Composite bodies
Most engineering shapes are built from rectangles, triangles, and circles. Their centroids combine through a simple weighted average, with holes counted as negative areas.
The method
Divide the shape into simple parts whose areas and centroids are known. Choose reference axes. Tabulate each part's area A, centroid coordinates, and first moments x times A and y times A. Then divide the summed first moments by the total area.
An L-shaped angle
An L-shape is made of a 100 by 20 mm horizontal rectangle at the bottom and a 20 by 80 mm vertical rectangle on its left end above it. Areas are 2,000 and 1,600 square millimetres, with centroids at (50, 10) and (10, 60). Then x bar = (100,000 + 16,000) / 3,600 and y bar = (20,000 + 96,000) / 3,600, both about 32.2 mm.
Holes and cut-outs
A hole is treated as a part with negative area. A 200 by 100 mm plate with a 40 mm diameter hole centred 50 mm from its left edge has total area 20,000 minus 1,257, about 18,743 square millimetres. With the hole also centred vertically and the origin at the bottom left corner, x bar = (20,000 x 100 - 1,257 x 50) / 18,743, about 103.4 mm, so the centroid shifts away from the hole to about (103.4, 50) mm.
Three dimensions
Composite volumes and bodies work the same way with volumes or masses and a z-coordinate. The CG of a loaded truck combines the chassis, engine, cab, and cargo, each with its own mass and CG.
Why centroids of sections matter
When a beam bends, its neutral axis, where there is neither tension nor compression, passes through the centroid of its cross-section. Designers must find the centroid of T-beams, channels, and built-up sections before calculating bending stresses.
The theorems of Pappus
The theorems of Pappus and Guldinus link centroids to surfaces and volumes of revolution: the volume generated by rotating an area about an axis equals the area times the distance its centroid travels. Rotating a 2 by 1 rectangle whose centroid is 3 units from an axis gives a volume of 2 x 2 pi x 3, about 37.7 cubic units.
Checking results
The centroid must lie within the overall bounding box and closer to the larger or heavier parts. For an L-shape, the centroid often lies outside the material itself, in the corner region, which is correct even though it can look surprising.
Worked example
A T-section has a 120 by 20 mm flange on top of a 20 by 100 mm web. With the origin at the bottom of the web, the web's area is 2,000 square millimetres at y = 50 and the flange's 2,400 at y = 110. So y bar = (100,000 + 264,000) / 4,400, about 82.7 mm above the bottom, much nearer the flange, which holds more area.
Key takeaways
Find composite centroids by splitting the shape into simple parts, tabulating each area and centroid, and dividing summed first moments by total area. Holes count as negative areas, shifting the centroid away from them. The same method applies to 3D bodies with volumes or masses. Beam neutral axes pass through section centroids, so they must be found before stress analysis. Pappus's theorems give volumes of revolution from areas and centroid paths. Centroids can lie outside the material, as in L-shapes.
Flashcard
Split, tabulate, sum first moments
Check your understanding
An L-shape combines 2,000 square mm at y = 10 mm and 1,600 square mm at y = 60 mm. What is y bar?