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.

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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.

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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.

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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.)

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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)

Rafter is the Best Medicine

Over at the timber frame, all the inner and outer leaves of the stud walls are up, and the open-web-joist rafters are on their way to forming the gable roof. Lifting these rafters into place is a real team effort, so it was nice to have seven guys on site for a busy day last week. We have set up temporary cleats and ladders to help us navigate the gable’s 16-on-12 pitch.

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Rowan and Chris lift rafters onto the roof, while Ben preps near the new dormer window.

Ben and I spent half a day prepping the shed roof for insulation. The job began with shoveling snow off the roof and casting off our protective tarp (which keeps the inside dry while construction is in progress). Then we cut and installed X-braces between the rafters at midspan. Bracing helps the rafters share loads, so if there’s a snow drift or a heavy carpenter standing in one spot then a single rafter doesn’t need to do all the work.

Next, we stapled a tight mesh called Insulweb on the side of each rafter, and along the bottom and top of the roof area. Insulweb encloses each bay of the roof area and enables us to blow cellulose insulation without stuff leaking out. Installation got tricky at the end, when the only place left to stand was balancing atop each rafter.

I remarked many months ago that I would hate to have the job of loading bags of cellulose into the blower machine. Well, that was precisely my job today… and it wasn’t half bad. The paper-fiber material doesn’t itch like fiberglass, but it does produce lots of dust, so wearing a respirator is essential (not so uncomfortable in today’s sub-freezing temperatures). Opening one 25-pound bag at a time with a utility knife, I fed the cellulose into the hopper and tried not to let too much fall on the ground.

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Loading a bag of cellulose into the blower machine.

The machine did most of the heavy lifting, pulling cellulose from the hopper into a hose and blowing it up to Ben at the other end. I don’t envy Ben’s job. Perching carefully atop the roof, Ben wrestled the hose into one bay at a time and then stood by until the bay was tightly packed with insulation. Ben also managed a switch that turned the blower on and off from afar. We operated at two speeds: a very fast initial pack and a more leisurely dense pack.

Apparently, in the early days of cellulose insulation, walls and ceilings were loose-packed, and R-values were satisfactory. But the material settled to the bottom over time, leaving air voids that offered no insulation at all. That’s why today’s standard practice is to fill the voids up front with as much material as possible, leaving no space for it to settle. Today it took 38 bags of material (just under 1000 pounds) to fill two thirds of the northeast gable roof (about 500 cubic feet) with dense-packed cellulose. That’s a lotta shredded newspaper between those rafters.

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Dudley Do-Right Ben looks down from the roof.

Woodshop: 4 Ways to Hold Up Your Roof

Beneath every roof is a structural system to support its weight, plus the force of all the snow, wind, and people that might stand atop it. Here are four support possibilities, along with lots of reasons to use them (or not).

1. Simple rafters. Dimensional lumber is easy to buy, cut, and erect. It might not be the cheapest way to span a given distance, and it’s not even an option for long spans: 2×12 is the maximum depth you can get, and anything longer than 16 feet requires a special order. But the convenience makes simple rafters a great choice when you need to build it NOW.

2. Built-up rafters. This category includes more sophisticated types of lumber, like glulam and LVLs. Another example is an open-web beam, which looks like a cross between an I-beam and a truss. This stuff is more expensive than dimensional lumber, and might require a longer lead time. But you can get a longer span with built-up rafters, as well as greater structural depth for placing insulation if that’s where you choose to insulate.

Triforce built-up rafters. (from https://www2.buildinggreen.com)
Triforce built-up rafters. (from https://www2.buildinggreen.com)

3. Trusses. Prefabricated and delivered in one piece, trusses are usually the most economical way to span a distance. They save you the trouble of cutting a bird’s mouth and a precise peak angle, assuming you order them correctly and the factory gets the order right. And they’re easy to work with, although hoisting them into place might require some mechanical help. The main disadvantage of trusses is that they chew into your headroom, making the attic strictly a storage space.

Trusses make the attic uninhabitable... no harm done if you planned to insulate at floor level.
Trusses make the attic uninhabitable… no harm done if you planned to insulate at floor level.

4. Sister beams. If you want to retrofit a roof without demolishing the existing structure, you can make the existing structure stronger instead. Cut dimensional lumber as you would for simple rafters and nail them to one or both sides of the rafters you already have. The advantage here is you get to keep your old rafters, and the sister beams can be shallow (say, 2×8 rather than 2×12) since they don’t do all the work themselves.

Bob's roof: old timbers, new sisters.
Bob’s roof: old timbers, new sisters.