Monthly Mechanics: Let’s Design a Roof, Part 1

Good news! You’ve been hired to design a roof! Let’s see if you can hone your Monthly Mechanics skills and bust out a great creation. I’ll link you back to the relevant articles as we proceed.

The roof is in Rochester, New York. It’s a symmetrical gable roof with a 12-on-12 pitch, spans 20 feet, and is 30 feet long. Assume a dead load of 17 pounds per square foot (psf) for shingles, sheathing, insulation, and drywall. You can treat “roof live load” as a regular live load… in reality it’s more complicated, but this is conservative and simplifies the calculation. Ignore earthquake forces. Your job is to design the rafters with dimensional lumber spaced 2 feet on center.

Schematic of the roof and the rafter layout.

Schematic of the roof and the rafter layout.

Ready? GO!

Step 1: Calculate loads. There are four loads we need to worry about: dead load, roof live load, snow load, and wind load. We already know the dead load is 17 psf… but thanks to our 12-on-12 pitch, the roof line is about 28.2 feet long even though it only spans 20 feet. So we multiply 17 psf by (28.2 feet / 20 feet) = 1.41 to get a projected horizontal dead load of 25 psf. The remaining loads are already defined as horizontal projections, so we don’t need to multiply by 1.41 again. Design Loads has a table that informs us the roof live load is 20 psf, and a map that shows the ground snow load in Rochester to be 40 psf. (Now that’s a doozy!) Wind loads… well, it turns out they’re in the neighborhood of 10-20 psf. Which means we can ignore them, and I’ll show you why.

Step 2: Choose the worst-case load combination. Referring again to Design Loads, we see three load combinations we need to worry about. Two of them have a “live load” term that corresponds to either wind load or regular live load, whichever is greater. Well, if the wind load isn’t more than 20 psf, that means we can use the roof live load. Plugging in the loads from Step 1:

Load Combination 1: 1.4 * (Dead Load) = 35 psf
Load Combination 2: 1.2 * (Dead Load) + 1.6 * (Live Load) + 0.5 * (Snow Load) = 82 psf
Load Combination 3: 1.2 * (Dead Load) + 1.0 * (Live Load) + 1.6 * (Snow Load) = 114 psf

Load Combination 3 is the worst case.

Step 3: Distribute the load to one rafter. As we learned in Tributary Area, this is pretty easy: just multiply the load by the rafter spacing. Here we have 114 psf * 2 feet = 228 pounds per foot. Notice that the length of the roof doesn’t matter. It could be 30 feet or 300; as long as the rafter spacing doesn’t change then neither does the distributed load.

Distributed uniform load on one rafter.

Distributed uniform load on one rafter.

Step 4: Determine the reactions. Our gable roof is supported by walls on both sides, and unsupported at the peak. Drawing now from Actions and Reactions, we have a uniform vertical load of w = 228 pounds per foot over a horizontal span of L = 20 feet. Therefore the two reactions are wL/2 = 2280 pounds.

Reactions atop each wall.

Reactions atop each wall.

We’re making progress! Now is a good time to step back and take a breather. Tune in next month for Part 2, in which we’ll determine the internal forces in our rafter and then select a beam.

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