Captain Astounding's NIGHTCLUB

x4 : What is it good for?
An example from our book The Power
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All illustrations and text ©Dan Welchman Productions (our publisher). Not to be used or copied without written permission from the publisher.

Here's a nice example of x4 in use. x4 simply means x * x * x * x and it's pronounced "x to the power 4" or even just "x to the four". It doesn't get a special name for itself unlike x squared or x cubed. Anyway, the problem is these I beams holding up our main dancefloor. They're allowing the floor to sag in the middle when lots of people are dancing on a busy night.

Now not many people know this, but as you scale up the cross section of a beam by a certain factor x then it's stiffness goes up by a factor of x4. Or, to look at it another way, the sag in the middle of the beam gets smaller by a factor of x4. So if you double the section of a beam (ie make it twice as wide and twice as high) then it will get 16 times stiffer, because 24 = 16. In other words it would only sag 1/16th as much under a certain load; or it would take 16 times as much weight before it would droop the same amount. This is assuming that the bigger beam is spanning the same gap and still made of the same kind of metal.

Now this is a quite astounding increase in stiffness. It means that even if you only increase the cross section of the beam by 20% (ie a factor of 1.2) then the beam will get stiffer by a factor of 1.2 * 1.2 * 1.2 * 1.2 = 2.0736. So you've improved the stiffness more than two-fold just by scaling up by 20%.

It's not easy to see why this is true, but try this explanation:
Imagine the beam as being made up of strands or fibres of material running end to end like dry spaghetti. Now when you put heavy load in the middle of the beam the fibres at the bottom get stretched and the fibres at the top get compressed. And these fibres act like springs resisting the stretch and trying to straighten the beam again. Well when you scale up the beam cross section by some factor x it's like adding extra fibres. You're adding extra fibres at the sides and adding extra fibres top and bottom. The sheer amount of metal in the beam goes up by a factor of x squared. So it shouldn't be difficult to convince you that the beam gets stiffer by at least a factor of x squared.

But the average fibre in the bigger beam is also twice as far out from the middle of the beam. and this gives you two separate additional advantages in terms of beam stiffness. Take a single fibre right at the bottom for example. In the scaled up beam it's now x times further away from the centre line. Now let's imagine that the beam is hinged in the middle and that this one fibre is doing all the work of opposing the sag. If this fibre gets stretched by 1cm then it will generate a certain opposing force (tension) trying to un-stretch itself again, just like a spring. It would have done exactly the same thing in the smaller beam. But because the fibre is now further away from the middle it will produce a stronger resistance to the bending of the hinge. It's just like putting longer handles on a pair of pliers; although the tension in the fibre is the same it exerts a bigger effect trying to straighten the beam, because it's got better leverage just by being further away for the hinge. That's our first advantage from the fibre being further from the centre line. We have been comparing a fibre stretched 1cm in the scaled-up beam with a fibre stretched 1cm in the original beam. But in reality, for a certain depth of sag, the bottom-most fibre in the bigger beam will get stretched x times further than before! This is the second independent effect of the fibre being further out from the centre line. So if the fibre was stretched 1cm in the original beam then it'll be stretched 2cm when we scale up by factor of x=2. And because the fibre is acting as a spring then the more you stretch it the harder it'll pull back. The fibre now has better leverage and it's pulling harder than before.

So there you have it. You've ended up with a beam with x2 times as many fibres in it, and each of those fibres is pulling x times as hard and with x times better leverage for counteracting the sag of the beam. So the net result is a beam that's x * x * x * x times stiffer. And that's x4!