Monthly Mechanics: Projected Loads

Roofs are weird. Whereas a floor is a horizontal surface with vertical loads on it, a roof is a sloping surface with vertical loads on it. This is a problem when you’re designing a rafter, because you really want your loads either perpendicular to the rafter (which causes bending) or parallel to it (tension and compression) – a problem we didn’t really explain when we designed a roof together.

Proj1

But there’s hope! We can easily break down a load pointing in a funny direction, into components pointing in useful directions. The method we use to break it down is called vector algebra, but I prefer to think of it as counting in 2D. Let me explain.

Start with a familiar number line. You can think of a number on the line, say 5, as the sum of two smaller numbers, say 2 and 3. If you count 2 to the right, and then count another 3 to the right, you end up at 5. In other words, 2+3 is equivalent to 5.

Proj2

Let’s try the same thing in 2D. Consider a point that’s 5 units away from zero, but instead of being directly to the right it’s at an angle of about 37 degrees. You can get to the same point by counting only along horizontal and vertical lines: 4 units to the right, and then 3 units up. (The three arrows together form a 3-4-5 right triangle, which you may remember from geometry class.) Since they start and end in the same places, the two component arrows are equivalent to the third arrow.

Proj3

And that’s how you can turn a vertical load on a sloping roof into a pair of component loads that you can analyze. For example, a 1500-pound dead load (from snow or an exhaust fan or a decoration) on a roof with a 9-on-12 pitch gets projected as a 900-pound weight acting parallel to the roof plus a 1200-pound weight acting perpendicular to the roof. A roof rafter under that 1500-pound weight is subject to 900 pounds of compression combined with bending due to 1200 pounds. (I chose a 9-on-12 roof pitch to form a 9-12-15 triangle, which is similar to a 3-4-5 triangle.)

Proj4

If it seems weird that the two component loads (900 and 1200 pounds) add up to more than the full load (1500 pounds), just remember that the shortest distance between two points is a straight line. If you walk only on paved paths instead of cutting diagonally across a lawn, you’ll have to walk farther, but you end up in the same place. Sometimes taking a different route than usual is the key to solving an engineering problem.

Projected2

(flickr – creative commons)

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