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Second Moment of Area (Moment of Intertia)
When I was at college, studying calculus, we spent a fair amount of time learning to calculate the second moment of area (also known as the moment of inertia) of cantilevers and beams having various cross-sectional shapes.
Our lecturer never explained to us what second moment of area actually was. I remember asking him, ‘What does “second moment of area” actually mean, sir?’ (yes, we actually called our lecturers, ‘sir’, in those days!), but never received a satisfactory answer… well, not one that many of us understood at the time, anyway! (I felt, at the time, that he didn’t know what it meant, either!)
And, in those days, there was no Google Search (indeed, no internet or personal computers, either, for that matter!), to look up its meaning! Just dusty old textbooks which told us how to calculate it, but not what it meant!
Years later, as a lecturer myself, I was tasked with developing a course, for Edmonton Power linesmen on ‘the mechanics of overhead-lines’. During my research, the topic of ‘second moment of area’, as it applied to distribution poles, came up and, finally, I was able to understand exactly what it meant.
So, consider a plastic or wooden ruler. If you lay it flat, with most of it projecting over the side of a table (like a cantilever), and apply pressure to the free end, it will bend very easily. Now, turn it through right-angles, so that its standing on its edge while projecting over the table. This time, applying pressure to the free end is unlikely to make the ruler bend at all.
What this tells us is that the cross-sectional shape of a cantilever (or a beam) determines how easily it can bend. In the case of our plastic ruler (acting as a cantilever), it bends easily when it is flat, but hardly bends at all when it lies on its edge.
So, calculating the second moment of area of a cantilever or beam allows us to quantify (apply a figure to) how its cross-sectional shape affects its ability to bend. The higher the second moment of area, the more difficult it is to bend.