The Centroid of a section (or Center of Gravity if we assume a uniform density section) is the intersection of its neutral axes. Under bending in any arbitrary direction across the section, this is the one point guaranteed to have zero axial stress, so long as the “plane sections remain plane” assumption is satisfied.
However, the centroid doesn’t have to be located within the physical material of the shape. For example, the centroid of a channel shape is usually in the void space between the flanges, and the centroid of a pipe is located exactly in the middle of the open space.
Centroid vs Center of Gravity
The centroid is technically the arithmetic mean of all the material in a section, while the center of gravity is the resultant point of action in the plane of all gravitational force acting on the object, perpendicular to the section.
However, for any section made up of a uniform-density material, the centroid and center of gravity are the same points. For this reason, they are often treated the same and used interchangeably in structural engineering.
The one major exception to this rule would be composites, such as steel-reinforced concrete, where the centroid and center of gravity are very much not in the same location.
How do you find the Centroid?
There are a few different ways to find the centroid, the choice of which to use depends on the required precision and the type of section we’re looking at.
Approximate Balancing Method
For quick, dirty, imprecise locating, this old-timey method is still valid, though a bit imprecise:
A simple approximate method of locating the center of gravity and neutral axis of a section is shown in Fig. 3. Draw the outline of the section, either full size or to some convenient scale, on a piece of heavy cardboard. Cut out the section and balance it carefully over a knife edge, as shown in the figure; the line along which it rests on the edge of the knife is a line passing through the center of gravity, and by locating two such lines in different directions, the center of gravity will be found at their point of intersection.
“Loads in Structures” – a 1905 Correspondence Course Manual
While this method may seem laughable in modern times, keep in mind that the first pocket-sized calculators were about 70 years from hitting the market.
That being said, for a quick first pass or approximation, this method is still very much viable, especially if coupled with a little modern technology. A Computer-Numerically Controlled (CNC) paper cutter, like the Cricut, can be used to cut out computer-generated shapes very precisely, and cheap digital calipers can then be used to get a much more accurate measurement of the balance points than used to be possible. This is so simple it could easily be used as an activity for kids!
Exact, Calculus-Based Method
Calculus might make you cringe in memory of late nights spent cramming during Freshman year, and you might not have touched it in years, but it really is the easiest method if you need the exact solution for a section bounded by something that can be described exactly with a function.
Mathematically, the definition of the coordinates defining the centroid is:
In reality, this usually reduces to a double integration over x & y within the function-defined boundary of the shape.
Shape Breakdown Method
This method sounds super complicated, but it’s not. Shape breakdown just means we’re breaking the overall section down into shapes we recognize, and using tabulated equations for those basic shapes to help us find the overall centroid of the shape.
Instead of trying to balance a shape on a butter knife and get an imprecise answer, or trying to integrate across a crazy boundary (imagine trying to write equations that fully describe the boundary of an I-beam with sloped flanges and fillets!), we can jump from integration to a summation of finite areas. Because these are all known shapes, no accuracy is lost in this jump.
Mathematically, that looks like this:
Consider this Wikipedia List of Centroids or the similar tables of centroids and areas for basic shapes found in the inside covers or appendices of any good Mechanics of Materials textbook.
Shapes like a simple T-beam can be quickly broken down into rectangles, and more complicated shapes can be split up into more pieces, such as the channel shape example analyzed in my article on the Parallel Axis Theorem.
Computer Tools
Most practicing engineers do not manually calculate centroids, or any other shape properties for that matter. They either use tabulated results (see next section) or computer tools for strange shapes.
All of these computer tools work in essentially the same way, meshing complex shapes down with a huge number of tiny basic shapes (usually triangles, though quadrilaterals, hexagons, and other shapes are used in some specialty applications) and then applying an automated routine similar to the shape breakdown method outlined in the previous section.
A few tools that are widely-used and capable of computing centroids, moments of inertia, and many other section properties include:
Standardized Section Property Tables
Not available for everything, but many standardized manufactured shapes in things like Steel or Aluminum do have tabulated section properties available. These are clearly the easiest and fastest way to get a hold of the centroid, among other properties.
Steel shapes can be found in your copy of the Steel Construction Manual, up in the first physically-tabbed section, or for computer use, there’s an Excel file freely available from AISC as well.
What is the Centroid used for?
The centroid is typically used for calculating more-complex section properties, like the Moment of Inertia, or for figuring the total applied moment from a force applied perpendicular to the section at some eccentricity from the centroid.
The “plane sections remain plane” assumption also gives us the ability to use our centroid / neutral axis to compute strains at any location in the section, given only the distance from the neutral axis out to the coordinate in question.
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