Monthly Mechanics: Deflection

If you stand in the middle of a pedestrian bridge and jump up and down, the bridge sags and bounces under your weight. You already knew that; you’ve probably tried it. But did you know that if you jump up and down on the floor of your house, the floor sags and bounces too? You might not feel the movement because it’s so small (unless you have an old house with a weaker floor), but I guarantee you it’s happening.

Everything in the world changes shape when a sideways force acts on it. Roofs sag under heavy snow. Trees and tall buildings sway in the wind. Trampolines stretch when you land on them. This is Newton’s Third Law at work again: every action has an equal and opposite reaction. The trampoline and the floor and the bridge are pushing back on you with a weight equal to your own, and this reaction is accompanied by a change of shape… a deflection, in physics parlance.

What’s cool is that if you know a few characteristics of the bridge (or floor or trampoline, but let’s stick with the bridge) and how much weight you’re putting on it, you can calculate EXACTLY how much the bridge deflects. In fact, we’ve already covered all the characteristics you need to know! They are the moment of inertia I, the elastic modulus E, and the length of the bridge L. That’s it.

The calculation is surprisingly elegant, but it involves calculus so instead of going through the math I’ll just tell you the two most useful results. If you put one weight P in the middle of the bridge, then the deflection is P*L3/48*E*I. And if you put a uniform load w (something like 15 pounds per foot) across the entire length of the bridge, then the deflection is 5*w*L4 / 384*E*I.

Deflection due to a single load and a uniform load.

Deflection due to a single load and a uniform load.

When you design a bridge or a building, you try to minimize deflection. (Not so much when you design a trampoline.) It’s not that deflection damages a structure – most building materials are fairly elastic and snap right back to their original shape – but simply that excessive deflection is uncomfortable for the users. Nobody wants to drive across a bridge and feel it sagging, or for that matter drive under a bridge and see it sagging. Of course, if your bridge is a 2×10 plank spanning between two scaffold towers, you might have less concern.

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