Calculating a Region or Load’s Center of Area: Centroids
The centroid or center of area of a geometric region is the geometric center of an object’s shape. Centroid calculations are very common in statics, whether you’re calculating the location of a distributed load’s resultant or determining an object’s center of mass. To compute the center of area of a region (or distributed load), you can compute the x-coordinate (and the other coordinates similarly) from the following equations:
For discrete regions: You can break discrete regions into simple shapes such as triangles, rectangles, circles, and so on.
For discrete shapes, creating a simple table such as the one that follows for each coordinate can be useful. You can create as many rows as you need for as many regions as you have.
|Region 1||x-distance for Region 1||Area of Region 1||Product of xi and Ai|
|TOTALS||—————–||Sum of all Ai rows||Sum of all xiAi rows|
For continuous regions: Continuous regions are usually defined by more complex boundaries, so you must define them with mathematical equations such as the one that follows: