Beams and Bridges…Expert speaks

Answer by Patrick Barry:

1) If you double a beam’s width, you increase the strength 2x.
2) If you double a beams depth you increase the strength 4x.
3) If you double the beams length, you decrease its strength 4x.
4) As well as everything else, the beam has to carry its own weight too.

Depth is how ‘thick’ a beam is vertically – from the top to the bottom.

Imagine a beam of strength C, 10% of which is needed to carry its own weight. We need to make a beam twice as long with the same strength.

If we make it twice as long and four times as wide, it has the same strength, but 80% of that strength is carrying its own weight, there’s not much left for traffic. Remember it is twice as long as it was as well as wider. If we make it even a bit longer, it would stay up, but need 100% of its strength to do so. Not much use for a bridge.

If we make it twice as long and twice as deep, we have a new beam with four times the strength (4xC) that weighs 4x as much as the old beam. This means it uses 10% of its full strength (4C) to carry itself. Since 90% of 4 is much larger than 90% of 1, the new bridge can actually carry much more traffic than the old one, or be made even longer safely.

Depth is much more important then width structurally, that’s why truss bridges are so strong for the amount of material used: the structural depth here reaches from the top of the top beam all the way to the skinny tension beam below the roadway. You can imagine the entire truss as a single beam with holes punched in it to save weight. The remaining material is just enough to hold the top material and bottom material apart and make sure they work together. The same trick is being used in the steel beam at the top of the page. It’s an easy way to save weight meaning more strength available to carry other stuff.

The ‘working together’ is an important part of the depth concept. You can’t just pile up loose boards and have the same benefit. In the sketch below, all five ‘beams’ are acting independently so the strength is just 5xC.
If we glue them together so they can’t slide they act like a single thick beam. This means the strength goes from 5xC to 25xC while the total weight stays the same!
(If you are following the maths, for rectangular cross-sections, the strength is proportional to the depth squared)

When a force (like the weight of a car or a person) acts on a material, the atoms resist the force by getting squished closer together or getting pulled a little further apart. Both cases are just like springs, the further you pull or push them, the more force is required.

If you look at the end of the ‘beams’ in the sketch do you see how much shorter the top of the glued beam is (compression)? The same thing is happening along the bottom of it- stretching (tension). We’re making the material at the top and bottom of the beam work much harder then before. They’d love to move closer to the center to relax a bit, but the other boards are holding them apart. The center board barely changes size at all, meaning it is not contributing much to the strength.
This is why both the Truss and the I-Beam get rid of most of the material near the center, it’s just more weight to carry.

EDITED.

Fixed the typos, and added a little more on depth, trusses and 2nd moment of area.
2nd edit: fixed the strength numbers as according to Jakob Friborg Nielsen‘s insightful comment. I increases 8x with 2xdepth, but the stress also increases 2x (deeper lever arm inside beam). It’s an important distinction.

What are some good abstractions regarding bridge engineering?

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