# Bending of beams

By Marcus Wilson 15/11/2011

The ceiling in our new house is held up by seven large, curved, steel beams. There are also steel beams holding up parts of the upper floor. These beams are I-beams – so-called because they resemble the capital letter ‘I’ in shape (except they don’t in a sans-serif format as this blog gets published in.) A better description for a sans serif format would be an ‘H’ beam – imagine the letter ‘H’ rotated by 90 degrees – that’s what the cross-section of the beams look like.

The beam has its distinctive shape for good reason. The idea is to put the a lot of steel into places where the stresses are greatest. If you imagine supporting a beam by its two ends, and then applying a load in the middle, the beam will bend slightly. The top surface will be on the inside of the bend, and become slightly shorter and so be in compression; the bottom surface will be on the outside, and become a bit longer and be in tension (stretching). A line running through the centre of the beam along its length won’t change in length at all. This is called the ‘neutral axis’. The stresses are largest on the inside and outside surfaces of the curve, and this is where most material needs to be. So an I-beam has two flanges – one on the innermost surface of the curve, one on the outermost.  In this way one can get the strength of a much thicker beam of uniform area, without the weight and cost.

The centre bit of the beam is only 3 or 4  mm thick – which makes the thing look quite precarious, but it’s quite sufficient for the task. (Amendment 16 Nov 11 – that’s probably not true – 5 or 6 mm I think is a better estimate. It’s a tricky thing to measure since you can’t put a ruler against it, except at the ends of the beams, which are outside the house a few metres off the ground. I wasn’t going up there last night for the sake of a blog entry.)

A mathematical way of describing this is through the ‘second moment of area‘. Basically, this is a measure of how far away the bulk of the beam’s cross-sectional area is away from the neutral axis. (Formally, its the mean square distance from the neutral axis). A stiff beam needs a large second moment of area.  There’s a whole mathematically intensive theory describing how beams deform when loaded (involving a fourth order differential equation) that architects would do well to have some knowledge about.