines one of the body's centroidal axes -- an axis coincident with its centroid or center of mass. For this example, since the material is homogenous, the center of mass is also the center of area.

Another way to think about this is to imagine balancing a planar area on a pyramid-shaped support. The only way to balance the area on a single support is to support it exactly at the centroid. If the pyramid-shaped support (or, a 3D pin connection) is moved to any other point, the planar area will not balance -- it will rotate.

note to self ... also show prismatic members, and how the application for centroids is for the cross-sectional plane.

🟦  6.4  Centroids we already know

We already know the location of the centroid for three common shapes. Please note the terminology in the drawings below:

When you need to find the centroid of a triangle, find the third-points of each leg. The centroid lies at the intersection of lines that are perpendicular to each side, located at the third-point, on the "heavy" or long side of the triangle.

Before spending time calculating a centroid, determine whether your area happens to be symmetric. If you find two perpendicular lines of symmetry, then the centroid lies at the intersection of those lines. These types of areas are sometimes called doubly-symmetric.

Some shapes are anti-symmetric. Anti-symmetry is not the same thing as asymmetry. The diagram below shows how an anti-symmetric shape can be folded over two lines of anti-symmetry

note to self: put in image of single-symmetry.

🟦  6.5  Integration method and composite area method

coming soon

🟦  6.6  Example problem

Interactive: click the image to access the interactive model

put in example problems in flipbooks or videos

solids and voids

🟦  6.7 Other applications for centroidal calculations

center of mass, center of length, center of weight...

➜ Practice Problems

coming soon