Monthly Mechanics: Cables

Cables are everywhere, holding up everything from chandeliers to chairlifts, rock climbers to suspension bridges. A cable is a type of structure, and one of the simplest, because it supports only one kind of load: tension.

Let’s do a quick review. Tension and compression are opposite forces. Tension occurs when you pull on something; compression occurs when you push on something. The chain of a hanging chandelier is in tension; the upright of a floor lamp is in compression. End of review.

So, cable structures can only work in tension. There’s no way to put a cable in compression – it crumples if you try. (As my college professor liked to say, “You can’t push on a string.”) This property of cables is a really great thing. It means that when you put weights on a cable, you can predict the exact deflected shape… and, consequently, the forces within the cable.

Consider a chairlift cable stretching between two towers 85 feet apart, with a quad chair hanging 35 feet from the left tower. Let’s say all four seats are filled, so the chair weighs something like 850 pounds. (That’s 150 pounds for each occupant and 250 pounds for the chair itself.) And let’s ignore the weight of the cable itself. Then the loaded cable looks like this:

Chairlift schematic.

Chairlift schematic.

The first step is to determine the vertical reactions at each tower. We know how to calculate reactions for a beam, and the same logic applies here. The reactions are 500 pounds at the left tower, and 350 pounds at the right tower.

Chairlift1

We need just one more bit of information to determine the cable’s deflected shape: either the horizontal pull, or the cable’s maximum sag. It doesn’t matter which. A chairlift typically has a counterweight at one end that puts a horizontal pull on the cable; a suspension bridge’s anchorages do basically the same thing. For the example at hand, let’s say the counterweight pulls the cable with 1200 pounds of horizontal force.

Chairlift2

Adding together the horizontal and vertical force vectors, we find that the cable between the left tower and the chair has a resultant force of 1300 pounds at an angle of about 23 degrees from horizontal. (Imagine a triangle with a horizontal leg of 1200 and a vertical leg of 500 – the resultant force is the hypotenuse.) But a cable can only work in tension. So it runs straight from the tower to the chair at an angle of 23 degrees. Likewise, the cable between the chair and the right tower holds a force of 1250 pounds and runs at an angle of 16 degrees. It has to.

Chairlift3

This shape is called the funicular form. Every cable you’ll ever see stretches automatically to its funicular form, keeping the entire cable in pure tension.

The same thing works with a two-dimensional fabric stretched in three dimensions, like a circus tent. The math is more complicated, but the same tenet applies: it’s possible to predict the deflected shape just from the loads and the horizontal tension. After all, fabric is just a bunch of strings woven together, and you can’t push on a string.

A circus (apparently named after me) showcases two tension structures: cables and fabric.

A circus (apparently named after me) showcases two tension structures: cables and fabric.

Thanks to Paul Kassabian, who actually said, “Don’t push on a rope.”

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